Daily Papers

Generative models can now produce photorealistic synthetic data which is virtually indistinguishable from the real data used to train it. This is a significant evolution over previous models which could produce reasonable facsimiles of the training data, but ones which could be visually distinguished from the training data by human evaluation. Recent work on OOD detection has raised doubts that generative model likelihoods are optimal OOD detectors due to issues involving likelihood misestimation, entropy in the generative process, and typicality. We speculate that generative OOD detectors also failed because their models focused on the pixels rather than the semantic content of the data, leading to failures in near-OOD cases where the pixels may be similar but the information content is significantly different. We hypothesize that estimating typical sets using self-supervised learners leads to better OOD detectors. We introduce a novel approach that leverages representation learning, and informative summary statistics based on manifold estimation, to address all of the aforementioned issues. Our method outperforms other unsupervised approaches and achieves state-of-the art performance on well-established challenging benchmarks, and new synthetic data detection tasks.

Generative Adversarial Networks (GANs) have shown compelling results in various tasks and applications in recent years. However, mode collapse remains a critical problem in GANs. In this paper, we propose a novel training pipeline to address the mode collapse issue of GANs. Different from existing methods, we propose to generalize the discriminator as feature embedding and maximize the entropy of distributions in the embedding space learned by the discriminator. Specifically, two regularization terms, i.e., Deep Local Linear Embedding (DLLE) and Deep Isometric feature Mapping (DIsoMap), are designed to encourage the discriminator to learn the structural information embedded in the data, such that the embedding space learned by the discriminator can be well-formed. Based on the well-learned embedding space supported by the discriminator, a non-parametric entropy estimator is designed to efficiently maximize the entropy of embedding vectors, playing as an approximation of maximizing the entropy of the generated distribution. By improving the discriminator and maximizing the distance of the most similar samples in the embedding space, our pipeline effectively reduces the mode collapse without sacrificing the quality of generated samples. Extensive experimental results show the effectiveness of our method, which outperforms the GAN baseline, MaF-GAN on CelebA (9.13 vs. 12.43 in FID) and surpasses the recent state-of-the-art energy-based model on the ANIME-FACE dataset (2.80 vs. 2.26 in Inception score). The code is available at https://github.com/HaozheLiu-ST/MEE

The success of machine learning algorithms generally depends on data representation, and we hypothesize that this is because different representations can entangle and hide more or less the different explanatory factors of variation behind the data. Although specific domain knowledge can be used to help design representations, learning with generic priors can also be used, and the quest for AI is motivating the design of more powerful representation-learning algorithms implementing such priors. This paper reviews recent work in the area of unsupervised feature learning and deep learning, covering advances in probabilistic models, auto-encoders, manifold learning, and deep networks. This motivates longer-term unanswered questions about the appropriate objectives for learning good representations, for computing representations (i.e., inference), and the geometrical connections between representation learning, density estimation and manifold learning.

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.

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.

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.

A significant gap exists between theory and practice in deep learning. Generalization and approximation error bounds are often derived for simplified models or are too loose to be informative. Many rely on the manifold hypothesis and on geometric regularity such as intrinsic dimension, curvature, and reach. Progress requires insight into data-manifold geometry and suitable benchmarks, yet existing options are polarized: analytic manifolds with known geometry but limited applicability, or real-world datasets where geometry is only coarsely estimable. We introduce a benchmarking framework for studying data geometry. We repurpose and extend dSprites and COIL-20 with additional transformation dimensions and dense, axis-aligned sampling, and pair them with finite-difference estimators that recover curvature, reach, and volume at near-ground-truth accuracy in a regime where general-purpose estimators are unreliable or difficult to deploy. The framework is intended as a controlled testbed, useful as a calibration environment for geometric estimators and a sandbox for probing theoretical assumptions. To illustrate its use, we present two application studies, namely assessing the scaling behavior of the bounds of Genovese et al. and Fefferman et al., and tracking the layer-wise geometry of a β-VAE, highlighting the behavior of current bounds and the value of controlled benchmarks for guiding and validating future theory. A reference implementation is available at https://github.com/koulakis/manifold-microscope.

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.

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.

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.

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.

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 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.

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.

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.

Event cameras provide several unique advantages over standard frame-based sensors, including high temporal resolution, low latency, and robustness to extreme lighting. However, existing learning-based approaches for event processing are typically confined to narrow, task-specific silos and lack the ability to generalize across modalities. We address this gap with REALM, a cross-modal framework that learns an RGB and Event Aligned Latent Manifold by projecting event representations into the pretrained latent space of RGB foundation models. Instead of task-specific training, we leverage low-rank adaptation (LoRA) to bridge the modality gap, effectively unlocking the geometric and semantic priors of frozen RGB backbones for asynchronous event streams. We demonstrate that REALM effectively maps events into the ViT-based foundation latent space. Our method allows us to perform downstream tasks like depth estimation and semantic segmentation by simply transferring linear heads trained on the RGB teacher. Most significantly, REALM enables the direct, zero-shot application of complex, frozen image-trained decoders, such as MASt3R, to raw event data. We demonstrate state-of-the-art performance in wide-baseline feature matching, significantly outperforming specialized architectures. Code and models are available upon acceptance.

Appearance-based gaze estimation (AGE) has achieved remarkable performance in constrained settings, yet we reveal a significant generalization gap where existing AGE models often fail in practical, unconstrained scenarios, particularly those involving facial wearables and poor lighting conditions. We attribute this failure to two core factors: limited image diversity and inconsistent label fidelity across different datasets, especially along the pitch axis. To address these, we propose a robust AGE framework that enhances generalization without requiring additional human-annotated data. First, we expand the image manifold via an ensemble of augmentation techniques, including synthesis of eyeglasses, masks, and varied lighting. Second, to mitigate the impact of anisotropic inter-dataset label deviation, we reformulate gaze regression as a multi-task learning problem, incorporating multi-view supervised contrastive (SupCon) learning, discretized label classification, and eye-region segmentation as auxiliary objectives. To rigorously validate our approach, we curate new benchmark datasets designed to evaluate gaze robustness under challenging conditions, a dimension largely overlooked by existing evaluation protocols. Our MobileNet-based lightweight model achieves generalization performance competitive with the state-of-the-art (SOTA) UniGaze-H, while utilizing less than 1\% of its parameters, enabling high-fidelity, real-time gaze tracking on mobile devices.

Existing video depth estimation faces a fundamental trade-off: generative models suffer from stochastic geometric hallucinations and scale drift, while discriminative models demand massive labeled datasets to resolve semantic ambiguities. To break this impasse, we present DVD, the first framework to deterministically adapt pre-trained video diffusion models into single-pass depth regressors. Specifically, DVD features three core designs: (i) repurposing the diffusion timestep as a structural anchor to balance global stability with high-frequency details; (ii) latent manifold rectification (LMR) to mitigate regression-induced over-smoothing, enforcing differential constraints to restore sharp boundaries and coherent motion; and (iii) global affine coherence, an inherent property bounding inter-window divergence, which enables seamless long-video inference without requiring complex temporal alignment. Extensive experiments demonstrate that DVD achieves state-of-the-art zero-shot performance across benchmarks. Furthermore, DVD successfully unlocks the profound geometric priors implicit in video foundation models using 163x less task-specific data than leading baselines. Notably, we fully release our pipeline, providing the whole training suite for SOTA video depth estimation to benefit the open-source community.

