Daily Papers

We investigate the geometric structure of learning dynamics in overparameterized transformer models through carefully controlled modular arithmetic tasks. Our primary finding is that despite operating in high-dimensional parameter spaces (d=128), transformer training trajectories rapidly collapse onto low-dimensional execution manifolds of dimension 3--4. This dimensional collapse is robust across random seeds and moderate task difficulties, though the orientation of the manifold in parameter space varies between runs. We demonstrate that this geometric structure underlies several empirically observed phenomena: (1) sharp attention concentration emerges as saturation along routing coordinates within the execution manifold, (2) SGD commutators are preferentially aligned with the execution subspace (up to 10times random baseline) early in training, with >92% of non-commutativity confined to orthogonal staging directions and this alignment decreasing as training converges, and (3) sparse autoencoders capture auxiliary routing structure but fail to isolate execution itself, which remains distributed across the low-dimensional manifold. Our results suggest a unifying geometric framework for understanding transformer learning, where the vast majority of parameters serve to absorb optimization interference while core computation occurs in a dramatically reduced subspace. These findings have implications for interpretability, training curriculum design, and understanding the role of overparameterization in neural network learning.

Sparse autoencoders (SAEs) are widely used to extract interpretable features from neural network representations, often under the implicit assumption that concepts correspond to independent linear directions. However, a growing body of evidence suggests that many concepts are instead organized along low-dimensional manifolds encoding continuous geometric relationships. This raises three basic questions: what does it mean for an SAE to capture a manifold, when do existing SAE architectures do so, and how? We develop a theoretical framework that answers these questions and show that SAEs can capture manifolds in two fundamentally different ways: globally, by allocating a compact group of atoms whose linear span contains the entire manifold, or locally, by distributing it across features that each selectively tile a restricted region of the underlying geometry. Empirically, we find that SAEs suboptimally recover continuous structures, mixing the global subspace and local tiling solutions in a fragmented regime we call dilution. This explains why manifold structure is rarely visible at the level of individual concepts and motivates post-hoc unsupervised discovery methods that search for coherent groups of atoms rather than isolated directions. More broadly, our results suggest that future representation learning methods should treat geometric objects, not just individual directions, as the basic units of interpretability.

We develop information geometric techniques to understand the representations learned by deep networks when they are trained on different tasks using supervised, meta-, semi-supervised and contrastive learning. We shed light on the following phenomena that relate to the structure of the space of tasks: (1) the manifold of probabilistic models trained on different tasks using different representation learning methods is effectively low-dimensional; (2) supervised learning on one task results in a surprising amount of progress even on seemingly dissimilar tasks; progress on other tasks is larger if the training task has diverse classes; (3) the structure of the space of tasks indicated by our analysis is consistent with parts of the Wordnet phylogenetic tree; (4) episodic meta-learning algorithms and supervised learning traverse different trajectories during training but they fit similar models eventually; (5) contrastive and semi-supervised learning methods traverse trajectories similar to those of supervised learning. We use classification tasks constructed from the CIFAR-10 and Imagenet datasets to study these phenomena.

Recently, diffusion models have achieved a great performance with a small dataset of size n and a fast optimization process. However, the estimation error of diffusion models suffers from the curse of dimensionality n^{-1/D} with the data dimension D. Since images are usually a union of low-dimensional manifolds, current works model the data as a union of linear subspaces with Gaussian latent and achieve a 1/n bound. Though this modeling reflects the multi-manifold property, the Gaussian latent can not capture the multi-modal property of the latent manifold. To bridge this gap, we propose the mixture subspace of low-rank mixture of Gaussian (MoLR-MoG) modeling, which models the target data as a union of K linear subspaces, and each subspace admits a mixture of Gaussian latent (n_k modals with dimension d_k). With this modeling, the corresponding score function naturally has a mixture of expert (MoE) structure, captures the multi-modal information, and contains nonlinear property. We first conduct real-world experiments to show that the generation results of MoE-latent MoG NN are much better than MoE-latent Gaussian score. Furthermore, MoE-latent MoG NN achieves a comparable performance with MoE-latent Unet with 10 times parameters. These results indicate that the MoLR-MoG modeling is reasonable and suitable for real-world data. After that, based on such MoE-latent MoG score, we provide a R^4Σ_{k=1^Kn_k}Σ_{k=1^Kn_kd_k}/n estimation error, which escapes the curse of dimensionality by using data structure. Finally, we study the optimization process and prove the convergence guarantee under the MoLR-MoG modeling. Combined with these results, under a setting close to real-world data, this work explains why diffusion models only require a small training sample and enjoy a fast optimization process to achieve a great performance.

Large Language Models (LLMs) show strong generalization across diverse tasks, yet the internal decision-making processes behind their predictions remain opaque. In this work, we study the geometry of hidden representations in LLMs through the lens of intrinsic dimension (ID), focusing specifically on decision-making dynamics in a multiple-choice question answering (MCQA) setting. We perform a large-scale study, with 28 open-weight transformer models and estimate ID across layers using multiple estimators, while also quantifying per-layer performance on MCQA tasks. Our findings reveal a consistent ID pattern across models: early layers operate on low-dimensional manifolds, middle layers expand this space, and later layers compress it again, converging to decision-relevant representations. Together, these results suggest LLMs implicitly learn to project linguistic inputs onto structured, low-dimensional manifolds aligned with task-specific decisions, providing new geometric insights into how generalization and reasoning emerge in language models.

We introduce a new generative model where samples are produced via Langevin dynamics using gradients of the data distribution estimated with score matching. Because gradients can be ill-defined and hard to estimate when the data resides on low-dimensional manifolds, we perturb the data with different levels of Gaussian noise, and jointly estimate the corresponding scores, i.e., the vector fields of gradients of the perturbed data distribution for all noise levels. For sampling, we propose an annealed Langevin dynamics where we use gradients corresponding to gradually decreasing noise levels as the sampling process gets closer to the data manifold. Our framework allows flexible model architectures, requires no sampling during training or the use of adversarial methods, and provides a learning objective that can be used for principled model comparisons. Our models produce samples comparable to GANs on MNIST, CelebA and CIFAR-10 datasets, achieving a new state-of-the-art inception score of 8.87 on CIFAR-10. Additionally, we demonstrate that our models learn effective representations via image inpainting experiments.

We consider the problem of simultaneously clustering and learning a linear representation of data lying close to a union of low-dimensional manifolds, a fundamental task in machine learning and computer vision. When the manifolds are assumed to be linear subspaces, this reduces to the classical problem of subspace clustering, which has been studied extensively over the past two decades. Unfortunately, many real-world datasets such as natural images can not be well approximated by linear subspaces. On the other hand, numerous works have attempted to learn an appropriate transformation of the data, such that data is mapped from a union of general non-linear manifolds to a union of linear subspaces (with points from the same manifold being mapped to the same subspace). However, many existing works have limitations such as assuming knowledge of the membership of samples to clusters, requiring high sampling density, or being shown theoretically to learn trivial representations. In this paper, we propose to optimize the Maximal Coding Rate Reduction metric with respect to both the data representation and a novel doubly stochastic cluster membership, inspired by state-of-the-art subspace clustering results. We give a parameterization of such a representation and membership, allowing efficient mini-batching and one-shot initialization. Experiments on CIFAR-10, -20, -100, and TinyImageNet-200 datasets show that the proposed method is much more accurate and scalable than state-of-the-art deep clustering methods, and further learns a latent linear representation of the data.

The superior representation capability of pre-trained vision foundation models (VFMs) has been harnessed for enhancing latent diffusion models (LDMs). These approaches inject the rich semantics from high-dimensional VFM representations (e.g., DINOv3) into LDMs at different phases, resulting in accelerated learning and better generation performance. However, the high-dimensionality of VFM representations may also lead to Information Overload, particularly when the VFM features exceed the size of the original image for decoding. To address this issue while preserving the utility of VFM features, we propose RePack (Representation Packing), a simple yet effective framework for improving Diffusion Transformers (DiTs). RePack transforms the VFM representation into a more compact, decoder-friendly representation by projecting onto low-dimensional manifolds. We find that RePack can effectively filter out non-semantic noise while preserving the core structural information needed for high-fidelity reconstruction. Experimental results show that RePack significantly accelerates DiT convergence and outperforms recent methods that directly inject raw VFM features into the decoder for image reconstruction. On DiT-XL/2, RePack achieves an FID of 3.66 in only 64 epochs, which is 35% faster than the state-of-the-art method. This demonstrates that RePack successfully extracts the core semantics of VFM representations while bypassing their high-dimensionality side effects.

Drawing motivation from the manifold hypothesis, which posits that most high-dimensional data lies on or near low-dimensional manifolds, we apply manifold learning to the space of neural networks. We learn manifolds where datapoints are neural networks by introducing a distance between the hidden layer representations of the neural networks. These distances are then fed to the non-linear dimensionality reduction algorithm PHATE to create a manifold of neural networks. We characterize this manifold using features of the representation, including class separation, hierarchical cluster structure, spectral entropy, and topological structure. Our analysis reveals that high-performing networks cluster together in the manifold, displaying consistent embedding patterns across all these features. Finally, we demonstrate the utility of this approach for guiding hyperparameter optimization and neural architecture search by sampling from the manifold.

Artificial Intelligence (AI) has steadily improved across a wide range of tasks. However, the development and deployment of AI are almost entirely controlled by a few powerful organizations that are racing to create Artificial General Intelligence (AGI). The centralized entities make decisions with little public oversight, shaping the future of humanity, often with unforeseen consequences. In this paper, we propose OML, which stands for Open, Monetizable, and Loyal AI, an approach designed to democratize AI development. OML is realized through an interdisciplinary framework spanning AI, blockchain, and cryptography. We present several ideas for constructing OML using technologies such as Trusted Execution Environments (TEE), traditional cryptographic primitives like fully homomorphic encryption and functional encryption, obfuscation, and AI-native solutions rooted in the sample complexity and intrinsic hardness of AI tasks. A key innovation of our work is introducing a new scientific field: AI-native cryptography. Unlike conventional cryptography, which focuses on discrete data and binary security guarantees, AI-native cryptography exploits the continuous nature of AI data representations and their low-dimensional manifolds, focusing on improving approximate performance. One core idea is to transform AI attack methods, such as data poisoning, into security tools. This novel approach serves as a foundation for OML 1.0 which uses model fingerprinting to protect the integrity and ownership of AI models. The spirit of OML is to establish a decentralized, open, and transparent platform for AI development, enabling the community to contribute, monetize, and take ownership of AI models. By decentralizing control and ensuring transparency through blockchain technology, OML prevents the concentration of power and provides accountability in AI development that has not been possible before.

Deep neural networks (DNNs) trained for image denoising are able to generate high-quality samples with score-based reverse diffusion algorithms. These impressive capabilities seem to imply an escape from the curse of dimensionality, but recent reports of memorization of the training set raise the question of whether these networks are learning the "true" continuous density of the data. Here, we show that two DNNs trained on non-overlapping subsets of a dataset learn nearly the same score function, and thus the same density, when the number of training images is large enough. In this regime of strong generalization, diffusion-generated images are distinct from the training set, and are of high visual quality, suggesting that the inductive biases of the DNNs are well-aligned with the data density. We analyze the learned denoising functions and show that the inductive biases give rise to a shrinkage operation in a basis adapted to the underlying image. Examination of these bases reveals oscillating harmonic structures along contours and in homogeneous regions. We demonstrate that trained denoisers are inductively biased towards these geometry-adaptive harmonic bases since they arise not only when the network is trained on photographic images, but also when it is trained on image classes supported on low-dimensional manifolds for which the harmonic basis is suboptimal. Finally, we show that when trained on regular image classes for which the optimal basis is known to be geometry-adaptive and harmonic, the denoising performance of the networks is near-optimal.

