Geodesics found in deep linear networks.
problem Finding shortest paths in deep neural networks.
method Derived ODEs and explicit solutions for geodesics.
result Horizontal straight lines are geodesics in invariant manifold.
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arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
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48 results for “Deep Riemannian Networks”
Entropy formula derived for deep linear networks using geometric analysis.
problem Thermodynamic description of learning in deep linear networks.
method Group actions, Riemannian submersion, foliation, Jacobi matrices.
result Entropy formula defined on the balanced manifold of DLN.
Unified framework for Riemannian deep learning across manifold-valued representations.
problem Deep learning on manifold-valued representations often relies on Euclidean approximations or costly geometric operations.
method Develops reusable neural modules, manifold-specific network architectures, and geometric designs.
result Generalizes batch normalization and multinomial logistic regression to broader classes of manifolds.
Unified framework for Riemannian deep learning across manifold-valued representations.
problem Deep learning on manifold-valued data lacks reusable modules, specific network architectures, and efficient geometric operations.
method Develops reusable neural modules, manifold-specific network architectures, and geometric designs for broad classes of Lie groups and gyrogroups.
result Generalizes batch normalization and multinomial logistic regression to Riemannian manifolds, including SPD and hyperbolic spaces.
Researchers extend ResNets to Riemannian manifolds, improving performance over existing methods.
problem Learning on Riemannian manifolds, especially for hierarchical graphs and manifold-valued data.
method Geometrically principled extension of ResNets to general Riemannian manifolds.
result Riemannian ResNets outperform existing manifold neural networks in relevant metrics and training dynamics.
Proposes eDNNs and iDNNs for deep learning on manifolds.
problem Deep learning on manifolds with geometric preservation and intrinsic geometry incorporation.
method Intrinsic and extrinsic deep neural networks (iDNNs and eDNNs) with geometric embeddings and maps.
result Empirical risk minimizers of eDNNs and iDNNs converge optimally.
Deep networks without non-linearities are equivalent to shallow ones.
problem Training deep orthogonal linear networks with no non-linearity.
method Riemannian gradient descent and gradient descent on factorization.
result Training deep overparametrized networks is equivalent to shallow ones.
End-to-end DRNs outperform ConvNets in EEG decoding.
problem Improving performance of Deep Riemannian Networks (DRNs) in EEG decoding.
method Wide, end-to-end DRN architecture designed and tested on five public EEG datasets.
result EE(G)-SPDNet outperforms state-of-the-art ConvNets in EEG decoding.
Regularization leads to balancedness in deep linear networks.
problem Balancedness in deep linear networks.
method Geometric invariant theory and Riemannian geometry of fibers.
result Balancing flows converge to the balanced manifold at a uniform exponential rate.
Proposes BN layers for neural networks on complex domains, improving training stability and accuracy.
problem Training stability and accuracy issues in neural networks on complex domains.
method Developed Riemannian batch normalization (BN) layers with connections to existing layers.
result Demonstrated improved performance on radar clutter classification, node classification, and action recognition.
We combine Riemannian geometry with the mean field theory of high dimensional chaos to study the nature of signal propagation in generic, deep neural networks with random weights. Our results reveal an order-to-chaos expressivity phase transition, with networks in the chaotic phase computing nonlinear functions whose g…
FiberNet integrates geometry into machine learning for clearer classification.
problem Lack of interpretability in traditional deep learning.
method Reformulates classification as geometric optimization on fiber bundles, introducing learnable Riemannian metrics and variational prototype optimization.
result Clear geometric interpretability and efficiency in classification.
New approximative kernels improve PDE-G-CNNs for geometric deep learning.
problem Inaccurate approximations of exact kernels in PDE-G-CNNs.
method Developed new approximative kernels that work regardless of spatial anisotropy.
result New kernels provide better error estimates and maintain reflectional symmetries.
This paper extends geometric study of neural networks to non-differentiable layers and random walks.
problem Understanding the geometric properties of neural networks, especially those with non-differentiable activation functions.
method Singular Riemannian geometry approach to convolutional, residual, and recursive neural networks.
result Illustrated geometric findings with numerical experiments on image classification and thermodynamic problems.
Develops a flexible deep autoencoding topic model with scalable hybrid Bayesian inference.
problem Flexible and interpretable document analysis models.
method DATM with hybrid Bayesian inference, including topic-layer-adaptive stochastic gradient Riemannian MCMC and Weibull variational encoder.
result Demonstrates scalability and efficacy on big corpora in unsupervised and supervised learning tasks.
Differentiates Fréchet mean for hyperbolic space applications.
problem Difficulty in applying Fréchet mean due to lack of closed-form derivative.
method Developed differentiation method and explicit gradient expressions for hyperbolic space.
result Fully integrated Fréchet mean into hyperbolic neural network pipeline.