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).

Rapidly increasing quality of AI-generated content makes it difficult to distinguish between human and AI-generated texts, which may lead to undesirable consequences for society. Therefore, it becomes increasingly important to study the properties of human texts that are invariant over different text domains and varying proficiency of human writers, can be easily calculated for any language, and can robustly separate natural and AI-generated texts regardless of the generation model and sampling method. In this work, we propose such an invariant for human-written texts, namely the intrinsic dimensionality of the manifold underlying the set of embeddings for a given text sample. We show that the average intrinsic dimensionality of fluent texts in a natural language is hovering around the value 9 for several alphabet-based languages and around 7 for Chinese, while the average intrinsic dimensionality of AI-generated texts for each language is approx 1.5 lower, with a clear statistical separation between human-generated and AI-generated distributions. This property allows us to build a score-based artificial text detector. The proposed detector's accuracy is stable over text domains, generator models, and human writer proficiency levels, outperforming SOTA detectors in model-agnostic and cross-domain scenarios by a significant margin.

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.

In meta reinforcement learning (meta RL), an agent learns from a set of training tasks how to quickly solve a new task, drawn from the same task distribution. The optimal meta RL policy, a.k.a. the Bayes-optimal behavior, is well defined, and guarantees optimal reward in expectation, taken with respect to the task distribution. The question we explore in this work is how many training tasks are required to guarantee approximately optimal behavior with high probability. Recent work provided the first such PAC analysis for a model-free setting, where a history-dependent policy was learned from the training tasks. In this work, we propose a different approach: directly learn the task distribution, using density estimation techniques, and then train a policy on the learned task distribution. We show that our approach leads to bounds that depend on the dimension of the task distribution. In particular, in settings where the task distribution lies in a low-dimensional manifold, we extend our analysis to use dimensionality reduction techniques and account for such structure, obtaining significantly better bounds than previous work, which strictly depend on the number of states and actions. The key of our approach is the regularization implied by the kernel density estimation method. We further demonstrate that this regularization is useful in practice, when `plugged in' the state-of-the-art VariBAD meta RL algorithm.

In this paper, we present a method for unconstrained end-to-end head pose estimation. We address the problem of ambiguous rotation labels by introducing the rotation matrix formalism for our ground truth data and propose a continuous 6D rotation matrix representation for efficient and robust direct regression. This way, our method can learn the full rotation appearance which is contrary to previous approaches that restrict the pose prediction to a narrow-angle for satisfactory results. In addition, we propose a geodesic distance-based loss to penalize our network with respect to the SO(3) manifold geometry. Experiments on the public AFLW2000 and BIWI datasets demonstrate that our proposed method significantly outperforms other state-of-the-art methods by up to 20\%. We open-source our training and testing code along with our pre-trained models: https://github.com/thohemp/6DRepNet.

In this paper, we introduce an SE(3) diffusion model-based point cloud registration framework for 6D object pose estimation in real-world scenarios. Our approach formulates the 3D registration task as a denoising diffusion process, which progressively refines the pose of the source point cloud to obtain a precise alignment with the model point cloud. Training our framework involves two operations: An SE(3) diffusion process and an SE(3) reverse process. The SE(3) diffusion process gradually perturbs the optimal rigid transformation of a pair of point clouds by continuously injecting noise (perturbation transformation). By contrast, the SE(3) reverse process focuses on learning a denoising network that refines the noisy transformation step-by-step, bringing it closer to the optimal transformation for accurate pose estimation. Unlike standard diffusion models used in linear Euclidean spaces, our diffusion model operates on the SE(3) manifold. This requires exploiting the linear Lie algebra se(3) associated with SE(3) to constrain the transformation transitions during the diffusion and reverse processes. Additionally, to effectively train our denoising network, we derive a registration-specific variational lower bound as the optimization objective for model learning. Furthermore, we show that our denoising network can be constructed with a surrogate registration model, making our approach applicable to different deep registration networks. Extensive experiments demonstrate that our diffusion registration framework presents outstanding pose estimation performance on the real-world TUD-L, LINEMOD, and Occluded-LINEMOD datasets.

Diffusion models are generative models that have recently demonstrated impressive performances in terms of sampling quality and density estimation in high dimensions. They rely on a forward continuous diffusion process and a backward continuous denoising process, which can be described by a time-dependent vector field and is used as a generative model. In the original formulation of the diffusion model, this vector field is assumed to be the score function (i.e. it is the gradient of the log-probability at a given time in the diffusion process). Curiously, on the practical side, most studies on diffusion models implement this vector field as a neural network function and do not constrain it be the gradient of some energy function (that is, most studies do not constrain the vector field to be conservative). Even though some studies investigated empirically whether such a constraint will lead to a performance gain, they lead to contradicting results and failed to provide analytical results. Here, we provide three analytical results regarding the extent of the modeling freedom of this vector field. {Firstly, we propose a novel decomposition of vector fields into a conservative component and an orthogonal component which satisfies a given (gauge) freedom. Secondly, from this orthogonal decomposition, we show that exact density estimation and exact sampling is achieved when the conservative component is exactly equals to the true score and therefore conservativity is neither necessary nor sufficient to obtain exact density estimation and exact sampling. Finally, we show that when it comes to inferring local information of the data manifold, constraining the vector field to be conservative is desirable.

Human pose and shape (HPS) estimation methods achieve remarkable results. However, current HPS benchmarks are mostly designed to test models in scenarios that are similar to the training data. This can lead to critical situations in real-world applications when the observed data differs significantly from the training data and hence is out-of-distribution (OOD). It is therefore important to test and improve the OOD robustness of HPS methods. To address this fundamental problem, we develop a simulator that can be controlled in a fine-grained manner using interpretable parameters to explore the manifold of images of human pose, e.g. by varying poses, shapes, and clothes. We introduce a learning-based testing method, termed PoseExaminer, that automatically diagnoses HPS algorithms by searching over the parameter space of human pose images to find the failure modes. Our strategy for exploring this high-dimensional parameter space is a multi-agent reinforcement learning system, in which the agents collaborate to explore different parts of the parameter space. We show that our PoseExaminer discovers a variety of limitations in current state-of-the-art models that are relevant in real-world scenarios but are missed by current benchmarks. For example, it finds large regions of realistic human poses that are not predicted correctly, as well as reduced performance for humans with skinny and corpulent body shapes. In addition, we show that fine-tuning HPS methods by exploiting the failure modes found by PoseExaminer improve their robustness and even their performance on standard benchmarks by a significant margin. The code are available for research purposes.

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.

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.

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 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.

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.

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.

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 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.

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.

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

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.

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.

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.

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.

In this paper, we tackle the problem of online semi-supervised learning (SSL). When data arrive in a stream, the dual problems of computation and data storage arise for any SSL method. We propose a fast approximate online SSL algorithm that solves for the harmonic solution on an approximate graph. We show, both empirically and theoretically, that good behavior can be achieved by collapsing nearby points into a set of local "representative points" that minimize distortion. Moreover, we regularize the harmonic solution to achieve better stability properties. We apply our algorithm to face recognition and optical character recognition applications to show that we can take advantage of the manifold structure to outperform the previous methods. Unlike previous heuristic approaches, we show that our method yields provable performance bounds.