Joint-Embedding Predictive Architectures (JEPAs) provide a simpleframework for learning world models by predicting future latent representations.However, JEPA training is subject to a bias-variance tradeoff.Without sufficient structural constraints, excessive representationalvariance causes the model to collapse to trivial solutions.The recent LeWorldModel (LeWM) shows that this issue can be alleviated bysimply constraining latent embeddings with an isotropic Gaussian prior.However, latent representations inherently lie on low-dimensional manifoldswithin a high-dimensional ambient space, and enforcing an isotropic Gaussianprior directly in this ambient space introduces an overly strong bias.In this work, we propose ame, which seeks a favorable operatingpoint on the bias-variance frontier by applying Gaussian constraints inmultiple random subspaces rather than in the originalembedding space.This design relaxes the global constraint while preserving itsanti-collapse effect, leading to a better balance between trainingstability and representation flexibility.Extensive experiments across fourcontinuous-control environments demonstrate that consistentlyoutperforms LeWM with very clear margins.Our method is simple yet effective, and serves as a strong baseline for future JEPA-based world model research.fdefinedeeemodeThe code is available at https://github.com/intcomp/Sub-JEPA.

Language models can perceive visual properties of text despite receiving only sequences of tokens-we mechanistically investigate how Claude 3.5 Haiku accomplishes one such task: linebreaking in fixed-width text. We find that character counts are represented on low-dimensional curved manifolds discretized by sparse feature families, analogous to biological place cells. Accurate predictions emerge from a sequence of geometric transformations: token lengths are accumulated into character count manifolds, attention heads twist these manifolds to estimate distance to the line boundary, and the decision to break the line is enabled by arranging estimates orthogonally to create a linear decision boundary. We validate our findings through causal interventions and discover visual illusions--character sequences that hijack the counting mechanism. Our work demonstrates the rich sensory processing of early layers, the intricacy of attention algorithms, and the importance of combining feature-based and geometric views of interpretability.

Despite their success, modern convolutional neural networks (CNNs) exhibit fundamental limitations, including data inefficiency, poor out-of-distribution generalization, and vulnerability to adversarial perturbations. These shortcomings can be traced to a lack of inductive biases that reflect the inherent geometric structure of the visual world. The primate visual system, in contrast, demonstrates superior efficiency and robustness, suggesting that its architectural and computational principles,which evolved to internalize these structures,may offer a blueprint for more capable artificial vision. This paper introduces Visual Cortex Network (VCNet), a novel neural network architecture whose design is informed by the macro-scale organization of the primate visual cortex. VCNet is framed as a geometric framework that emulates key biological mechanisms, including hierarchical processing across distinct cortical areas, dual-stream information segregation for learning disentangled representations, and top-down predictive feedback for representation refinement. We interpret these mechanisms through the lens of geometry and dynamical systems, positing that they guide the learning of structured, low-dimensional neural manifolds. We evaluate VCNet on two specialized benchmarks: the Spots-10 animal pattern dataset, which probes sensitivity to natural textures, and a light field image classification task, which requires processing higher-dimensional visual data. Our results show that VCNet achieves state-of-the-art accuracy of 92.1\% on Spots-10 and 74.4\% on the light field dataset, surpassing contemporary models of comparable size. This work demonstrates that integrating high-level neuroscientific principles, viewed through a geometric lens, can lead to more efficient and robust models, providing a promising direction for addressing long-standing challenges in machine learning.

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.

Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent manifolds of arbitrary topology, we propose to learn a mixture model of variational autoencoders. Here, every encoder-decoder pair represents one chart of a manifold. We propose a loss function for maximum likelihood estimation of the model weights and choose an architecture that provides us the analytical expression of the charts and of their inverses. Once the manifold is learned, we use it for solving inverse problems by minimizing a data fidelity term restricted to the learned manifold. To solve the arising minimization problem we propose a Riemannian gradient descent algorithm on the learned manifold. We demonstrate the performance of our method for low-dimensional toy examples as well as for deblurring and electrical impedance tomography on certain image manifolds.

Machine unlearning (MU) for large language models (LLMs), commonly referred to as LLM unlearning, seeks to remove specific undesirable data or knowledge from a trained model, while maintaining its performance on standard tasks. While unlearning plays a vital role in protecting data privacy, enforcing copyright, and mitigating sociotechnical harms in LLMs, we identify a new vulnerability post-unlearning: unlearning trace detection. We discover that unlearning leaves behind persistent "fingerprints" in LLMs, detectable traces in both model behavior and internal representations. These traces can be identified from output responses, even when prompted with forget-irrelevant inputs. Specifically, even a simple supervised classifier can determine whether a model has undergone unlearning, using only its prediction logits or even its textual outputs. Further analysis shows that these traces are embedded in intermediate activations and propagate nonlinearly to the final layer, forming low-dimensional, learnable manifolds in activation space. Through extensive experiments, we demonstrate that unlearning traces can be detected with over 90% accuracy even under forget-irrelevant inputs, and that larger LLMs exhibit stronger detectability. These findings reveal that unlearning leaves measurable signatures, introducing a new risk of reverse-engineering forgotten information when a model is identified as unlearned, given an input query.

Galaxies can be characterized by many internal properties such as stellar mass, gas metallicity, and star-formation rate. We quantify the amount of cosmological and astrophysical information that the internal properties of individual galaxies and their host dark matter halos contain. We train neural networks using hundreds of thousands of galaxies from 2,000 state-of-the-art hydrodynamic simulations with different cosmologies and astrophysical models of the CAMELS project to perform likelihood-free inference on the value of the cosmological and astrophysical parameters. We find that knowing the internal properties of a single galaxy allow our models to infer the value of Omega_{rm m}, at fixed Omega_{rm b}, with a sim10% precision, while no constraint can be placed on sigma_8. Our results hold for any type of galaxy, central or satellite, massive or dwarf, at all considered redshifts, zleq3, and they incorporate uncertainties in astrophysics as modeled in CAMELS. However, our models are not robust to changes in subgrid physics due to the large intrinsic differences the two considered models imprint on galaxy properties. We find that the stellar mass, stellar metallicity, and maximum circular velocity are among the most important galaxy properties to determine the value of Omega_{rm m}. We believe that our results can be explained taking into account that changes in the value of Omega_{rm m}, or potentially Omega_{rm b}/Omega_{rm m}, affect the dark matter content of galaxies. That effect leaves a distinct signature in galaxy properties to the one induced by galactic processes. Our results suggest that the low-dimensional manifold hosting galaxy properties provides a tight direct link between cosmology and astrophysics.

We investigate statistical properties of a likelihood approach to nonparametric estimation of a singular distribution using deep generative models. More specifically, a deep generative model is used to model high-dimensional data that are assumed to concentrate around some low-dimensional structure. Estimating the distribution supported on this low-dimensional structure, such as a low-dimensional manifold, is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. In the considered model, a usual likelihood approach can fail to estimate the target distribution consistently due to the singularity. We prove that a novel and effective solution exists by perturbing the data with an instance noise, which leads to consistent estimation of the underlying distribution with desirable convergence rates. We also characterize the class of distributions that can be efficiently estimated via deep generative models. This class is sufficiently general to contain various structured distributions such as product distributions, classically smooth distributions and distributions supported on a low-dimensional manifold. Our analysis provides some insights on how deep generative models can avoid the curse of dimensionality for nonparametric distribution estimation. We conduct a thorough simulation study and real data analysis to empirically demonstrate that the proposed data perturbation technique improves the estimation performance significantly.

Today's denoising diffusion models do not "denoise" in the classical sense, i.e., they do not directly predict clean images. Rather, the neural networks predict noise or a noised quantity. In this paper, we suggest that predicting clean data and predicting noised quantities are fundamentally different. According to the manifold assumption, natural data should lie on a low-dimensional manifold, whereas noised quantities do not. With this assumption, we advocate for models that directly predict clean data, which allows apparently under-capacity networks to operate effectively in very high-dimensional spaces. We show that simple, large-patch Transformers on pixels can be strong generative models: using no tokenizer, no pre-training, and no extra loss. Our approach is conceptually nothing more than "Just image Transformers", or JiT, as we call it. We report competitive results using JiT with large patch sizes of 16 and 32 on ImageNet at resolutions of 256 and 512, where predicting high-dimensional noised quantities can fail catastrophically. With our networks mapping back to the basics of the manifold, our research goes back to basics and pursues a self-contained paradigm for Transformer-based diffusion on raw natural data.

Deep neural networks can approximate functions on different types of data, from images to graphs, with varied underlying structure. This underlying structure can be viewed as the geometry of the data manifold. By extending recent advances in the theoretical understanding of neural networks, we study how a randomly initialized neural network with piece-wise linear activation splits the data manifold into regions where the neural network behaves as a linear function. We derive bounds on the density of boundary of linear regions and the distance to these boundaries on the data manifold. This leads to insights into the expressivity of randomly initialized deep neural networks on non-Euclidean data sets. We empirically corroborate our theoretical results using a toy supervised learning problem. Our experiments demonstrate that number of linear regions varies across manifolds and the results hold with changing neural network architectures. We further demonstrate how the complexity of linear regions is different on the low dimensional manifold of images as compared to the Euclidean space, using the MetFaces dataset.

Flow matching with x-prediction -- regressing the clean data point rather than the ambient velocity -- is known to exploit low-dimensional manifold structure effectively in pixel space li2025back. We ask whether a pretrained representation space, while containing a low-dimensional data manifold of comparable intrinsic dimensionality, offers a distribution more favorable for flow-matching learning. Comparing pixel, SD-VAE, and DINOv2 features along four geometric axes, we find that pixel and DINOv2 share nearly identical intrinsic dimensionalities (both d!approx!33) yet DINOv2 exhibits 7.3times higher effective rank, 35times better covariance conditioning, 11.5times lower excess kurtosis, and 1.7times lower on-manifold interpolation error; SD-VAE latents are consistently intermediate, indicating that the advantage stems from representation-learning objectives rather than mere compression. These statistical properties render the flow-matching regression well-conditioned and remove the need for the specialized prediction heads or Riemannian transport used by prior DINOv2 diffusion methods. We propose the Representation Image Transformer (RiT): a vanilla Diffusion Transformer trained by x-prediction on frozen DINOv2 features, augmented only by a dimension-aware noise schedule and joint [CLS]-patch modeling. On ImageNet 256{times}256, RiT attains FID 1.45 without guidance and 1.14 with classifier-free guidance, outperforming DiT^DH-XL with 19% fewer parameters (676M vs.\ 839M). The resulting ODE is efficiently solvable at coarse discretizations: with classifier-free guidance, 5 Heun steps already reach FID 2.0 and 10 steps reach 1.25, without distillation or consistency training. Code at https://github.com/lezhang7/RiT.

The success of deep learning based face recognition systems has given rise to serious privacy concerns due to their ability to enable unauthorized tracking of users in the digital world. Existing methods for enhancing privacy fail to generate naturalistic images that can protect facial privacy without compromising user experience. We propose a novel two-step approach for facial privacy protection that relies on finding adversarial latent codes in the low-dimensional manifold of a pretrained generative model. The first step inverts the given face image into the latent space and finetunes the generative model to achieve an accurate reconstruction of the given image from its latent code. This step produces a good initialization, aiding the generation of high-quality faces that resemble the given identity. Subsequently, user-defined makeup text prompts and identity-preserving regularization are used to guide the search for adversarial codes in the latent space. Extensive experiments demonstrate that faces generated by our approach have stronger black-box transferability with an absolute gain of 12.06% over the state-of-the-art facial privacy protection approach under the face verification task. Finally, we demonstrate the effectiveness of the proposed approach for commercial face recognition systems. Our code is available at https://github.com/fahadshamshad/Clip2Protect.