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.

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

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.

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.

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.

Understanding how language and embedding models encode semantic relationships is fundamental to model interpretability. While early word embeddings exhibited intuitive vector arithmetic (''king'' - ''man'' + ''woman'' = ''queen''), modern high-dimensional text representations lack straightforward interpretable geometric properties. We introduce Rotor-Invariant Shift Estimation (RISE), a geometric approach that represents semantic-syntactic transformations as consistent rotational operations in embedding space, leveraging the manifold structure of modern language representations. RISE operations have the ability to operate across both languages and models without reducing performance, suggesting the existence of analogous cross-lingual geometric structure. We compare and evaluate RISE using two baseline methods, three embedding models, three datasets, and seven morphologically diverse languages in five major language groups. Our results demonstrate that RISE consistently maps discourse-level semantic-syntactic transformations with distinct grammatical features (e.g., negation and conditionality) across languages and models. This work provides the first demonstration that discourse-level semantic-syntactic transformations correspond to consistent geometric operations in multilingual embedding spaces, empirically supporting the linear representation hypothesis at the sentence level.

Measuring geometric similarity between high-dimensional network representations is a topic of longstanding interest to neuroscience and deep learning. Although many methods have been proposed, only a few works have rigorously analyzed their statistical efficiency or quantified estimator uncertainty in data-limited regimes. Here, we derive upper and lower bounds on the worst-case convergence of standard estimators of shape distancex2014a measure of representational dissimilarity proposed by Williams et al. (2021).These bounds reveal the challenging nature of the problem in high-dimensional feature spaces. To overcome these challenges, we introduce a new method-of-moments estimator with a tunable bias-variance tradeoff. We show that this estimator achieves substantially lower bias than standard estimators in simulation and on neural data, particularly in high-dimensional settings. Thus, we lay the foundation for a rigorous statistical theory for high-dimensional shape analysis, and we contribute a new estimation method that is well-suited to practical scientific settings.

Dependency graph, as a heterogeneous graph representing the intrinsic relationships between different pairs of system entities, is essential to many data analysis applications, such as root cause diagnosis, intrusion detection, etc. Given a well-trained dependency graph from a source domain and an immature dependency graph from a target domain, how can we extract the entity and dependency knowledge from the source to enhance the target? One way is to directly apply a mature dependency graph learned from a source domain to the target domain. But due to the domain variety problem, directly using the source dependency graph often can not achieve good performance. Traditional transfer learning methods mainly focus on numerical data and are not applicable. In this paper, we propose ACRET, a knowledge transfer based model for accelerating dependency graph learning from heterogeneous categorical event streams. In particular, we first propose an entity estimation model to filter out irrelevant entities from the source domain based on entity embedding and manifold learning. Only the entities with statistically high correlations are transferred to the target domain. On the surviving entities, we propose a dependency construction model for constructing the unbiased dependency relationships by solving a two-constraint optimization problem. The experimental results on synthetic and real-world datasets demonstrate the effectiveness and efficiency of ACRET. We also apply ACRET to a real enterprise security system for intrusion detection. Our method is able to achieve superior detection performance at least 20 days lead lag time in advance with more than 70% accuracy.

We introduce methods for obtaining pretrained Geometric Neural Operators (GNPs) that can serve as basal foundation models for use in obtaining geometric features. These can be used within data processing pipelines for machine learning tasks and numerical methods. We show how our GNPs can be trained to learn robust latent representations for the differential geometry of point-clouds to provide estimates of metric, curvature, and other shape-related features. We demonstrate how our pre-trained GNPs can be used (i) to estimate the geometric properties of surfaces of arbitrary shape and topologies with robustness in the presence of noise, (ii) to approximate solutions of geometric partial differential equations (PDEs) on manifolds, and (iii) to solve equations for shape deformations such as curvature driven flows. We release codes and weights for using GNPs in the package geo_neural_op. This allows for incorporating our pre-trained GNPs as components for reuse within existing and new data processing pipelines. The GNPs also can be used as part of numerical solvers involving geometry or as part of methods for performing inference and other geometric tasks.

Generative Models (GMs) have attracted considerable attention due to their tremendous success in various domains, such as computer vision where they are capable to generate impressive realistic-looking images. Likelihood-based GMs are attractive due to the possibility to generate new data by a single model evaluation. However, they typically achieve lower sample quality compared to state-of-the-art score-based diffusion models (DMs). This paper provides a significant step in the direction of addressing this limitation. The idea is to borrow one of the strengths of score-based DMs, which is the ability to perform accurate density estimation in low-density regions and to address manifold overfitting by means of data mollification. We connect data mollification through the addition of Gaussian noise to Gaussian homotopy, which is a well-known technique to improve optimization. Data mollification can be implemented by adding one line of code in the optimization loop, and we demonstrate that this provides a boost in generation quality of likelihood-based GMs, without computational overheads. We report results on image data sets with popular likelihood-based GMs, including variants of variational autoencoders and normalizing flows, showing large improvements in FID score.

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).

Leveraging the priors of 2D diffusion models for 3D editing has emerged as a promising paradigm. However, maintaining multi-view consistency in edited results remains challenging, and the extreme scarcity of 3D-consistent editing paired data renders supervised fine-tuning (SFT), the most effective training strategy for editing tasks, infeasible. In this paper, we observe that, while generating multi-view consistent 3D content is highly challenging, verifying 3D consistency is tractable, naturally positioning reinforcement learning (RL) as a feasible solution. Motivated by this, we propose RL3DEdit, a single-pass framework driven by RL optimization with novel rewards derived from the 3D foundation model, VGGT. Specifically, we leverage VGGT's robust priors learned from massive real-world data, feed the edited images, and utilize the output confidence maps and pose estimation errors as reward signals, effectively anchoring the 2D editing priors onto a 3D-consistent manifold via RL. Extensive experiments demonstrate that RL3DEdit achieves stable multi-view consistency and outperforms state-of-the-art methods in editing quality with high efficiency. To promote the development of 3D editing, we will release the code and model.

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.

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.

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.

k-nearest neighbor (k-NN) search is a fundamental primitive in geometry processing and computer graphics. While spatial partitioning structures such as kd-trees are standard, they are often manifold-blind, failing to exploit the intrinsic low-dimensional structure of points sampled from 2-manifolds. Recent advances in dynamic programming-based nearest neighbor search (DP-NNS) leverage incrementally constructed Voronoi diagrams to accelerate queries, where each site p maintains a list of successors that progressively refine its Voronoi cell. However, DP-NNS is restricted to single nearest neighbor (k=1) searches, precluding their adoption in applications that require local neighborhood statistics. In this paper, we generalize the DP-NNS framework to support arbitrary k-NN queries for manifold-aligned data. Our approach is founded on the geometric observation that if p_i is the nearest neighbor of a query q in P, then the second nearest neighbor of q must reside either within the prefix set P_{1:i-1} = {p_1, \dots, p_{i-1}} or within p_i's successor list. By recursively extending this principle, we introduce Manifold k-NN, a recursive algorithmic scheme that significantly outperforms conventional kd-trees for manifold-aligned data. Our method achieves a 1\times--10\times speedup in volume-to-surface query scenarios and inherently supports dynamic prefix queries -- enabling k-NN searches within any subset P_{1:m} (m \leq n) with zero overhead. Furthermore, we extend the framework to support point deletion via local Delaunay updates, providing a complete suite of dynamic operations for point set modification. Comprehensive experiments on diverse geometric datasets demonstrate the efficiency and broad applicability of our approach for modern graphics pipelines. Source code is available at https://github.com/sssomeone/manifold-knn.

We propose a new approach for generative modeling based on training a neural network to be idempotent. An idempotent operator is one that can be applied sequentially without changing the result beyond the initial application, namely f(f(z))=f(z). The proposed model f is trained to map a source distribution (e.g, Gaussian noise) to a target distribution (e.g. realistic images) using the following objectives: (1) Instances from the target distribution should map to themselves, namely f(x)=x. We define the target manifold as the set of all instances that f maps to themselves. (2) Instances that form the source distribution should map onto the defined target manifold. This is achieved by optimizing the idempotence term, f(f(z))=f(z) which encourages the range of f(z) to be on the target manifold. Under ideal assumptions such a process provably converges to the target distribution. This strategy results in a model capable of generating an output in one step, maintaining a consistent latent space, while also allowing sequential applications for refinement. Additionally, we find that by processing inputs from both target and source distributions, the model adeptly projects corrupted or modified data back to the target manifold. This work is a first step towards a ``global projector'' that enables projecting any input into a target data distribution.

We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.

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.

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.

Remarkable advances have been achieved recently in learning neural representations that characterize object geometry, while generating textured objects suitable for downstream applications and 3D rendering remains at an early stage. In particular, reconstructing textured geometry from images of real objects is a significant challenge -- reconstructed geometry is often inexact, making realistic texturing a significant challenge. We present Mesh2Tex, which learns a realistic object texture manifold from uncorrelated collections of 3D object geometry and photorealistic RGB images, by leveraging a hybrid mesh-neural-field texture representation. Our texture representation enables compact encoding of high-resolution textures as a neural field in the barycentric coordinate system of the mesh faces. The learned texture manifold enables effective navigation to generate an object texture for a given 3D object geometry that matches to an input RGB image, which maintains robustness even under challenging real-world scenarios where the mesh geometry approximates an inexact match to the underlying geometry in the RGB image. Mesh2Tex can effectively generate realistic object textures for an object mesh to match real images observations towards digitization of real environments, significantly improving over previous state of the art.

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.

Self-supervised learning (SSL) has emerged as a powerful paradigm for learning representations without labeled data. Most SSL approaches rely on strong, well-established, handcrafted data augmentations to generate diverse views for representation learning. However, designing such augmentations requires domain-specific knowledge and implicitly imposes representational invariances on the model, which can limit generalization. In this work, we propose an unsupervised representation learning method that replaces augmentations by generating views using orthonormal bases and overcomplete frames. We show that embeddings learned from orthonormal and overcomplete spaces reside on distinct manifolds, shaped by the geometric biases introduced by representing samples in different spaces. By jointly leveraging the complementary geometry of these distinct manifolds, our approach achieves superior performance without artificially increasing data diversity through strong augmentations. We demonstrate the effectiveness of our method on nine datasets across five temporal sequence tasks, where signal-specific characteristics make data augmentations particularly challenging. Without relying on augmentation-induced diversity, our method achieves performance gains of up to 15--20\% over existing self-supervised approaches. Source code: https://github.com/eth-siplab/Learning-with-FrameProjections

State-of-the-art flow models generate stunning images from text or image prompts. However, they suffer from diversity collapse when generating multiple samples under the same conditioning. Existing methods address this issue via either latent guidance, which has limited effectiveness, or sample selection, which relies on external reward models that incur significant inference-time overhead. In this work, we introduce an efficient, training-free self-guidance mechanism to mitigate diversity collapse without requiring additional reward models. Specifically, we disperse the internal features of the flow model during batch generation with feature self-guidance. Further, to keep the features close to the manifold, we introduce a manifold regularization step that projects these dispersed features back onto the data manifold, ensuring diverse generation without sacrificing alignment with the input conditions. Our method integrates seamlessly as a plug-and-play module into pretrained flow models, adding only a marginal inference cost. Experiments demonstrate significant improvements in diversity while preserving fidelity across several conditional flow models, including multi-step and few-step text-to-image, depth-to-image, and reference image generation.

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).

Dimensionality reduction is a fundamental task that aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation. To preserve the underlying data structure, multi-dimensional scaling (MDS) methods focus on preserving pairwise dissimilarities, such as distances. They optimise the embedding to have pairwise distances as close as possible to the data dissimilarities. However, the current standard is limited to embedding data in Riemannian manifolds. Motivated by the lack of asymmetry in the Riemannian metric of the embedding space, this paper extends the MDS problem to a natural asymmetric generalisation of Riemannian manifolds called Finsler manifolds. Inspired by Euclidean space, we define a canonical Finsler space for embedding asymmetric data. Due to its simplicity with respect to geodesics, data representation in this space is both intuitive and simple to analyse. We demonstrate that our generalisation benefits from the same theoretical convergence guarantees. We reveal the effectiveness of our Finsler embedding across various types of non-symmetric data, highlighting its value in applications such as data visualisation, dimensionality reduction, directed graph embedding, and link prediction.

Image analysis in the euclidean space through linear hyperspaces is well studied. However, in the quest for more effective image representations, we turn to hyperbolic manifolds. They provide a compelling alternative to capture complex hierarchical relationships in images with remarkably small dimensionality. To demonstrate hyperbolic embeddings' competence, we introduce a light-weight hyperbolic graph neural network for image segmentation, encompassing patch-level features in a very small embedding size. Our solution, Seg-HGNN, surpasses the current best unsupervised method by 2.5\%, 4\% on VOC-07, VOC-12 for localization, and by 0.8\%, 1.3\% on CUB-200, ECSSD for segmentation, respectively. With less than 7.5k trainable parameters, Seg-HGNN delivers effective and fast (approx 2 images/second) results on very standard GPUs like the GTX1650. This empirical evaluation presents compelling evidence of the efficacy and potential of hyperbolic representations for vision tasks.

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.

What is the geometry of a visual percept? The most widely used protocols for decomposing neural network representations into interpretable parts treat concepts as isolated directions, yet recent work shows that concepts are often realized as geometric structures in low dimensional regions of activation space. We turn to the literature of Structured sparsity to close this gap, and show that block sparsity, which groups directions into blocks, is the prior matched to a generative model in which a representation is a sparse sum of low-dimensional manifolds: the modern, learned form of a classical idea in visual neuroscience, where a visual feature is carried by a coordinated group of neurons rather than a single tuned one. We implement three variants of block-sparse featurizers (BSFs) and, through a minimum-description-length analysis, show that all three describe activations more compactly than direction-based featurizers, with the recovered concepts typically two- to four-dimensional. We then use BSFs to (i) recontextualize prior work, showing that curve detectors in InceptionV1 actually read from a single continuous curve manifold, (ii) discover novel manifolds including shadows and lighting in DINOv3, and (iii) support interpretable control of image generation in diffusion models (SDXL) via manifold steering.

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.

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.

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

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 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.