Large Language Models achieve next-token prediction by transporting a vectorized piece of text (prompt) across an accompanying embedding space under the action of successive transformer layers. The resulting high-dimensional trajectories realize different contextualization, or 'thinking', steps, and fully determine the output probability distribution. We aim to characterize the statistical properties of ensembles of these 'lines of thought.' We observe that independent trajectories cluster along a low-dimensional, non-Euclidean manifold, and that their path can be well approximated by a stochastic equation with few parameters extracted from data. We find it remarkable that the vast complexity of such large models can be reduced to a much simpler form, and we reflect on implications.

Refusal refers to the functional behavior enabling safety-aligned language models to reject harmful or unethical prompts. Following the growing scientific interest in mechanistic interpretability, recent work encoded refusal behavior as a single direction in the model's latent space; e.g., computed as the difference between the centroids of harmful and harmless prompt representations. However, emerging evidence suggests that concepts in LLMs often appear to be encoded as a low-dimensional manifold embedded in the high-dimensional latent space. Motivated by these findings, we propose a novel method leveraging Self-Organizing Maps (SOMs) to extract multiple refusal directions. To this end, we first prove that SOMs generalize the prior work's difference-in-means technique. We then train SOMs on harmful prompt representations to identify multiple neurons. By subtracting the centroid of harmless representations from each neuron, we derive a set of multiple directions expressing the refusal concept. We validate our method on an extensive experimental setup, demonstrating that ablating multiple directions from models' internals outperforms not only the single-direction baseline but also specialized jailbreak algorithms, leading to an effective suppression of refusal. Finally, we conclude by analyzing the mechanistic implications of our approach.

Zero-Shot Learning (ZSL) promises to scale visual recognition by bypassing the conventional model training requirement of annotated examples for every category. This is achieved by establishing a mapping connecting low-level features and a semantic description of the label space, referred as visual-semantic mapping, on auxiliary data. Reusing the learned mapping to project target videos into an embedding space thus allows novel-classes to be recognised by nearest neighbour inference. However, existing ZSL methods suffer from auxiliary-target domain shift intrinsically induced by assuming the same mapping for the disjoint auxiliary and target classes. This compromises the generalisation accuracy of ZSL recognition on the target data. In this work, we improve the ability of ZSL to generalise across this domain shift in both model- and data-centric ways by formulating a visual-semantic mapping with better generalisation properties and a dynamic data re-weighting method to prioritise auxiliary data that are relevant to the target classes. Specifically: (1) We introduce a multi-task visual-semantic mapping to improve generalisation by constraining the semantic mapping parameters to lie on a low-dimensional manifold, (2) We explore prioritised data augmentation by expanding the pool of auxiliary data with additional instances weighted by relevance to the target domain. The proposed new model is applied to the challenging zero-shot action recognition problem to demonstrate its advantages over existing ZSL models.

Vision-language segmentation models such as SAM3 enable flexible, prompt-driven visual grounding, but inherit large, general-purpose text encoders originally designed for open-ended language understanding. In practice, segmentation prompts are short, structured, and semantically constrained, leading to substantial over-provisioning in text encoder capacity and persistent computational and memory overhead. In this paper, we perform a large-scale anatomical analysis of text prompting in vision-language segmentation, covering 404,796 real prompts across multiple benchmarks. Our analysis reveals severe redundancy: most context windows are underutilized, vocabulary usage is highly sparse, and text embeddings lie on low-dimensional manifold despite high-dimensional representations. Motivated by these findings, we propose SAM3-LiteText, a lightweight text encoding framework that replaces the original SAM3 text encoder with a compact MobileCLIP student that is optimized by knowledge distillation. Extensive experiments on image and video segmentation benchmarks show that SAM3-LiteText reduces text encoder parameters by up to 88%, substantially reducing static memory footprint, while maintaining segmentation performance comparable to the original model. Code: https://github.com/SimonZeng7108/efficientsam3/tree/sam3_litetext.

Kolmogorov-Arnold Networks (KANs) replace scalar weights with per-edge vectors of basis coefficients, thereby boosting expressivity and accuracy but at the same time resulting in a multiplicative increase in parameters and memory. We propose MetaCluster, a framework that makes KANs highly compressible without sacrificing accuracy. Specifically, a lightweight meta-learner, trained jointly with the KAN, is used to map low-dimensional embedding to coefficient vectors, shaping them to lie on a low-dimensional manifold that is amenable to clustering. We then run K-means in coefficient space and replace per-edge vectors with shared centroids. Afterwards, the meta-learner can be discarded, and a brief fine-tuning of the centroid codebook recovers any residual accuracy loss. The resulting model stores only a small codebook and per-edge indices, exploiting the vector nature of KAN parameters to amortize storage across multiple coefficients. On MNIST, CIFAR-10, and CIFAR-100, across standard KANs and ConvKANs using multiple basis functions, MetaCluster achieves a reduction of up to 80times in parameter storage, with no loss in accuracy. Code will be released upon publication.

Recently, unsupervised representation learning (URL) has improved the sample efficiency of Reinforcement Learning (RL) by pretraining a model from a large unlabeled dataset. The underlying principle of these methods is to learn temporally predictive representations by predicting future states in the latent space. However, an important challenge of this approach is the representational collapse, where the subspace of the latent representations collapses into a low-dimensional manifold. To address this issue, we propose a novel URL framework that causally predicts future states while increasing the dimension of the latent manifold by decorrelating the features in the latent space. Through extensive empirical studies, we demonstrate that our framework effectively learns predictive representations without collapse, which significantly improves the sample efficiency of state-of-the-art URL methods on the Atari 100k benchmark. The code is available at https://github.com/dojeon-ai/SimTPR.

Normalizing flows, a popular class of deep generative models, often fail to represent extreme phenomena observed in real-world processes. In particular, existing normalizing flow architectures struggle to model multivariate extremes, characterized by heavy-tailed marginal distributions and asymmetric tail dependence among variables. In light of this shortcoming, we propose COMET (COpula Multivariate ExTreme) Flows, which decompose the process of modeling a joint distribution into two parts: (i) modeling its marginal distributions, and (ii) modeling its copula distribution. COMET Flows capture heavy-tailed marginal distributions by combining a parametric tail belief at extreme quantiles of the marginals with an empirical kernel density function at mid-quantiles. In addition, COMET Flows capture asymmetric tail dependence among multivariate extremes by viewing such dependence as inducing a low-dimensional manifold structure in feature space. Experimental results on both synthetic and real-world datasets demonstrate the effectiveness of COMET Flows in capturing both heavy-tailed marginals and asymmetric tail dependence compared to other state-of-the-art baseline architectures. All code is available on GitHub at https://github.com/andrewmcdonald27/COMETFlows.

Target encoding plays a central role when learning Convolutional Neural Networks. In this realm, One-hot encoding is the most prevalent strategy due to its simplicity. However, this so widespread encoding schema assumes a flat label space, thus ignoring rich relationships existing among labels that can be exploited during training. In large-scale datasets, data does not span the full label space, but instead lies in a low-dimensional output manifold. Following this observation, we embed the targets into a low-dimensional space, drastically improving convergence speed while preserving accuracy. Our contribution is two fold: (i) We show that random projections of the label space are a valid tool to find such lower dimensional embeddings, boosting dramatically convergence rates at zero computational cost; and (ii) we propose a normalized eigenrepresentation of the class manifold that encodes the targets with minimal information loss, improving the accuracy of random projections encoding while enjoying the same convergence rates. Experiments on CIFAR-100, CUB200-2011, Imagenet, and MIT Places demonstrate that the proposed approach drastically improves convergence speed while reaching very competitive accuracy rates.

This work investigates how large language models (LLMs) internally represent emotion by analyzing the geometry of their hidden-state space. The paper identifies a low-dimensional emotional manifold and shows that emotional representations are directionally encoded, distributed across layers, and aligned with interpretable dimensions. These structures are stable across depth and generalize to eight real-world emotion datasets spanning five languages. Cross-domain alignment yields low error and strong linear probe performance, indicating a universal emotional subspace. Within this space, internal emotion perception can be steered while preserving semantics using a learned intervention module, with especially strong control for basic emotions across languages. These findings reveal a consistent and manipulable affective geometry in LLMs and offer insight into how they internalize and process emotion.

Modern diffusion/flow-based models for image generation typically exhibit two core characteristics: (i) using multi-step sampling, and (ii) operating in a latent space. Recent advances have made encouraging progress on each aspect individually, paving the way toward one-step diffusion/flow without latents. In this work, we take a further step towards this goal and propose "pixel MeanFlow" (pMF). Our core guideline is to formulate the network output space and the loss space separately. The network target is designed to be on a presumed low-dimensional image manifold (i.e., x-prediction), while the loss is defined via MeanFlow in the velocity space. We introduce a simple transformation between the image manifold and the average velocity field. In experiments, pMF achieves strong results for one-step latent-free generation on ImageNet at 256x256 resolution (2.22 FID) and 512x512 resolution (2.48 FID), filling a key missing piece in this regime. We hope that our study will further advance the boundaries of diffusion/flow-based generative models.

Recent empirical studies have demonstrated that diffusion models can effectively learn the image distribution and generate new samples. Remarkably, these models can achieve this even with a small number of training samples despite a large image dimension, circumventing the curse of dimensionality. In this work, we provide theoretical insights into this phenomenon by leveraging key empirical observations: (i) the low intrinsic dimensionality of image data, (ii) a union of manifold structure of image data, and (iii) the low-rank property of the denoising autoencoder in trained diffusion models. These observations motivate us to assume the underlying data distribution of image data as a mixture of low-rank Gaussians and to parameterize the denoising autoencoder as a low-rank model according to the score function of the assumed distribution. With these setups, we rigorously show that optimizing the training loss of diffusion models is equivalent to solving the canonical subspace clustering problem over the training samples. Based on this equivalence, we further show that the minimal number of samples required to learn the underlying distribution scales linearly with the intrinsic dimensions under the above data and model assumptions. This insight sheds light on why diffusion models can break the curse of dimensionality and exhibit the phase transition in learning distributions. Moreover, we empirically establish a correspondence between the subspaces and the semantic representations of image data, facilitating image editing. We validate these results with corroborated experimental results on both simulated distributions and image datasets.

When a language model asserts that "the capital of Australia is Sydney," does it know this is wrong? We characterize the geometry of correctness representations across 9 models from 5 architecture families. The structure is simple: the discriminative signal occupies 3-8 dimensions, performance degrades with additional dimensions, and no nonlinear classifier improves over linear separation. Centroid distance in the low-dimensional subspace matches trained probe performance (0.90 AUC), enabling few-shot detection: on GPT-2, 25 labeled examples achieve 89% of full-data accuracy. We validate causally through activation steering: the learned direction produces 10.9 percentage point changes in error rates while random directions show no effect. Internal probes achieve 0.80-0.97 AUC; output-based methods (P(True), semantic entropy) achieve only 0.44-0.64 AUC. The correctness signal exists internally but is not expressed in outputs. That centroid distance matches probe performance indicates class separation is a mean shift, making detection geometric rather than learned.

Understanding how Large Language Models (LLMs) perform complex reasoning and their failure mechanisms is a challenge in interpretability research. To provide a measurable geometric analysis perspective, we define the concept of the Reasoning Manifold, a latent low-dimensional geometric structure formed by the internal representations corresponding to all correctly reasoned generations. This structure can be conceptualized as the embodiment of the effective thinking paths that the model has learned to successfully solve a given task. Based on this concept, we build REMA, a framework that explains the origins of failures by quantitatively comparing the spatial relationships of internal model representations corresponding to both erroneous and correct reasoning samples. Specifically, REMA first quantifies the geometric deviation of each erroneous representation by calculating its k-nearest neighbors distance to the approximated manifold formed by correct representations, thereby providing a unified failure signal. It then localizes the divergence points where these deviations first become significant by tracking this deviation metric across the model's layers and comparing it against a baseline of internal fluctuations from correct representations, thus identifying where the reasoning chain begins to go off-track. Our extensive experiments on diverse language and multimodal models and tasks demonstrate the low-dimensional nature of the reasoning manifold and the high separability between erroneous and correct reasoning representations. The results also validate the effectiveness of the REMA framework in analyzing the origins of reasoning failures. This research connects abstract reasoning failures to measurable geometric deviations in representations, providing new avenues for in-depth understanding and diagnosis of the internal computational processes of black-box models.