The efficient coding hypothesis proposes that the response properties of sensory systems are adapted to the statistics of their inputs such that they capture maximal information about the environment, subject to biological constraints. While elegant, information theoretic properties are notoriously difficult to measure in practical settings or to employ as objective functions in optimization. This difficulty has necessitated that computational models designed to test the hypothesis employ several different information metrics ranging from approximations and lower bounds to proxy measures like reconstruction error. Recent theoretical advances have characterized a novel and ecologically relevant efficiency metric, the manifold capacity, which is the number of object categories that may be represented in a linearly separable fashion. However, calculating manifold capacity is a computationally intensive iterative procedure that until now has precluded its use as an objective. Here we outline the simplifying assumptions that allow manifold capacity to be optimized directly, yielding Maximum Manifold Capacity Representations (MMCR). The resulting method is closely related to and inspired by advances in the field of self supervised learning (SSL), and we demonstrate that MMCRs are competitive with state of the art results on standard SSL benchmarks. Empirical analyses reveal differences between MMCRs and representations learned by other SSL frameworks, and suggest a mechanism by which manifold compression gives rise to class separability. Finally we evaluate a set of SSL methods on a suite of neural predictivity benchmarks, and find MMCRs are higly competitive as models of the ventral stream.

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.

Estimating the pose of an object from a monocular image is an inverse problem fundamental in computer vision. The ill-posed nature of this problem requires incorporating deformation priors to solve it. In practice, many materials do not perceptibly shrink or extend when manipulated, constituting a powerful and well-known prior. Mathematically, this translates to the preservation of the Riemannian metric. Neural networks offer the perfect playground to solve the surface reconstruction problem as they can approximate surfaces with arbitrary precision and allow the computation of differential geometry quantities. This paper presents an approach to inferring continuous deformable surfaces from a sequence of images, which is benchmarked against several techniques and obtains state-of-the-art performance without the need for offline training.

Recent text-to-video (T2V) diffusion models rely heavily on auxiliary reward signals (e.g., via reward models or DPO) to align generated content with human aesthetics and improve realism. These signals, however, incur substantial computational overhead, require costly human annotations, and often yield limited improvement in fine-grained local details. In this paper, we argue that your data manifold is secretly a reward model. By explicitly modeling the manifold structure of high-quality Supervised Fine-Tuning (SFT) data and encouraging video latents to lie on this manifold, we derive dense, differentiable, and nearly cost-free reward signals that significantly improve video quality, particularly in mitigating low-level distortions. Our modeling builds upon Local Coordinate Coding (LCC), which captures the `skeleton' of the manifold. However, directly applying LCC suffers from mean regression, pulling latents toward the geometric mean and losing high-frequency details. We therefore extend it to Shell Local Coordinate Coding (Shell-LCC), which models the manifold `surface' as an isotropic shell to align with the true high-density region. Experiments demonstrate that our approach improves realism, enhances high-frequency details, reduces over-smoothing artifacts, and alleviates motion blur.

While large language models (LLMs) have emerged as a significant advancement in artificial intelligence, the hardware and computational costs for training LLMs are also significantly burdensome. Among the state-of-the-art optimizers, AdamW relies on diagonal curvature estimates and ignores structural properties, while Muon applies global spectral normalization at the expense of losing curvature information. In this study, we restriked manifold optimization methods for training LLMs, which may address both optimizers' limitations, while conventional manifold optimization methods have been largely overlooked due to the poor performance in large-scale model optimization. By innovatively projecting the momentum onto the tangent space of model parameters and constraining it on a rotational Oblique manifold, we propose a novel, powerful, and efficient optimizer **Mano** that is the first to bridge the performance gap between manifold optimization and modern optimizers. Extensive experiments on the LLaMA and Qwen3 models demonstrate that Mano consistently and significantly outperforms AdamW and Muon even with less memory consumption and computational complexity, respectively, suggesting an expanded Pareto frontier in terms of space and time efficiency.

With diffusion and flow matching models achieving state-of-the-art generating performance, the interest of the community now turned to reducing the inference time without sacrificing sample quality. Consistency Models (CMs), which are trained to be consistent on diffusion or probability flow ordinary differential equation (PF-ODE) trajectories, enable one or two-step flow or diffusion sampling. However, CMs typically require prolonged training with large batch sizes to obtain competitive sample quality. In this paper, we examine the training dynamics of CMs near convergence and discover that CM tangents -- CM output update directions -- are quite oscillatory, in the sense that they move parallel to the data manifold, not towards the manifold. To mitigate oscillatory tangents, we propose a new loss function, called the manifold feature distance (MFD), which provides manifold-aligned tangents that point toward the data manifold. Consequently, our method -- dubbed Align Your Tangent (AYT) -- can accelerate CM training by orders of magnitude and even out-perform the learned perceptual image patch similarity metric (LPIPS). Furthermore, we find that our loss enables training with extremely small batch sizes without compromising sample quality. Code: https://github.com/1202kbs/AYT

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 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.

Geometric foundation models, such as the Visual Geometry Grounded Transformer (VGGT), provide strong 3D priors from unposed images. However, such models operate purely in a feed-forward, deterministic regime, \ie~they cannot generate plausible geometry beyond what the input views directly support. Generative models for 3D scenes, on the other hand, must rely on strong geometric priors to produce coherent outputs from sparse inputs. We bridge these two paradigms by performing flow matching directly in VGGT's latent space, leveraging its learned 3D priors without committing to any explicit downstream representation such as Gaussians, meshes, or video-VAE latents. This requires respecting the latent geometry: VGGT tokens occupy a product of high-dimensional hyperspheres on which standard Euclidean flow matching fails. We address this with a Riemannian Flow Matching framework defined on a product manifold of four hyperspheres, aligned with VGGT's multi-scale encoder, which keeps generated tokens on the valid data manifold required by the frozen decoding heads. On RealEstate10K, ScanNet++ and ETH3D, our method achieves strong performance against recent scene generation baselines in both per-view appearance and aggregated 3D geometry, establishing latent-space flow matching on geometric foundation models as a viable paradigm for 3D generation. The project page can be found https://lisaweijler.github.io/geometry-grounded-rfm/{here}.

The Distributional Principal Autoencoder (DPA) combines distributionally correct reconstruction with principal-component-like interpretability of the encodings. In this work, we provide exact theoretical guarantees on both fronts. First, we derive a closed-form relation linking each optimal level-set geometry to the data-distribution score. This result explains DPA's empirical ability to disentangle factors of variation of the data, as well as allows the score to be recovered directly from samples. When the data follows the Boltzmann distribution, we demonstrate that this relation yields an approximation of the minimum free-energy path for the Mueller-Brown potential in a single fit. Second, we prove that if the data lies on a manifold that can be approximated by the encoder, latent components beyond the manifold dimension are conditionally independent of the data distribution - carrying no additional information - and thus reveal the intrinsic dimension. Together, these results show that a single model can learn the data distribution and its intrinsic dimension with exact guarantees simultaneously, unifying two longstanding goals of unsupervised learning.

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.

We present MoGe, a powerful model for recovering 3D geometry from monocular open-domain images. Given a single image, our model directly predicts a 3D point map of the captured scene with an affine-invariant representation, which is agnostic to true global scale and shift. This new representation precludes ambiguous supervision in training and facilitate effective geometry learning. Furthermore, we propose a set of novel global and local geometry supervisions that empower the model to learn high-quality geometry. These include a robust, optimal, and efficient point cloud alignment solver for accurate global shape learning, and a multi-scale local geometry loss promoting precise local geometry supervision. We train our model on a large, mixed dataset and demonstrate its strong generalizability and high accuracy. In our comprehensive evaluation on diverse unseen datasets, our model significantly outperforms state-of-the-art methods across all tasks, including monocular estimation of 3D point map, depth map, and camera field of view. Code and models can be found on our project page.

We propose a spherical kernel for efficient graph convolution of 3D point clouds. Our metric-based kernels systematically quantize the local 3D space to identify distinctive geometric relationships in the data. Similar to the regular grid CNN kernels, the spherical kernel maintains translation-invariance and asymmetry properties, where the former guarantees weight sharing among similar local structures in the data and the latter facilitates fine geometric learning. The proposed kernel is applied to graph neural networks without edge-dependent filter generation, making it computationally attractive for large point clouds. In our graph networks, each vertex is associated with a single point location and edges connect the neighborhood points within a defined range. The graph gets coarsened in the network with farthest point sampling. Analogous to the standard CNNs, we define pooling and unpooling operations for our network. We demonstrate the effectiveness of the proposed spherical kernel with graph neural networks for point cloud classification and semantic segmentation using ModelNet, ShapeNet, RueMonge2014, ScanNet and S3DIS datasets. The source code and the trained models can be downloaded from https://github.com/hlei-ziyan/SPH3D-GCN.

Despite the recent advancements, conditional image generation still faces challenges of cost, generalizability, and the need for task-specific training. In this paper, we propose Manifold Preserving Guided Diffusion (MPGD), a training-free conditional generation framework that leverages pretrained diffusion models and off-the-shelf neural networks with minimal additional inference cost for a broad range of tasks. Specifically, we leverage the manifold hypothesis to refine the guided diffusion steps and introduce a shortcut algorithm in the process. We then propose two methods for on-manifold training-free guidance using pre-trained autoencoders and demonstrate that our shortcut inherently preserves the manifolds when applied to latent diffusion models. Our experiments show that MPGD is efficient and effective for solving a variety of conditional generation applications in low-compute settings, and can consistently offer up to 3.8x speed-ups with the same number of diffusion steps while maintaining high sample quality compared to the baselines.

We introduce the Information-Estimation Metric (IEM), a novel form of distance function derived from an underlying continuous probability density over a domain of signals. The IEM is rooted in a fundamental relationship between information theory and estimation theory, which links the log-probability of a signal with the errors of an optimal denoiser, applied to noisy observations of the signal. In particular, the IEM between a pair of signals is obtained by comparing their denoising error vectors over a range of noise amplitudes. Geometrically, this amounts to comparing the score vector fields of the blurred density around the signals over a range of blur levels. We prove that the IEM is a valid global distance metric and derive a closed-form expression for its local second-order approximation, which yields a Riemannian metric. For Gaussian-distributed signals, the IEM coincides with the Mahalanobis distance. But for more complex distributions, it adapts, both locally and globally, to the geometry of the distribution. In practice, the IEM can be computed using a learned denoiser (analogous to generative diffusion models) and solving a one-dimensional integral. To demonstrate the value of our framework, we learn an IEM on the ImageNet database. Experiments show that this IEM is competitive with or outperforms state-of-the-art supervised image quality metrics in predicting human perceptual judgments.

In this paper we introduce a novel family of attributed graphs for the purpose of shape discrimination. Our graphs typically arise from variations on the Mapper graph construction, which is an approximation of the Reeb graph for point cloud data. Our attributions enrich these constructions with (persistent) homology in ways that are provably stable, thereby recording extra topological information that is typically lost in these graph constructions. We provide experiments which illustrate the use of these invariants for shape representation and classification. In particular, we obtain competitive shape classification results when using our topologically attributed graphs as inputs to a simple graph neural network classifier.

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.

We present EuLearn, the first surface datasets equitably representing a diversity of topological types. We designed our embedded surfaces of uniformly varying genera relying on random knots, thus allowing our surfaces to knot with themselves. EuLearn contributes new topological datasets of meshes, point clouds, and scalar fields in 3D. We aim to facilitate the training of machine learning systems that can discern topological features. We experimented with specific emblematic 3D neural network architectures, finding that their vanilla implementations perform poorly on genus classification. To enhance performance, we developed a novel, non-Euclidean, statistical sampling method adapted to graph and manifold data. We also introduce adjacency-informed adaptations of PointNet and Transformer architectures that rely on our non-Euclidean sampling strategy. Our results demonstrate that incorporating topological information into deep learning workflows significantly improves performance on these otherwise challenging EuLearn datasets.

Entropy and mutual information in neural networks provide rich information on the learning process, but they have proven difficult to compute reliably in high dimensions. Indeed, in noisy and high-dimensional data, traditional estimates in ambient dimensions approach a fixed entropy and are prohibitively hard to compute. To address these issues, we leverage data geometry to access the underlying manifold and reliably compute these information-theoretic measures. Specifically, we define diffusion spectral entropy (DSE) in neural representations of a dataset as well as diffusion spectral mutual information (DSMI) between different variables representing data. First, we show that they form noise-resistant measures of intrinsic dimensionality and relationship strength in high-dimensional simulated data that outperform classic Shannon entropy, nonparametric estimation, and mutual information neural estimation (MINE). We then study the evolution of representations in classification networks with supervised learning, self-supervision, or overfitting. We observe that (1) DSE of neural representations increases during training; (2) DSMI with the class label increases during generalizable learning but stays stagnant during overfitting; (3) DSMI with the input signal shows differing trends: on MNIST it increases, while on CIFAR-10 and STL-10 it decreases. Finally, we show that DSE can be used to guide better network initialization and that DSMI can be used to predict downstream classification accuracy across 962 models on ImageNet. The official implementation is available at https://github.com/ChenLiu-1996/DiffusionSpectralEntropy.

Tokenizers are a crucial component of latent diffusion models, as they define the latent space in which diffusion models operate. However, existing tokenizers are primarily designed to improve reconstruction fidelity or inherit pretrained representations, leaving unclear what kind of latent space is truly friendly for generative modeling. In this paper, we study this question from the perspective of latent manifold organization. By constructing controlled tokenizer variants, we identify three key properties of a diffusion-friendly latent manifold: coherent spatial structure, local manifold continuity, and global manifold semantics. We find that these properties are more consistent with downstream generation quality than reconstruction fidelity. Motivated by this finding, we propose the Prior-Aligned AutoEncoder (PAE), which explicitly shapes the latent manifold instead of leaving diffusion-friendly manifold to emerge indirectly from reconstruction or inheritance. Specifically, PAE leverages refined priors derived from VFMs and perturbation-based regularization to turn spatial structure, local continuity, and global semantics into explicit training objectives. On ImageNet 256x256, PAE improves both training efficiency and generation quality over existing tokenizers, reaching performance comparable to RAE with up to 13x faster convergence under the same training setup and achieving a new state-of-the-art gFID of 1.03. These results highlight the importance of organizing the latent manifold for latent diffusion models.