In this paper, we adopt the Universal Manifold Embedding (UME) framework for the estimation of rigid transformations and extend it, so that it can accommodate scenarios involving partial overlap and differently sampled point clouds. UME is a methodology designed for mapping observations of the same object, related by rigid transformations, into a single low-dimensional linear subspace. This process yields a transformation-invariant representation of the observations, with its matrix form representation being covariant (i.e. equivariant) with the transformation. We extend the UME framework by introducing a UME-compatible feature extraction method augmented with a unique UME contrastive loss and a sampling equalizer. These components are integrated into a comprehensive and robust registration pipeline, named UMERegRobust. We propose the RotKITTI registration benchmark, specifically tailored to evaluate registration methods for scenarios involving large rotations. UMERegRobust achieves better than state-of-the-art performance on the KITTI benchmark, especially when strict precision of (1{\deg}, 10cm) is considered (with an average gain of +9%), and notably outperform SOTA methods on the RotKITTI benchmark (with +45% gain compared the most recent SOTA method).

Building general-purpose embodied agents across diverse hardware remains a central challenge in robotics, often framed as the ''one-brain, many-forms'' paradigm. Progress is hindered by fragmented data, inconsistent representations, and misaligned training objectives. We present ABot-M0, a framework that builds a systematic data curation pipeline while jointly optimizing model architecture and training strategies, enabling end-to-end transformation of heterogeneous raw data into unified, efficient representations. From six public datasets, we clean, standardize, and balance samples to construct UniACT-dataset, a large-scale dataset with over 6 million trajectories and 9,500 hours of data, covering diverse robot morphologies and task scenarios. Unified pre-training improves knowledge transfer and generalization across platforms and tasks, supporting general-purpose embodied intelligence. To improve action prediction efficiency and stability, we propose the Action Manifold Hypothesis: effective robot actions lie not in the full high-dimensional space but on a low-dimensional, smooth manifold governed by physical laws and task constraints. Based on this, we introduce Action Manifold Learning (AML), which uses a DiT backbone to predict clean, continuous action sequences directly. This shifts learning from denoising to projection onto feasible manifolds, improving decoding speed and policy stability. ABot-M0 supports modular perception via a dual-stream mechanism that integrates VLM semantics with geometric priors and multi-view inputs from plug-and-play 3D modules such as VGGT and Qwen-Image-Edit, enhancing spatial understanding without modifying the backbone and mitigating standard VLM limitations in 3D reasoning. Experiments show components operate independently with additive benefits. We will release all code and pipelines for reproducibility and future research.

Large-scale vision-language models (VLMs) such as CLIP have gained popularity for their generalizable and expressive multimodal representations. By leveraging large-scale training data with diverse textual metadata, VLMs acquire open-vocabulary capabilities, solving tasks beyond their training scope. This paper investigates the temporal awareness of VLMs, assessing their ability to position visual content in time. We introduce TIME10k, a benchmark dataset of over 10,000 images with temporal ground truth, and evaluate the time-awareness of 37 VLMs by a novel methodology. Our investigation reveals that temporal information is structured along a low-dimensional, non-linear manifold in the VLM embedding space. Based on this insight, we propose methods to derive an explicit ``timeline'' representation from the embedding space. These representations model time and its chronological progression and thereby facilitate temporal reasoning tasks. Our timeline approaches achieve competitive to superior accuracy compared to a prompt-based baseline while being computationally efficient. All code and data are available at https://tekayanidham.github.io/timeline-page/.

Model order reduction has been extensively studied over the last two decades. Projection-based methods such as the Proper Orthogonal Decomposition and the Reduced Basis Method enjoy the important advantages of Galerkin methods in the derivation of the reduced problem, but are limited to linear data compression for which the reduced solution is sought as a linear combination of spatial modes. Nonlinear data compression must be used when the solution manifold is not embedded in a low-dimensional subspace. Early methods involve piecewise linear data compression, by constructing a dictionary of reduced-order models tailored to a partition of the solution manifold. In this work, we introduce the concept of optimal partition of the solution manifold in terms of normalized Kolmogorov widths, and prove that the optimal partitions can be found by means of a representative-based clustering algorithm using the sine dissimilarity measure on the solution manifold.

Creating new visual concepts often requires connecting distinct ideas through their most relevant shared attributes -- their vibe. We introduce Vibe Blending, a novel task for generating coherent and meaningful hybrids that reveals these shared attributes between images. Achieving such blends is challenging for current methods, which struggle to identify and traverse nonlinear paths linking distant concepts in latent space. We propose Vibe Space, a hierarchical graph manifold that learns low-dimensional geodesics in feature spaces like CLIP, enabling smooth and semantically consistent transitions between concepts. To evaluate creative quality, we design a cognitively inspired framework combining human judgments, LLM reasoning, and a geometric path-based difficulty score. We find that Vibe Space produces blends that humans consistently rate as more creative and coherent than current methods.

We revisit a basic question in sequence modeling: is explicit self-attention actually necessary for strong performance and reasoning? We argue that standard multi-head attention is best seen as a form of tensor lifting: hidden vectors are mapped into a high-dimensional space of pairwise interactions, and learning proceeds by constraining this lifted tensor through gradient descent. This mechanism is extremely expressive but mathematically opaque, because after many layers it becomes very hard to describe the model with a small family of explicit invariants. To explore an alternative, we propose an attention-free architecture based on Grassmann flows. Instead of forming an L by L attention matrix, our Causal Grassmann layer (i) linearly reduces token states, (ii) encodes local token pairs as two-dimensional subspaces on a Grassmann manifold via Plucker coordinates, and (iii) fuses these geometric features back into the hidden states through gated mixing. Information therefore propagates by controlled deformations of low-rank subspaces over multi-scale local windows, so the core computation lives on a finite-dimensional manifold rather than in an unstructured tensor space. On the Wikitext-2 language modeling benchmark, purely Grassmann-based models with 13 to 18 million parameters achieve validation perplexities within about 10 to 15 percent of size-matched Transformers. On the SNLI natural language inference task, a Grassmann-Plucker head on top of DistilBERT slightly outperforms a Transformer head, with best validation and test accuracies of 0.8550 and 0.8538 compared to 0.8545 and 0.8511. We analyze the complexity of Grassmann mixing, show linear scaling in sequence length for fixed rank, and argue that such manifold-based designs offer a more structured route toward geometric and invariant-based interpretations of neural reasoning.

Decomposing a scene into its shape, reflectance and illumination is a fundamental problem in computer vision and graphics. Neural approaches such as NeRF have achieved remarkable success in view synthesis, but do not explicitly perform decomposition and instead operate exclusively on radiance (the product of reflectance and illumination). Extensions to NeRF, such as NeRD, can perform decomposition but struggle to accurately recover detailed illumination, thereby significantly limiting realism. We propose a novel reflectance decomposition network that can estimate shape, BRDF, and per-image illumination given a set of object images captured under varying illumination. Our key technique is a novel illumination integration network called Neural-PIL that replaces a costly illumination integral operation in the rendering with a simple network query. In addition, we also learn deep low-dimensional priors on BRDF and illumination representations using novel smooth manifold auto-encoders. Our decompositions can result in considerably better BRDF and light estimates enabling more accurate novel view-synthesis and relighting compared to prior art. Project page: https://markboss.me/publication/2021-neural-pil/

Non-linear manifold learning enables high-dimensional data analysis, but requires out-of-sample-extension methods to process new data points. In this paper, we propose a manifold learning algorithm based on deep learning to create an encoder, which maps a high-dimensional dataset and its low-dimensional embedding, and a decoder, which takes the embedded data back to the high-dimensional space. Stacking the encoder and decoder together constructs an autoencoder, which we term a diffusion net, that performs out-of-sample-extension as well as outlier detection. We introduce new neural net constraints for the encoder, which preserves the local geometry of the points, and we prove rates of convergence for the encoder. Also, our approach is efficient in both computational complexity and memory requirements, as opposed to previous methods that require storage of all training points in both the high-dimensional and the low-dimensional spaces to calculate the out-of-sample-extension and the pre-image.

We study the problem of 3D object generation. We propose a novel framework, namely 3D Generative Adversarial Network (3D-GAN), which generates 3D objects from a probabilistic space by leveraging recent advances in volumetric convolutional networks and generative adversarial nets. The benefits of our model are three-fold: first, the use of an adversarial criterion, instead of traditional heuristic criteria, enables the generator to capture object structure implicitly and to synthesize high-quality 3D objects; second, the generator establishes a mapping from a low-dimensional probabilistic space to the space of 3D objects, so that we can sample objects without a reference image or CAD models, and explore the 3D object manifold; third, the adversarial discriminator provides a powerful 3D shape descriptor which, learned without supervision, has wide applications in 3D object recognition. Experiments demonstrate that our method generates high-quality 3D objects, and our unsupervisedly learned features achieve impressive performance on 3D object recognition, comparable with those of supervised learning methods.

Faithful visualizations of data residing on manifolds must take the underlying geometry into account when producing a flat planar view of the data. In this paper, we extend the classic stochastic neighbor embedding (SNE) algorithm to data on general Riemannian manifolds. We replace standard Gaussian assumptions with Riemannian diffusion counterparts and propose an efficient approximation that only requires access to calculations of Riemannian distances and volumes. We demonstrate that the approach also allows for mapping data from one manifold to another, e.g. from a high-dimensional sphere to a low-dimensional one.

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank k tangent subbundle on R^d, k<d, which we call a principal subbundle. This determines a sub-Riemannian metric on R^d. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold M, construction of a representation of the point-cloud in R^k, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the 'manifoldness' of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning algorithms as functors that map pseudometric spaces to optimization objectives and that factor through hierarchical clustering functors. We then use this characterization to prove refinement bounds on manifold learning loss functions and construct a hierarchy of manifold learning algorithms based on their equivariants. We express several popular manifold learning algorithms as functors at different levels of this hierarchy, including Metric Multidimensional Scaling, IsoMap, and UMAP. Next, we use interleaving distance to study the stability of a broad class of manifold learning algorithms. We present bounds on how closely the embeddings these algorithms produce from noisy data approximate the embeddings they would learn from noiseless data. Finally, we use our framework to derive a set of novel manifold learning algorithms, which we experimentally demonstrate are competitive with the state of the art.

Matrix manifolds, such as manifolds of Symmetric Positive Definite (SPD) matrices and Grassmann manifolds, appear in many applications. Recently, by applying the theory of gyrogroups and gyrovector spaces that is a powerful framework for studying hyperbolic geometry, some works have attempted to build principled generalizations of Euclidean neural networks on matrix manifolds. However, due to the lack of many concepts in gyrovector spaces for the considered manifolds, e.g., the inner product and gyroangles, techniques and mathematical tools provided by these works are still limited compared to those developed for studying hyperbolic geometry. In this paper, we generalize some notions in gyrovector spaces for SPD and Grassmann manifolds, and propose new models and layers for building neural networks on these manifolds. We show the effectiveness of our approach in two applications, i.e., human action recognition and knowledge graph completion.

Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible in closed form, so prior methods resort to imprecise approximations of the score matching training objective that degrade performance and preclude applications in high dimensions. In this work, we reexamine these approximations and propose several practical improvements. Our key observation is that most relevant manifolds are symmetric spaces, which are much more amenable to computation. By leveraging and combining various ans\"{a}tze, we can quickly compute relevant quantities to high precision. On low dimensional datasets, our correction produces a noticeable improvement, allowing diffusion to compete with other methods. Additionally, we show that our method enables us to scale to high dimensional tasks on nontrivial manifolds. In particular, we model QCD densities on SU(n) lattices and contrastively learned embeddings on high dimensional hyperspheres.

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

The rapidly growing field of single-cell transcriptomic sequencing (scRNAseq) presents challenges for data analysis due to its massive datasets. A common method in manifold learning consists in hypothesizing that datasets lie on a lower dimensional manifold. This allows to study the geometry of point clouds by extracting meaningful descriptors like curvature. In this work, we will present Adaptive Local PCA (AdaL-PCA), a data-driven method for accurately estimating various notions of intrinsic curvature on data manifolds, in particular principal curvatures for surfaces. The model relies on local PCA to estimate the tangent spaces. The evaluation of AdaL-PCA on sampled surfaces shows state-of-the-art results. Combined with a PHATE embedding, the model applied to single-cell RNA sequencing data allows us to identify key variations in the cellular differentiation.

UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.

Several first order stochastic optimization methods commonly used in the Euclidean domain such as stochastic gradient descent (SGD), accelerated gradient descent or variance reduced methods have already been adapted to certain Riemannian settings. However, some of the most popular of these optimization tools - namely Adam , Adagrad and the more recent Amsgrad - remain to be generalized to Riemannian manifolds. We discuss the difficulty of generalizing such adaptive schemes to the most agnostic Riemannian setting, and then provide algorithms and convergence proofs for geodesically convex objectives in the particular case of a product of Riemannian manifolds, in which adaptivity is implemented across manifolds in the cartesian product. Our generalization is tight in the sense that choosing the Euclidean space as Riemannian manifold yields the same algorithms and regret bounds as those that were already known for the standard algorithms. Experimentally, we show faster convergence and to a lower train loss value for Riemannian adaptive methods over their corresponding baselines on the realistic task of embedding the WordNet taxonomy in the Poincare ball.

Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalization from few examples. A small child can recognize a dog after extremely limited exposure because the perceptual manifold of "dog" is compact, structured, and low-dimensional. We formalize this principle through a deterministic functional-topological framework in which the set of valid realizations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite Hausdorff radius, and an induced continuous perceptual functional. This geometry provides explicit limits on knowledge, conditions for identifiability, and guarantees for generalization from sparse evidence -- properties fundamental to both natural and artificial intelligence. Across electromechanical, electrochemical, and physiological domains, we show that real-world processes consistently generate compact perceptual manifolds with the same geometric characteristics. Their boundaries can be discovered in a fully self-supervised manner as the empirical radius saturates with increasing sampling, even when the governing equations are unknown. These results demonstrate that deterministic functional topology offers a unified mathematical foundation for perception, representation, and world-model construction. It provides a geometric explanation for why biological learners and self-supervised AI systems can generalize from few observations, and establishes compact perceptual manifolds as a fundamental building block for future AI architectures. Finally, this work unifies biological perception and modern self-supervised models under a single geometric principle: both derive their generalization ability from the compactness and invariants of real-world perceptual manifolds.

Practitioners prune neural networks for efficiency gains and generalization improvements, but few scrutinize the factors determining the prunability of a neural network the maximum fraction of weights that pruning can remove without compromising the model's test accuracy. In this work, we study the properties of input data that may contribute to the prunability of a neural network. For high dimensional input data such as images, text, and audio, the manifold hypothesis suggests that these high dimensional inputs approximately lie on or near a significantly lower dimensional manifold. Prior work demonstrates that the underlying low dimensional structure of the input data may affect the sample efficiency of learning. In this paper, we investigate whether the low dimensional structure of the input data affects the prunability of a neural network.

Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer this question by training networks not in their native parameter space, but instead in a smaller, randomly oriented subspace. We slowly increase the dimension of this subspace, note at which dimension solutions first appear, and define this to be the intrinsic dimension of the objective landscape. The approach is simple to implement, computationally tractable, and produces several suggestive conclusions. Many problems have smaller intrinsic dimensions than one might suspect, and the intrinsic dimension for a given dataset varies little across a family of models with vastly different sizes. This latter result has the profound implication that once a parameter space is large enough to solve a problem, extra parameters serve directly to increase the dimensionality of the solution manifold. Intrinsic dimension allows some quantitative comparison of problem difficulty across supervised, reinforcement, and other types of learning where we conclude, for example, that solving the inverted pendulum problem is 100 times easier than classifying digits from MNIST, and playing Atari Pong from pixels is about as hard as classifying CIFAR-10. In addition to providing new cartography of the objective landscapes wandered by parameterized models, the method is a simple technique for constructively obtaining an upper bound on the minimum description length of a solution. A byproduct of this construction is a simple approach for compressing networks, in some cases by more than 100 times.

Diffusion-based manifold learning methods have proven useful in representation learning and dimensionality reduction of modern high dimensional, high throughput, noisy datasets. Such datasets are especially present in fields like biology and physics. While it is thought that these methods preserve underlying manifold structure of data by learning a proxy for geodesic distances, no specific theoretical links have been established. Here, we establish such a link via results in Riemannian geometry explicitly connecting heat diffusion to manifold distances. In this process, we also formulate a more general heat kernel based manifold embedding method that we call heat geodesic embeddings. This novel perspective makes clearer the choices available in manifold learning and denoising. Results show that our method outperforms existing state of the art in preserving ground truth manifold distances, and preserving cluster structure in toy datasets. We also showcase our method on single cell RNA-sequencing datasets with both continuum and cluster structure, where our method enables interpolation of withheld timepoints of data. Finally, we show that parameters of our more general method can be configured to give results similar to PHATE (a state-of-the-art diffusion based manifold learning method) as well as SNE (an attraction/repulsion neighborhood based method that forms the basis of t-SNE).

The manifold hypothesis posits that high-dimensional data often lies on a lower-dimensional manifold and that utilizing this manifold as the target space yields more efficient representations. While numerous traditional manifold-based techniques exist for dimensionality reduction, their application in self-supervised learning has witnessed slow progress. The recent MSimCLR method combines manifold encoding with SimCLR but requires extremely low target encoding dimensions to outperform SimCLR, limiting its applicability. This paper introduces a novel learning paradigm using an unbalanced atlas (UA), capable of surpassing state-of-the-art self-supervised learning approaches. We investigated and engineered the DeepInfomax with an unbalanced atlas (DIM-UA) method by adapting the Spatiotemporal DeepInfomax (ST-DIM) framework to align with our proposed UA paradigm. The efficacy of DIM-UA is demonstrated through training and evaluation on the Atari Annotated RAM Interface (AtariARI) benchmark, a modified version of the Atari 2600 framework that produces annotated image samples for representation learning. The UA paradigm improves existing algorithms significantly as the number of target encoding dimensions grows. For instance, the mean F1 score averaged over categories of DIM-UA is ~75% compared to ~70% of ST-DIM when using 16384 hidden units.

Learning the distribution of data on Riemannian manifolds is crucial for modeling data from non-Euclidean space, which is required by many applications in diverse scientific fields. Yet, existing generative models on manifolds suffer from expensive divergence computation or rely on approximations of heat kernel. These limitations restrict their applicability to simple geometries and hinder scalability to high dimensions. In this work, we introduce the Riemannian Diffusion Mixture, a principled framework for building a generative diffusion process on manifolds. Instead of following the denoising approach of previous diffusion models, we construct a diffusion process using a mixture of bridge processes derived on general manifolds without requiring heat kernel estimations. We develop a geometric understanding of the mixture process, deriving the drift as a weighted mean of tangent directions to the data points that guides the process toward the data distribution. We further propose a scalable training objective for learning the mixture process that readily applies to general manifolds. Our method achieves superior performance on diverse manifolds with dramatically reduced number of in-training simulation steps for general manifolds.

We present Manifold Diffusion Fields (MDF), an approach to learn generative models of continuous functions defined over Riemannian manifolds. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on the manifold via the eigen-functions of the Laplace-Beltrami Operator. MDF represents functions using an explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. Empirical results on several datasets and manifolds show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.

We propose Riemannian Flow Matching (RFM), a simple yet powerful framework for training continuous normalizing flows on manifolds. Existing methods for generative modeling on manifolds either require expensive simulation, are inherently unable to scale to high dimensions, or use approximations for limiting quantities that result in biased training objectives. Riemannian Flow Matching bypasses these limitations and offers several advantages over previous approaches: it is simulation-free on simple geometries, does not require divergence computation, and computes its target vector field in closed-form. The key ingredient behind RFM is the construction of a relatively simple premetric for defining target vector fields, which encompasses the existing Euclidean case. To extend to general geometries, we rely on the use of spectral decompositions to efficiently compute premetrics on the fly. Our method achieves state-of-the-art performance on many real-world non-Euclidean datasets, and we demonstrate tractable training on general geometries, including triangular meshes with highly non-trivial curvature and boundaries.

A neural network with the widely-used ReLU activation has been shown to partition the sample space into many convex polytopes for prediction. However, the parameterized way a neural network and other machine learning models use to partition the space has imperfections, e.g., the compromised interpretability for complex models, the inflexibility in decision boundary construction due to the generic character of the model, and the risk of being trapped into shortcut solutions. In contrast, although the non-parameterized models can adorably avoid or downplay these issues, they are usually insufficiently powerful either due to over-simplification or the failure to accommodate the manifold structures of data. In this context, we first propose a new type of machine learning models referred to as Manifoldron that directly derives decision boundaries from data and partitions the space via manifold structure discovery. Then, we systematically analyze the key characteristics of the Manifoldron such as manifold characterization capability and its link to neural networks. The experimental results on 4 synthetic examples, 20 public benchmark datasets, and 1 real-world application demonstrate that the proposed Manifoldron performs competitively compared to the mainstream machine learning models. We have shared our code in https://github.com/wdayang/Manifoldron for free download and evaluation.

We study the problem of visualizing large-scale and high-dimensional data in a low-dimensional (typically 2D or 3D) space. Much success has been reported recently by techniques that first compute a similarity structure of the data points and then project them into a low-dimensional space with the structure preserved. These two steps suffer from considerable computational costs, preventing the state-of-the-art methods such as the t-SNE from scaling to large-scale and high-dimensional data (e.g., millions of data points and hundreds of dimensions). We propose the LargeVis, a technique that first constructs an accurately approximated K-nearest neighbor graph from the data and then layouts the graph in the low-dimensional space. Comparing to t-SNE, LargeVis significantly reduces the computational cost of the graph construction step and employs a principled probabilistic model for the visualization step, the objective of which can be effectively optimized through asynchronous stochastic gradient descent with a linear time complexity. The whole procedure thus easily scales to millions of high-dimensional data points. Experimental results on real-world data sets demonstrate that the LargeVis outperforms the state-of-the-art methods in both efficiency and effectiveness. The hyper-parameters of LargeVis are also much more stable over different data sets.

Normalization layers are crucial for deep learning, but their Euclidean formulations are inadequate for data on manifolds. On the other hand, many Riemannian manifolds in machine learning admit gyro-structures, enabling principled extensions of Euclidean neural networks to non-Euclidean domains. Inspired by this, we introduce GyroBN, a principled Riemannian batch normalization framework for gyrogroups. We establish two necessary conditions, namely pseudo-reduction and gyroisometric gyrations, that guarantee GyroBN with theoretical control over sample statistics, and show that these conditions hold for all known gyrogroups in machine learning. Our framework also incorporates several existing Riemannian normalization methods as special cases. We further instantiate GyroBN on seven representative geometries, including the Grassmannian, five constant curvature spaces, and the correlation manifold, and derive novel gyro and Riemannian structures to enable these instantiations. Experiments across these geometries demonstrate the effectiveness of GyroBN. The code is available at https://github.com/GitZH-Chen/GyroBN.git.