Dimensionality reduction methods are unsupervised approaches which learn low-dimensional spaces where some properties of the initial space, typically the notion of "neighborhood", are preserved. Such methods usually require propagation on large k-NN graphs or complicated optimization solvers. On the other hand, self-supervised learning approaches, typically used to learn representations from scratch, rely on simple and more scalable frameworks for learning. In this paper, we propose TLDR, a dimensionality reduction method for generic input spaces that is porting the recent self-supervised learning framework of Zbontar et al. (2021) to the specific task of dimensionality reduction, over arbitrary representations. We propose to use nearest neighbors to build pairs from a training set and a redundancy reduction loss to learn an encoder that produces representations invariant across such pairs. TLDR is a method that is simple, easy to train, and of broad applicability; it consists of an offline nearest neighbor computation step that can be highly approximated, and a straightforward learning process. Aiming for scalability, we focus on improving linear dimensionality reduction, and show consistent gains on image and document retrieval tasks, e.g. gaining +4% mAP over PCA on ROxford for GeM- AP, improving the performance of DINO on ImageNet or retaining it with a 10x compression.

In the graph node embedding problem, embedding spaces can vary significantly for different data types, leading to the need for different GNN model types. In this paper, we model the embedding update of a node feature as a Hamiltonian orbit over time. Since the Hamiltonian orbits generalize the exponential maps, this approach allows us to learn the underlying manifold of the graph in training, in contrast to most of the existing literature that assumes a fixed graph embedding manifold with a closed exponential map solution. Our proposed node embedding strategy can automatically learn, without extensive tuning, the underlying geometry of any given graph dataset even if it has diverse geometries. We test Hamiltonian functions of different forms and verify the performance of our approach on two graph node embedding downstream tasks: node classification and link prediction. Numerical experiments demonstrate that our approach adapts better to different types of graph datasets than popular state-of-the-art graph node embedding GNNs. The code is available at https://github.com/zknus/Hamiltonian-GNN.

We address the problem of performing backpropagation for computation graphs involving 3D transformation groups SO(3), SE(3), and Sim(3). 3D transformation groups are widely used in 3D vision and robotics, but they do not form vector spaces and instead lie on smooth manifolds. The standard backpropagation approach, which embeds 3D transformations in Euclidean spaces, suffers from numerical difficulties. We introduce a new library, which exploits the group structure of 3D transformations and performs backpropagation in the tangent spaces of manifolds. We show that our approach is numerically more stable, easier to implement, and beneficial to a diverse set of tasks. Our plug-and-play PyTorch library is available at https://github.com/princeton-vl/lietorch.

Graph representation of objects and their relations in a scene, known as a scene graph, provides a precise and discernible interface to manipulate a scene by modifying the nodes or the edges in the graph. Although existing works have shown promising results in modifying the placement and pose of objects, scene manipulation often leads to losing some visual characteristics like the appearance or identity of objects. In this work, we propose DisPositioNet, a model that learns a disentangled representation for each object for the task of image manipulation using scene graphs in a self-supervised manner. Our framework enables the disentanglement of the variational latent embeddings as well as the feature representation in the graph. In addition to producing more realistic images due to the decomposition of features like pose and identity, our method takes advantage of the probabilistic sampling in the intermediate features to generate more diverse images in object replacement or addition tasks. The results of our experiments show that disentangling the feature representations in the latent manifold of the model outperforms the previous works qualitatively and quantitatively on two public benchmarks. Project Page: https://scenegenie.github.io/DispositioNet/

In this paper we propose a novel learning framework called Supervised and Weakly Supervised Learning where the goal is to learn simultaneously from weakly and strongly labeled data. Strongly labeled data can be simply understood as fully supervised data where all labeled instances are available. In weakly supervised learning only data is weakly labeled which prevents one from directly applying supervised learning methods. Our proposed framework is motivated by the fact that a small amount of strongly labeled data can give considerable improvement over only weakly supervised learning. The primary problem domain focus of this paper is acoustic event and scene detection in audio recordings. We first propose a naive formulation for leveraging labeled data in both forms. We then propose a more general framework for Supervised and Weakly Supervised Learning (SWSL). Based on this general framework, we propose a graph based approach for SWSL. Our main method is based on manifold regularization on graphs in which we show that the unified learning can be formulated as a constraint optimization problem which can be solved by iterative concave-convex procedure (CCCP). Our experiments show that our proposed framework can address several concerns of audio content analysis using weakly labeled data.

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.

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.

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.

Classic 3D scene graph generation approaches fail to work in real-time due to the heavy computational cost of environment mapping and the need to generate intermediate point-cloud representations. To alleviate this issue, a recent work eschews point clouds in favor of a lightweight Gaussian distribution for each object. This approximation drastically speeds up inference and enables real-time 3D scene graph generation. However, the representation has two key weaknesses. 1) Each object is approximated by a single 3D Gaussian, which causes a severe loss of 3D geometric detail. 2) The discrepancy between this approximation and the true object geometry exacerbates the inaccurate merging of object candidates during online inference. To address these issues, we propose NoPA, which represents each object as a separate non-parametric distribution. This formulation retains 3D geometric information while preserving real-time inference of the parametric Gaussian formulation. To build upon our novel object representation, we propose a tailored merging strategy to recover coherent object instances. Specifically, we leverage maximum mean discrepancy on kernel density estimates to enable robust merging of object candidates during online exploration while minimizing added computational complexity. The key is to maintain a fixed particle set per object. Furthermore, to rectify the relation loss caused by misclassified objects, NoPA propagates relationships between objects with high affinity. Experiments show that NoPA substantially outperforms current methods without sacrificing real-time inference speed.

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.

For non-rigid objects, predicting the 3D shape from 2D keypoint observations is ill-posed due to occlusions, and the need to disentangle changes in viewpoint and changes in shape. This challenge has often been addressed by embedding low-rank constraints into specialized models. These models can be hard to train, as they depend on finding a canonical way of aligning observations, before they can learn detailed geometry. These constraints have limited the reconstruction quality. We show that generic, high capacity models, trained with an unsupervised loss, allow for more accurate predicted shapes. In particular, applying low-rank constraints to localized subsets of the full shape allows the high capacity to be suitably constrained. We reduce the state-of-the-art reconstruction error on the S-Up3D dataset by over 70%.

Dimensionality reduction is crucial both for visualization and preprocessing high dimensional data for machine learning. We introduce a novel method based on a hierarchy built on 1-nearest neighbor graphs in the original space which is used to preserve the grouping properties of the data distribution on multiple levels. The core of the proposal is an optimization-free projection that is competitive with the latest versions of t-SNE and UMAP in performance and visualization quality while being an order of magnitude faster in run-time. Furthermore, its interpretable mechanics, the ability to project new data, and the natural separation of data clusters in visualizations make it a general purpose unsupervised dimension reduction technique. In the paper, we argue about the soundness of the proposed method and evaluate it on a diverse collection of datasets with sizes varying from 1K to 11M samples and dimensions from 28 to 16K. We perform comparisons with other state-of-the-art methods on multiple metrics and target dimensions highlighting its efficiency and performance. Code is available at https://github.com/koulakis/h-nne

This paper proposes a novel algorithm for semisupervised learning. This algorithm learns graph cuts that maximize the margin with respect to the labels induced by the harmonic function solution. We motivate the approach, compare it to existing work, and prove a bound on its generalization error. The quality of our solutions is evaluated on a synthetic problem and three UCI ML repository datasets. In most cases, we outperform manifold regularization of support vector machines, which is a state-of-the-art approach to semi-supervised max-margin learning.