Finding meaningful representations and distances of hierarchical data is important in many fields. This paper presents a new method for hierarchical data embedding and distance. Our method relies on combining diffusion geometry, a central approach to manifold learning, and hyperbolic geometry. Specifically, using diffusion geometry, we build multi-scale densities on the data, aimed to reveal their hierarchical structure, and then embed them into a product of hyperbolic spaces. We show theoretically that our embedding and distance recover the underlying hierarchical structure. In addition, we demonstrate the efficacy of the proposed method and its advantages compared to existing methods on graph embedding benchmarks and hierarchical datasets.

Recent advances in generative models highlight the power of geometry-aware modeling in manifold-constrained settings. Yet, for natural images, the field remains confined to Euclidean assumptions, failing to exploit the potential of intrinsic geometric structures within the data. In this work, we investigate the geometry of natural images and observe that semantic information is predominantly encoded in directional components, while norm components can be approximated by the global average. This property holds across both RGB and latent spaces, suggesting that natural images can be effectively modeled on a hypersphere. Building on this finding, we introduce Spherical Optimal Transport Flow Matching (SOT-CFM), which utilizes angular distance, and Spherical Flow Matching (SFM), which constrains dynamics directly on the manifold. Our experiments demonstrate that these geometry-aware methods achieve superior performance against Euclidean baselines. Ultimately, this work provides a novel perspective that bridges the gap between Riemannian manifold-based modeling and natural image generation.

A standard Variational Autoencoder, with a Euclidean latent space, is structurally incapable of capturing topological properties of certain datasets. To remove topological obstructions, we introduce Diffusion Variational Autoencoders with arbitrary manifolds as a latent space. A Diffusion Variational Autoencoder uses transition kernels of Brownian motion on the manifold. In particular, it uses properties of the Brownian motion to implement the reparametrization trick and fast approximations to the KL divergence. We show that the Diffusion Variational Autoencoder is capable of capturing topological properties of synthetic datasets. Additionally, we train MNIST on spheres, tori, projective spaces, SO(3), and a torus embedded in R3. Although a natural dataset like MNIST does not have latent variables with a clear-cut topological structure, training it on a manifold can still highlight topological and geometrical properties.

We present Manify, an open-source Python library for non-Euclidean representation learning. Leveraging manifold learning techniques, Manify provides tools for learning embeddings in (products of) non-Euclidean spaces, performing classification and regression with data that lives in such spaces, and estimating the curvature of a manifold. Manify aims to advance research and applications in machine learning by offering a comprehensive suite of tools for manifold-based data analysis. Our source code, examples, datasets, results, and documentation are available at https://github.com/pchlenski/manify

Nonlinear manifolds are widespread in deep visual features, where Euclidean distances often fail to capture true similarity. This limitation becomes particularly severe in prototype-based interpretable fine-grained recognition, where subtle semantic distinctions are essential. To address this challenge, we propose a novel paradigm for prototype-based recognition that anchors similarity within the intrinsic geometry of deep features. Specifically, we distill the latent manifold structure of each class into a diffusion space and introduce a differentiable Nystr\"om interpolation, making the geometry accessible to both unseen samples and learnable prototypes. To ensure efficiency, we employ compact per-class landmark sets with periodic updates. This design keeps the embedding aligned with the evolving backbone, enabling fast and scalable inference. Extensive experiments on the CUB-200-2011 and Stanford Cars datasets show that our GeoProto framework produces prototypes focusing on semantically aligned parts, significantly outperforming Euclidean prototype networks.

We design a physics-aware auto-encoder to specifically reduce the dimensionality of solutions arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by a large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Moreover, the realized latent variables are often hard to interpret. Therefore, many of these methods are often dismissed in the reduced order modeling of dynamical systems governed by the partial differential equations (PDEs). Accordingly, we propose an auto-encoder type nonlinear dimensionality reduction algorithm. The unsupervised learning problem trains a diffeomorphic spatio-temporal grid, that registers the output sequence of the PDEs on a non-uniform parameter/time-varying grid, such that the Kolmogorov n-width of the mapped data on the learned grid is minimized. We demonstrate the efficacy and interpretability of our approach to separate convection/advection from diffusion/scaling on various manufactured and physical systems.

This paper presents a new, provably-convergent algorithm for computing the flag-mean and flag-median of a set of points on a flag manifold under the chordal metric. The flag manifold is a mathematical space consisting of flags, which are sequences of nested subspaces of a vector space that increase in dimension. The flag manifold is a superset of a wide range of known matrix spaces, including Stiefel and Grassmanians, making it a general object that is useful in a wide variety computer vision problems. To tackle the challenge of computing first order flag statistics, we first transform the problem into one that involves auxiliary variables constrained to the Stiefel manifold. The Stiefel manifold is a space of orthogonal frames, and leveraging the numerical stability and efficiency of Stiefel-manifold optimization enables us to compute the flag-mean effectively. Through a series of experiments, we show the competence of our method in Grassmann and rotation averaging, as well as principal component analysis. We release our source code under https://github.com/nmank/FlagAveraging.

Recent advances in diffusion and flow matching models have highlighted a shift in the preferred prediction target -- moving from noise (varepsilon) and velocity (v) to direct data (x) prediction -- particularly in high-dimensional settings. However, a formal explanation of why the optimal target depends on the specific properties of the data remains elusive. In this work, we provide a theoretical framework based on a generalized prediction formulation that accommodates arbitrary output targets, of which varepsilon-, v-, and x-prediction are special cases. We derive the analytical relationship between data's geometry and the optimal prediction target, offering a rigorous justification for why x-prediction becomes superior when the ambient dimension significantly exceeds the data's intrinsic dimension. Furthermore, while our theory identifies dimensionality as the governing factor for the optimal prediction target, the intrinsic dimension of manifold-bound data is typically intractable to estimate in practice. To bridge this gap, we propose k-Diff, a framework that employs a data-driven approach to learn the optimal prediction parameter k directly from data, bypassing the need for explicit dimension estimation. Extensive experiments in both latent-space and pixel-space image generation demonstrate that k-Diff consistently outperforms fixed-target baselines across varying architectures and data scales, providing a principled and automated approach to enhancing generative performance.

Simulations are the best approximation to experimental laboratories in astrophysics and cosmology. However, the complexity, richness, and large size of their outputs severely limit the interpretability of their predictions. We describe a new, unbiased, and machine learning based approach to obtaining useful scientific insights from a broad range of simulations. The method can be used on today's largest simulations and will be essential to solve the extreme data exploration and analysis challenges posed by the Exascale era. Furthermore, this concept is so flexible, that it will also enable explorative access to observed data. Our concept is based on applying nonlinear dimensionality reduction to learn compact representations of the data in a low-dimensional space. The simulation data is projected onto this space for interactive inspection, visual interpretation, sample selection, and local analysis. We present a prototype using a rotational invariant hyperspherical variational convolutional autoencoder, utilizing a power distribution in the latent space, and trained on galaxies from IllustrisTNG simulation. Thereby, we obtain a natural Hubble tuning fork like similarity space that can be visualized interactively on the surface of a sphere by exploiting the power of HiPS tilings in Aladin Lite.

In a graph convolutional network, we assume that the graph G is generated wrt some observation noise. During learning, we make small random perturbations ΔG of the graph and try to improve generalization. Based on quantum information geometry, ΔG can be characterized by the eigendecomposition of the graph Laplacian matrix. We try to minimize the loss wrt the perturbed G+Δ{G} while making Δ{G} to be effective in terms of the Fisher information of the neural network. Our proposed model can consistently improve graph convolutional networks on semi-supervised node classification tasks with reasonable computational overhead. We present three different geometries on the manifold of graphs: the intrinsic geometry measures the information theoretic dynamics of a graph; the extrinsic geometry characterizes how such dynamics can affect externally a graph neural network; the embedding geometry is for measuring node embeddings. These new analytical tools are useful in developing a good understanding of graph neural networks and fostering new techniques.

Normalizing Flows explicitly maximize a full-dimensional likelihood on the training data. However, real data is typically only supported on a lower-dimensional manifold leading the model to expend significant compute on modeling noise. Injective Flows fix this by jointly learning a manifold and the distribution on it. So far, they have been limited by restrictive architectures and/or high computational cost. We lift both constraints by a new efficient estimator for the maximum likelihood loss, compatible with free-form bottleneck architectures. We further show that naively learning both the data manifold and the distribution on it can lead to divergent solutions, and use this insight to motivate a stable maximum likelihood training objective. We perform extensive experiments on toy, tabular and image data, demonstrating the competitive performance of the resulting model.

We develop data-driven methods for incorporating physical information for priors to learn parsimonious representations of nonlinear systems arising from parameterized PDEs and mechanics. Our approach is based on Variational Autoencoders (VAEs) for learning from observations nonlinear state space models. We develop ways to incorporate geometric and topological priors through general manifold latent space representations. We investigate the performance of our methods for learning low dimensional representations for the nonlinear Burgers equation and constrained mechanical systems.

Deep neural networks have been demonstrated to achieve phenomenal success in many domains, and yet their inner mechanisms are not well understood. In this paper, we investigate the curvature of image manifolds, i.e., the manifold deviation from being flat in its principal directions. We find that state-of-the-art trained convolutional neural networks for image classification have a characteristic curvature profile along layers: an initial steep increase, followed by a long phase of a plateau, and followed by another increase. In contrast, this behavior does not appear in untrained networks in which the curvature flattens. We also show that the curvature gap between the last two layers has a strong correlation with the generalization capability of the network. Moreover, we find that the intrinsic dimension of latent codes is not necessarily indicative of curvature. Finally, we observe that common regularization methods such as mixup yield flatter representations when compared to other methods. Our experiments show consistent results over a variety of deep learning architectures and multiple data sets. Our code is publicly available at https://github.com/azencot-group/CRLM

Many deep generative models are defined as a push-forward of a Gaussian measure by a continuous generator, such as Generative Adversarial Networks (GANs) or Variational Auto-Encoders (VAEs). This work explores the latent space of such deep generative models. A key issue with these models is their tendency to output samples outside of the support of the target distribution when learning disconnected distributions. We investigate the relationship between the performance of these models and the geometry of their latent space. Building on recent developments in geometric measure theory, we prove a sufficient condition for optimality in the case where the dimension of the latent space is larger than the number of modes. Through experiments on GANs, we demonstrate the validity of our theoretical results and gain new insights into the latent space geometry of these models. Additionally, we propose a truncation method that enforces a simplicial cluster structure in the latent space and improves the performance of GANs.

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a ``noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a ``denoising'' process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.

Riemannian representation learning typically relies on approximating densities on chosen manifolds. This involves optimizing difficult objectives, potentially harming models. To completely circumvent this issue, we introduce the Riemannian generative decoder which finds manifold-valued maximum likelihood latents with a Riemannian optimizer while training a decoder network. By discarding the encoder, we vastly simplify the manifold constraint compared to current approaches which often only handle few specific manifolds. We validate our approach on three case studies -- a synthetic branching diffusion process, human migrations inferred from mitochondrial DNA, and cells undergoing a cell division cycle -- each showing that learned representations respect the prescribed geometry and capture intrinsic non-Euclidean structure. Our method requires only a decoder, is compatible with existing architectures, and yields interpretable latent spaces aligned with data geometry.