The ability to automatically estimate the quality and coverage of the samples produced by a generative model is a vital requirement for driving algorithm research. We present an evaluation metric that can separately and reliably measure both of these aspects in image generation tasks by forming explicit, non-parametric representations of the manifolds of real and generated data. We demonstrate the effectiveness of our metric in StyleGAN and BigGAN by providing several illustrative examples where existing metrics yield uninformative or contradictory results. Furthermore, we analyze multiple design variants of StyleGAN to better understand the relationships between the model architecture, training methods, and the properties of the resulting sample distribution. In the process, we identify new variants that improve the state-of-the-art. We also perform the first principled analysis of truncation methods and identify an improved method. Finally, we extend our metric to estimate the perceptual quality of individual samples, and use this to study latent space interpolations.

We propose MoGe-2, an advanced open-domain geometry estimation model that recovers a metric scale 3D point map of a scene from a single image. Our method builds upon the recent monocular geometry estimation approach, MoGe, which predicts affine-invariant point maps with unknown scales. We explore effective strategies to extend MoGe for metric geometry prediction without compromising the relative geometry accuracy provided by the affine-invariant point representation. Additionally, we discover that noise and errors in real data diminish fine-grained detail in the predicted geometry. We address this by developing a unified data refinement approach that filters and completes real data from different sources using sharp synthetic labels, significantly enhancing the granularity of the reconstructed geometry while maintaining the overall accuracy. We train our model on a large corpus of mixed datasets and conducted comprehensive evaluations, demonstrating its superior performance in achieving accurate relative geometry, precise metric scale, and fine-grained detail recovery -- capabilities that no previous methods have simultaneously achieved.

A common approach for sequence tagging tasks based on contextual word representations is to train a machine learning classifier directly on these embedding vectors. This approach has two shortcomings. First, such methods consider single input sequences in isolation and are unable to put an individual embedding vector in relation to vectors outside the current local context of use. Second, the high performance of these models relies on fine-tuning the embedding model in conjunction with the classifier, which may not always be feasible due to the size or inaccessibility of the underlying feature-generation model. It is thus desirable, given a collection of embedding vectors of a corpus, i.e., a datastore, to find features of each vector that describe its relation to other, similar vectors in the datastore. With this in mind, we introduce complexity measures of the local topology of the latent space of a contextual language model with respect to a given datastore. The effectiveness of our features is demonstrated through their application to dialogue term extraction. Our work continues a line of research that explores the manifold hypothesis for word embeddings, demonstrating that local structure in the space carved out by word embeddings can be exploited to infer semantic properties.

Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we choose a filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose a framework that learns a filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invariant, we develop a neural network architecture with such invariance. Additionally, we show a theoretical result on a finite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.

Data augmentation is critical to the empirical success of modern self-supervised representation learning, such as contrastive learning and masked language modeling. However, a theoretical understanding of the exact role of augmentation remains limited. Recent work has built the connection between self-supervised learning and the approximation of the top eigenspace of a graph Laplacian operator, suggesting that learning a linear probe atop such representation can be connected to RKHS regression. Building on this insight, this work delves into a statistical analysis of augmentation-based pretraining. Starting from the isometry property, a geometric characterization of the target function given by the augmentation, we disentangle the effects of the model and the augmentation, and prove two generalization bounds that are free of model complexity. Our first bound works for an arbitrary encoder, where the prediction error is decomposed as the sum of an estimation error incurred by fitting a linear probe with RKHS regression, and an approximation error entailed by RKHS approximation. Our second bound specifically addresses the case where the encoder is near-optimal, that is it approximates the top-d eigenspace of the RKHS induced by the augmentation. A key ingredient in our analysis is the augmentation complexity, which we use to quantitatively compare different augmentations and analyze their impact on downstream performance.

One-class learning is the classic problem of fitting a model to data for which annotations are available only for a single class. In this paper, we propose a novel objective for one-class learning. Our key idea is to use a pair of orthonormal frames -- as subspaces -- to "sandwich" the labeled data via optimizing for two objectives jointly: i) minimize the distance between the origins of the two subspaces, and ii) to maximize the margin between the hyperplanes and the data, either subspace demanding the data to be in its positive and negative orthant respectively. Our proposed objective however leads to a non-convex optimization problem, to which we resort to Riemannian optimization schemes and derive an efficient conjugate gradient scheme on the Stiefel manifold. To study the effectiveness of our scheme, we propose a new dataset~Dash-Cam-Pose, consisting of clips with skeleton poses of humans seated in a car, the task being to classify the clips as normal or abnormal; the latter is when any human pose is out-of-position with regard to say an airbag deployment. Our experiments on the proposed Dash-Cam-Pose dataset, as well as several other standard anomaly/novelty detection benchmarks demonstrate the benefits of our scheme, achieving state-of-the-art one-class accuracy.

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.

We address the fundamental question of why deep neural networks generalize by establishing a pointwise generalization theory for fully connected networks. This framework resolves long-standing barriers to characterizing the rich nonlinear feature-learning regime and builds a new statistical foundation for representation learning. For each trained model, we characterize the hypothesis via a pointwise Riemannian Dimension, derived from the eigenvalues of the learned feature representations across layers. This establishes a principled framework for deriving hypothesis-dependent, representation-aware generalization bounds. These bounds offer a systematic upgrade over approaches based on model size, products of norms, and infinite-width linearizations, yielding guarantees that are orders of magnitude tighter in both theory and experiment. Analytically, we identify the structural properties and mathematical principles that explain the tractability of deep networks. Empirically, the pointwise Riemannian Dimension exhibits substantial feature compression, decreases with increased over-parameterization, and captures the implicit bias of optimizers. Taken together, our results indicate that deep networks are mathematically tractable in practical regimes and that their generalization is sharply explained by pointwise, feature-spectrum-aware complexity.

Capsule networks (see e.g. Hinton et al., 2018) aim to encode knowledge of and reason about the relationship between an object and its parts. In this paper we specify a generative model for such data, and derive a variational algorithm for inferring the transformation of each model object in a scene, and the assignments of observed parts to the objects. We derive a learning algorithm for the object models, based on variational expectation maximization (Jordan et al., 1999). We also study an alternative inference algorithm based on the RANSAC method of Fischler and Bolles (1981). We apply these inference methods to (i) data generated from multiple geometric objects like squares and triangles ("constellations"), and (ii) data from a parts-based model of faces. Recent work by Kosiorek et al. (2019) has used amortized inference via stacked capsule autoencoders (SCAEs) to tackle this problem -- our results show that we significantly outperform them where we can make comparisons (on the constellations data).

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.

We present a geometric framework for filtered approximate nearest neighbor (ANN) search. Filtering a proximity graph by a metadata predicate produces a subgraph, a fiber, whose connectivity and geometry can differ sharply from the full graph. Using local signals, we propose a two-phase search algorithm that combines full-graph exploration with filtered-neighbor descent when the local geometry is favorable. These signals also classify search failures into three regimes: topological cuts, geometric folds, and genuine basins. A key observation is that all three share a common resolution: restarting the search in a fiber-present cluster near the query. To support this, we introduce a lightweight anchor structure that identifies such regions and restarts the search accordingly. We show empirically that the method outperforms FAISS HNSW on filtered search and the three failure regimes separate cleanly and shift predictably with filter selectivity.