We introduce a novel framework that directly learns a spectral basis for shape and manifold analysis from unstructured data, eliminating the need for traditional operator selection, discretization, and eigensolvers. Grounded in optimal-approximation theory, we train a network to decompose an implicit approximation operator by minimizing the reconstruction error in the learned basis over a chosen distribution of probe functions. For suitable distributions, they can be seen as an approximation of the Laplacian operator and its eigendecomposition, which are fundamental in geometry processing. Furthermore, our method recovers in a unified manner not only the spectral basis, but also the implicit metric's sampling density and the eigenvalues of the underlying operator. Notably, our unsupervised method makes no assumption on the data manifold, such as meshing or manifold dimensionality, allowing it to scale to arbitrary datasets of any dimension. On point clouds lying on surfaces in 3D and high-dimensional image manifolds, our approach yields meaningful spectral bases, that can resemble those of the Laplacian, without explicit construction of an operator. By replacing the traditional operator selection, construction, and eigendecomposition with a learning-based approach, our framework offers a principled, data-driven alternative to conventional pipelines. This opens new possibilities in geometry processing for unstructured data, particularly in high-dimensional spaces.

Manifold-valued measurements exist in numerous applications within computer vision and machine learning. Recent studies have extended Deep Neural Networks (DNNs) to manifolds, and concomitantly, normalization techniques have also been adapted to several manifolds, referred to as Riemannian normalization. Nonetheless, most of the existing Riemannian normalization methods have been derived in an ad hoc manner and only apply to specific manifolds. This paper establishes a unified framework for Riemannian Batch Normalization (RBN) techniques on Lie groups. Our framework offers the theoretical guarantee of controlling both the Riemannian mean and variance. Empirically, we focus on Symmetric Positive Definite (SPD) manifolds, which possess three distinct types of Lie group structures. Using the deformation concept, we generalize the existing Lie groups on SPD manifolds into three families of parameterized Lie groups. Specific normalization layers induced by these Lie groups are then proposed for SPD neural networks. We demonstrate the effectiveness of our approach through three sets of experiments: radar recognition, human action recognition, and electroencephalography (EEG) classification. The code is available at https://github.com/GitZH-Chen/LieBN.git.

We develop data-driven methods incorporating geometric and topological information to learn parsimonious representations of nonlinear dynamics from observations. The approaches learn nonlinear state-space models of the dynamics for general manifold latent spaces using training strategies related to Variational Autoencoders (VAEs). Our methods are referred to as Geometric Dynamic (GD) Variational Autoencoders (GD-VAEs). We learn encoders and decoders for the system states and evolution based on deep neural network architectures that include general Multilayer Perceptrons (MLPs), Convolutional Neural Networks (CNNs), and other architectures. Motivated by problems arising in parameterized PDEs and physics, we investigate the performance of our methods on tasks for learning reduced dimensional representations of the nonlinear Burgers Equations, Constrained Mechanical Systems, and spatial fields of Reaction-Diffusion Systems. GD-VAEs provide methods that can be used to obtain representations in manifold latent spaces for diverse learning tasks involving dynamics.

In this article, we develop an asymptotic method for constructing confidence regions for the set of all linear subspaces arising from PCA, from which we derive hypothesis tests on this set. Our method is based on the geometry of Riemannian manifolds with which some sets of linear subspaces are endowed.

This paper introduces Least Volume-a simple yet effective regularization inspired by geometric intuition-that can reduce the necessary number of latent dimensions needed by an autoencoder without requiring any prior knowledge of the intrinsic dimensionality of the dataset. We show that the Lipschitz continuity of the decoder is the key to making it work, provide a proof that PCA is just a linear special case of it, and reveal that it has a similar PCA-like importance ordering effect when applied to nonlinear models. We demonstrate the intuition behind the regularization on some pedagogical toy problems, and its effectiveness on several benchmark problems, including MNIST, CIFAR-10 and CelebA.

Representation learning has become an invaluable approach for learning from symbolic data such as text and graphs. However, while complex symbolic datasets often exhibit a latent hierarchical structure, state-of-the-art methods typically learn embeddings in Euclidean vector spaces, which do not account for this property. For this purpose, we introduce a new approach for learning hierarchical representations of symbolic data by embedding them into hyperbolic space -- or more precisely into an n-dimensional Poincar\'e ball. Due to the underlying hyperbolic geometry, this allows us to learn parsimonious representations of symbolic data by simultaneously capturing hierarchy and similarity. We introduce an efficient algorithm to learn the embeddings based on Riemannian optimization and show experimentally that Poincar\'e embeddings outperform Euclidean embeddings significantly on data with latent hierarchies, both in terms of representation capacity and in terms of generalization ability.

Generalization, the ability to perform well beyond the training context, is a hallmark of biological and artificial intelligence, yet anticipating unseen failures remains a central challenge. Conventional approaches often take a ``bottom-up'' mechanistic route by reverse-engineering interpretable features or circuits to build explanatory models. While insightful, these methods often struggle to provide the high-level, predictive signals for anticipating failure in real-world deployment. Here, we propose using a ``top-down'' approach to studying generalization failures inspired by medical biomarkers: identifying system-level measurements that serve as robust indicators of a model's future performance. Rather than mapping out detailed internal mechanisms, we systematically design and test network markers to probe structure, function links, identify prognostic indicators, and validate predictions in real-world settings. In image classification, we find that task-relevant geometric properties of in-distribution (ID) object manifolds consistently forecast poor out-of-distribution (OOD) generalization. In particular, reductions in two geometric measures, effective manifold dimensionality and utility, predict weaker OOD performance across diverse architectures, optimizers, and datasets. We apply this finding to transfer learning with ImageNet-pretrained models. We consistently find that the same geometric patterns predict OOD transfer performance more reliably than ID accuracy. This work demonstrates that representational geometry can expose hidden vulnerabilities, offering more robust guidance for model selection and AI interpretability.

Riemannian submanifold optimization with momentum is computationally challenging because, to ensure that the iterates remain on the submanifold, we often need to solve difficult differential equations. Here, we simplify such difficulties for a class of structured symmetric positive-definite matrices with the affine-invariant metric. We do so by proposing a generalized version of the Riemannian normal coordinates that dynamically orthonormalizes the metric and locally converts the problem into an unconstrained problem in the Euclidean space. We use our approach to simplify existing approaches for structured covariances and develop matrix-inverse-free 2^nd-order optimizers for deep learning in low precision settings. Code: https://github.com/yorkerlin/StructuredNGD-DL

Diffusion and flow models have become the dominant paradigm for generative modeling on Riemannian manifolds, with successful applications in protein backbone generation and DNA sequence design. However, these methods require tens to hundreds of neural network evaluations at inference time, which can become a computational bottleneck in large-scale scientific sampling workflows. We introduce Riemannian MeanFlow~(RMF), a framework for learning flow maps directly on manifolds, enabling high-quality generations with as few as one forward pass. We derive three equivalent characterizations of the manifold average velocity (Eulerian, Lagrangian, and semigroup identities), and analyze parameterizations and stabilization techniques to improve training on high-dimensional manifolds. In promoter DNA design and protein backbone generation settings, RMF achieves comparable sample quality to prior methods while requiring up to 10times fewer function evaluations. Finally, we show that few-step flow maps enable efficient reward-guided design through reward look-ahead, where terminal states can be predicted from intermediate steps at minimal additional cost.

Model merging aims to combine multiple models into one without additional training. Naïve parameter-space averaging can be fragile under architectural symmetries, as their geometry does not take them into account. In this work we show that not only the geometry, but also the averaging procedure itself, must be symmetry-invariant to achieve symmetry-aware merges. Consequently, we propose a general solution: merging as Fréchet averaging, i.e., selecting parameters that minimize a sum of geodesic distances on an appropriate manifold. In this view, the key design choice is the overall geometry, i.e., the choice of metric, manifold, and distance approximation, that determines what it means for two models to be "close". We show that Fréchet averaging, combined with simplifying assumptions, contains Fisher merging. Building on this, we examine the particular case of low-rank adapters (LoRA), whose symmetries induce a distinct geometry: that of a quotient manifold. We outline the limitations of current LoRA merging methods, propose a practical algorithm for this setting, and show how they compare with other commonly used approaches.

This paper introduces the Manifold Probe, a supervised method for discovering representation manifolds in superposition. The method generalizes linear regression probes by learning the space of features of a concept that can be linearly predicted from the representations, and then learning the directions used to encode them. We demonstrate the probe on representations of time and space in Llama 2-7b, finding manifolds which linearly represent an interpretable set of features in each case. In the case of time, we show that by steering along the manifold, we can influence the model's completions about the years in which famous songs, movies and books were released, providing evidence that the Manifold Probe can discover manifolds which are causally involved in model behaviour.

We address the computational challenges of solving parametric PDEs with non parametrized geometric variations and non-reducible problems, such as those involving shocks and discontinuities of variable positions. Traditional dimensionality reduction methods like POD struggle with these scenarios due to slowly decaying Kolmogorov widths. To overcome this, we propose a novel non-linear dimensionality reduction technique to reduce the required modes for representation. The non-linear reduction is obtained through a POD after applying a transformation on the fields, which we call optimal mappings, and is a solution to an optimization problem in infinite dimension. The proposed learning framework combines morphing techniques, non-linear dimensionality reduction, and Gaussian Process Regression (GPR). The problem is reformulated on a reference geometry before applying the dimensionality reduction. Our method learns both the optimal mapping, and the solution fields, using a series of GPR models, enabling efficient and accurate modeling of complex parametric PDEs with geometrical variability. The results obtained concur with current state-of-the-art models. We mainly compare our method with the winning solution of the ML4CFD NeurIPS 2024 competition.

We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

Currently, high-dimensional data is ubiquitous in data science, which necessitates the development of techniques to decompose and interpret such multidimensional (aka tensor) datasets. Finding a low dimensional representation of the data, that is, its inherent structure, is one of the approaches that can serve to understand the dynamics of low dimensional latent features hidden in the data. Nonnegative RESCAL is one such technique, particularly well suited to analyze self-relational data, such as dynamic networks found in international trade flows. Nonnegative RESCAL computes a low dimensional tensor representation by finding the latent space containing multiple modalities. Estimating the dimensionality of this latent space is crucial for extracting meaningful latent features. Here, to determine the dimensionality of the latent space with nonnegative RESCAL, we propose a latent dimension determination method which is based on clustering of the solutions of multiple realizations of nonnegative RESCAL decompositions. We demonstrate the performance of our model selection method on synthetic data and then we apply our method to decompose a network of international trade flows data from International Monetary Fund and validate the resulting features against empirical facts from economic literature.

The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.

We study optimization problems on Hadamard manifolds, motivated by recent advances in geometric approaches to optimization on curved spaces, particularly those involving the structure of Busemann functions. We introduce a projection based variant of the proximal point algorithm, termed the Busemann hybrid projection proximal point algorithm, which replaces Euclidean hyperplanes with horospheres defined via convex Busemann functions. The algorithm performs projections in closed form using the gradients of these functions, resulting in a geometrically intrinsic scheme that requires no tangent space linear solves. We allow for inexact subgradient evaluations and prove global convergence under controlled inexactness, with a relative error level strictly below one. We establish a Fejér type descent and sublinear complexity with a rate proportional to the inverse square root of the iteration count, and show that the exact variant coincides with the classical Riemannian proximal point algorithm. The framework clarifies the role of Busemann based subdifferentials in optimization on spaces of nonpositive curvature.

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning representations without labeled data, often by enforcing invariance to input transformations such as rotations or blurring. Recent studies have highlighted two pivotal properties for effective representations: (i) avoiding dimensional collapse-where the learned features occupy only a low-dimensional subspace, and (ii) enhancing uniformity of the induced distribution. In this work, we introduce T-REGS, a simple regularization framework for SSL based on the length of the Minimum Spanning Tree (MST) over the learned representation. We provide theoretical analysis demonstrating that T-REGS simultaneously mitigates dimensional collapse and promotes distribution uniformity on arbitrary compact Riemannian manifolds. Several experiments on synthetic data and on classical SSL benchmarks validate the effectiveness of our approach at enhancing representation quality.

Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while Nickel et al.'s recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable.

The Koopman operator provides a powerful framework for modeling dynamical systems and has attracted growing interest from the machine learning community. However, its infinite-dimensional nature makes identifying suitable finite-dimensional subspaces challenging, especially for deep architectures. We argue that these difficulties come from suboptimal representation learning, where latent variables fail to balance expressivity and simplicity. This tension is closely related to the information bottleneck (IB) dilemma: constructing compressed representations that are both compact and predictive. Rethinking Koopman learning through this lens, we demonstrate that latent mutual information promotes simplicity, yet an overemphasis on simplicity may cause latent space to collapse onto a few dominant modes. In contrast, expressiveness is sustained by the von Neumann entropy, which prevents such collapse and encourages mode diversity. This insight leads us to propose an information-theoretic Lagrangian formulation that explicitly balances this tradeoff. Furthermore, we propose a new algorithm based on the Lagrangian formulation that encourages both simplicity and expressiveness, leading to a stable and interpretable Koopman representation. Beyond quantitative evaluations, we further visualize the learned manifolds under our representations, observing empirical results consistent with our theoretical predictions. Finally, we validate our approach across a diverse range of dynamical systems, demonstrating improved performance over existing Koopman learning methods. The implementation is publicly available at https://github.com/Wenxuan52/InformationKoopman.

Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.

Attention-based graph neural networks (GNNs), such as graph attention networks (GATs), have become popular neural architectures for processing graph-structured data and learning node embeddings. Despite their empirical success, these models rely on labeled data and the theoretical properties of these models have yet to be fully understood. In this work, we propose a novel attention-based node embedding framework for graphs. Our framework builds upon a hierarchical kernel for multisets of subgraphs around nodes (e.g. neighborhoods) and each kernel leverages the geometry of a smooth statistical manifold to compare pairs of multisets, by "projecting" the multisets onto the manifold. By explicitly computing node embeddings with a manifold of Gaussian mixtures, our method leads to a new attention mechanism for neighborhood aggregation. We provide theoretical insights into generalizability and expressivity of our embeddings, contributing to a deeper understanding of attention-based GNNs. We propose both efficient unsupervised and supervised methods for learning the embeddings. Through experiments on several node classification benchmarks, we demonstrate that our proposed method outperforms existing attention-based graph models like GATs. Our code is available at https://github.com/BorgwardtLab/fisher_information_embedding.

Representations learned via self-supervised learning (SSL) can be susceptible to dimensional collapse, where the learned representation subspace is of extremely low dimensionality and thus fails to represent the full data distribution and modalities. Dimensional collapse also known as the "underfilling" phenomenon is one of the major causes of degraded performance on downstream tasks. Previous work has investigated the dimensional collapse problem of SSL at a global level. In this paper, we demonstrate that representations can span over high dimensional space globally, but collapse locally. To address this, we propose a method called local dimensionality regularization (LDReg). Our formulation is based on the derivation of the Fisher-Rao metric to compare and optimize local distance distributions at an asymptotically small radius for each data point. By increasing the local intrinsic dimensionality, we demonstrate through a range of experiments that LDReg improves the representation quality of SSL. The results also show that LDReg can regularize dimensionality at both local and global levels.

Identifying low-dimensional latent structures within high-dimensional data has long been a central topic in the machine learning community, driven by the need for data compression, storage, transmission, and deeper data understanding. Traditional methods, such as principal component analysis (PCA) and autoencoders (AE), operate in an unsupervised manner, ignoring label information even when it is available. In this work, we introduce a unified method capable of learning latent spaces in both unsupervised and supervised settings. We formulate the problem as a nonlinear multiple-response regression within an index model context. By applying the generalized Stein's lemma, the latent space can be estimated without knowing the nonlinear link functions. Our method can be viewed as a nonlinear generalization of PCA. Moreover, unlike AE and other neural network methods that operate as "black boxes", our approach not only offers better interpretability but also reduces computational complexity while providing strong theoretical guarantees. Comprehensive numerical experiments and real data analyses demonstrate the superior performance of our method.

Neural networks have achieved tremendous success in a large variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. In this work, we propose a novel algorithm to find efficient low-rank subnetworks. Remarkably, these subnetworks are determined and adapted already during the training phase and the overall time and memory resources required by both training and evaluating them are significantly reduced. The main idea is to restrict the weight matrices to a low-rank manifold and to update the low-rank factors rather than the full matrix during training. To derive training updates that are restricted to the prescribed manifold, we employ techniques from dynamic model order reduction for matrix differential equations. This allows us to provide approximation, stability, and descent guarantees. Moreover, our method automatically and dynamically adapts the ranks during training to achieve the desired approximation accuracy. The efficiency of the proposed method is demonstrated through a variety of numerical experiments on fully-connected and convolutional networks.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data, and on estimating local Hessian matrices of neural network loss landscapes.

Hyperbolic geometry is gaining traction in machine learning for its effectiveness at capturing hierarchical structures in real-world data. Hyperbolic spaces, where neighborhoods grow exponentially, offer substantial advantages and consistently deliver state-of-the-art results across diverse applications. However, hyperbolic classifiers often grapple with computational challenges. Methods reliant on Riemannian optimization frequently exhibit sluggishness, stemming from the increased computational demands of operations on Riemannian manifolds. In response to these challenges, we present hyperDT, a novel extension of decision tree algorithms into hyperbolic space. Crucially, hyperDT eliminates the need for computationally intensive Riemannian optimization, numerically unstable exponential and logarithmic maps, or pairwise comparisons between points by leveraging inner products to adapt Euclidean decision tree algorithms to hyperbolic space. Our approach is conceptually straightforward and maintains constant-time decision complexity while mitigating the scalability issues inherent in high-dimensional Euclidean spaces. Building upon hyperDT we introduce hyperRF, a hyperbolic random forest model. Extensive benchmarking across diverse datasets underscores the superior performance of these models, providing a swift, precise, accurate, and user-friendly toolkit for hyperbolic data analysis.

We give an alternative proof of the Schoen--Simon--Yau curvature estimates and associated Bernstein-type theorems (1975), and extend the original result by including the case of 6-dimensional (stable minimal) immersions. The key step is an ε-regularity theorem, that assumes smallness of the scale-invariant L^2 norm of the second fundamental form. Further, we obtain a graph description, in the Lipschitz multi-valued sense, for any stable minimal immersion of dimension ngeq 2, that may have a singular set Σ of locally finite H^{n-2}-measure, and that is weakly close to a hyperplane. (In fact, if H^{n-2}(Σ)=0, the conclusion is strengthened to a union of smooth graphs.) This follows directly from an ε-regularity theorem, that assumes smallness of the scale-invariant L^2 tilt-excess (verified when the hypersurface is weakly close to a hyperplane). Specialising the multi-valued decomposition to the case of embeddings, we recover the Schoen--Simon theorem (1981). In both ε-regularity theorems the relevant quantity (respectively, length of the second fundamental form and tilt function) solves a non-linear PDE on the immersed minimal hypersurface. The proof is carried out intrinsically (without linearising the PDE) by implementing an iteration method à la De Giorgi (from the linear De Giorgi--Nash--Moser theory). Stability implies estimates (intrinsic weak Caccioppoli inequalities) that make the iteration effective despite the non-linear framework. (In both ε-regularity theorems the method gives explicit constants that quantify the required smallness.)

Recently, diffusion models have been used to solve various inverse problems in an unsupervised manner with appropriate modifications to the sampling process. However, the current solvers, which recursively apply a reverse diffusion step followed by a projection-based measurement consistency step, often produce suboptimal results. By studying the generative sampling path, here we show that current solvers throw the sample path off the data manifold, and hence the error accumulates. To address this, we propose an additional correction term inspired by the manifold constraint, which can be used synergistically with the previous solvers to make the iterations close to the manifold. The proposed manifold constraint is straightforward to implement within a few lines of code, yet boosts the performance by a surprisingly large margin. With extensive experiments, we show that our method is superior to the previous methods both theoretically and empirically, producing promising results in many applications such as image inpainting, colorization, and sparse-view computed tomography. Code available https://github.com/HJ-harry/MCG_diffusion

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this work, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each item we can recommend is a node of an undirected graph and its expected rating is similar to the one of its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret with respect to the optimal policy would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose three algorithms for solving our problem that scale linearly and sublinearly in this dimension. Our experiments on content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens of node evaluations.

We introduce Geometric Neural Operators (GNPs) for accounting for geometric contributions in data-driven deep learning of operators. We show how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures, (ii) to approximate Partial Differential Equations (PDEs) on manifolds, (iii) learn solution maps for Laplace-Beltrami (LB) operators, and (iv) to solve Bayesian inverse problems for identifying manifold shapes. The methods allow for handling geometries of general shape including point-cloud representations. The developed GNPs provide approaches for incorporating the roles of geometry in data-driven learning of operators.

We show that two properly embedded compact surfaces in an orientable 4-manifold are cobordant if and only if they are Z/2-homologous and either the 4-manifold has boundary or the surfaces have the same normal Euler number. If the 4-manifold is simply-connected and the surfaces are closed, non-orientable, and cobordant, we show that they are in fact concordant. This completes the classification of closed surfaces in simply-connected 4-manifolds up to concordance. Our methods give new constructions of cobordisms with prescribed boundaries, and completely determine when a given cobordism between the boundaries extends to a cobordism or concordance between the surfaces. We obtain our concordance results by extending Sunukjian's method of ambient surgery to the unoriented case using Pin^--structures. We also discuss conditions for an arbitrary codimension 2 properly embedded submanifold to admit an unoriented spanning manifold with prescribed boundary. All results hold in both the smooth and topological categories.

Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, mostly because of the absence of corresponding hyperbolic neural network layers. This makes it hard to use hyperbolic embeddings in downstream tasks. Here, we bridge this gap in a principled manner by combining the formalism of Möbius gyrovector spaces with the Riemannian geometry of the Poincaré model of hyperbolic spaces. As a result, we derive hyperbolic versions of important deep learning tools: multinomial logistic regression, feed-forward and recurrent neural networks such as gated recurrent units. This allows to embed sequential data and perform classification in the hyperbolic space. Empirically, we show that, even if hyperbolic optimization tools are limited, hyperbolic sentence embeddings either outperform or are on par with their Euclidean variants on textual entailment and noisy-prefix recognition tasks.

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely referred to as "positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. In this paper we initiate an exploration of "positive geometries" and "canonical forms" as objects of study in their own right in a more general mathematical setting. We give a precise definition of positive geometries and canonical forms, introduce general methods for finding forms for more complicated positive geometries from simpler ones, and present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties. We also illustrate a number of strategies for computing canonical forms which yield interesting representations for the forms associated with wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

Collections of probability distributions arise in a variety of applications ranging from user activity pattern analysis to brain connectomics. In practice these distributions can be defined over diverse domain types including finite intervals, circles, cylinders, spheres, other manifolds, and graphs. This paper introduces an approach for detecting differences between two collections of distributions over such general domains. To this end, we propose the intrinsic slicing construction that yields a novel class of Wasserstein distances on manifolds and graphs. These distances are Hilbert embeddable, allowing us to reduce the distribution collection comparison problem to a more familiar mean testing problem in a Hilbert space. We provide two testing procedures one based on resampling and another on combining p-values from coordinate-wise tests. Our experiments in various synthetic and real data settings show that the resulting tests are powerful and the p-values are well-calibrated.

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this paper, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each item we can recommend is a node and its expected rating is similar to its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret with respect to the optimal policy would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose two algorithms for solving our problem that scale linearly and sublinearly in this dimension. Our experiments on real-world content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens of nodes evaluations.