New SPD metrics improve stability and efficiency in neural networks.
problem Designing stable and efficient Riemannian metrics on SPD manifolds.
method Cholesky decomposition to derive SPD metrics.
result Proposed metrics provide closed-form operators, computational efficiency, and improved numerical stability.
Real world data often exhibit low-dimensional geometric structures, and can be viewed as samples near a low-dimensional manifold. This paper studies nonparametric regression of Hölder functions on low-dimensional manifolds using deep ReLU networks. Suppose training data are sampled from a Hölder function in $\mathc…
Gradient descent with geometrically adapted metrics drives cost to global minimum at uniform rate.
problem Minimizing cost in deep learning networks.
method Adapting gradient descent to output layer metric in deep learning.
result Uniform exponential convergence to global minimum in cost.
Neural collapse occurs in normalized features over a Riemannian manifold.
problem Understanding neural collapse in normalized feature models.
method Simplified multi-class classification task to a nonconvex optimization problem over the Riemannian manifold, analyzing the landscape of critical points.
result The only global minimizers are neural collapse solutions, with all other critical points being strict saddles.
Deep Latent Dirichlet Allocation with Topic-Layer-Adaptive Stochastic Gradient Riemannian MCMCstat.ML
It is challenging to develop stochastic gradient based scalable inference for deep discrete latent variable models (LVMs), due to the difficulties in not only computing the gradients, but also adapting the step sizes to different latent factors and hidden layers. For the Poisson gamma belief network (PGBN), a recently …
A new metric mav offers a practical alternative to costly Riemannian distance.
problem Efficiently compute Riemannian distance on SE(3) invariant metrics.
method Propose mav distance, defined as Riemannian length of a curve.
result Mav distance offers a trainable invariant for geometric deep learning.
Survey of spectral, probabilistic, and deep metric learning methods.
problem Developing effective distance metrics for various machine learning tasks.
method Divided into spectral, probabilistic, and deep approaches, covering various techniques and their applications.
result Comprehensive overview of metric learning methods, including new developments and applications.
Training shapes the geometry of neural network feature maps, revealing local area magnification.
problem Understanding how training affects the geometric structure of neural network feature maps.
method Analyzing the Riemannian geometry induced by neural network feature maps at infinite width and after training.
result Training breaks the symmetry of the geometry induced by random neural network feature maps, magnifying local areas along decision boundaries.
Deep learning methods are reviewed for preserving structure in neural networks.
problem Challenges in applying deep learning, especially in preserving structure.
method Review of existing deep learning methods and new algorithmic frameworks.
result Mathematical understanding and systematic design of deep learning methods to preserve structure.
New insights into how neural networks learn features, especially when they are very wide.
problem Understanding how gradient flow in wide neural networks selects solutions, especially in the feature-learning regime.
method Axiomatizing the canonical regularizer as a function-space energy and lift, and deriving geodesic ridge for the feature-learning regime.
result Gradient flow in feature-learning networks biases towards ridge regularization, distorting the inductive bias and damaging pretrained networks.
NeuroPMD estimates densities on complex product manifolds.
problem Density estimation on high-dimensional product manifolds.
method Neural network directly parameterizes density, trained with manifold differential operators.
result NeuroPMD outperforms traditional methods in density estimation.
Deep neural networks reveal a low-dimensional manifold structure in data.
problem Understanding the structure of data for better model performance.
method Model-centric analysis of the data manifold using the local data matrix and Fisher information matrix.
result The dataset lies on a data leaf with a dimension bounded by the number of labels.
Geometry-aware similarity metrics for neural representations on Riemannian and statistical manifoldscs.LG
MSA compares neural representations' intrinsic geometry for better understanding.
problem Existing similarity measures fail to capture subtle distinctions between neural network solutions.
method Metric similarity analysis (MSA) using Riemannian geometry.
result MSA can disentangle features of neural computations and compare nonlinear dynamics.
ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groupscs.LG
ChebLieNet uses Lie groups to create invariant spectral graph networks.
problem Handling anisotropic data in graph neural networks.
method Develops anisotropic convolutional layers on Lie groups with Riemannian metrics.
result Demonstrates the effectiveness of balancing equivariance and invariance.
Covariance matrices have attracted attention for machine learning applications due to their capacity to capture interesting structure in the data. The main challenge is that one needs to take into account the particular geometry of the Riemannian manifold of symmetric positive definite (SPD) matrices they belong to. In…
Proposes Geodesic Integrated Gradients (GIG) for more accurate feature attributions in deep networks.
problem Flawed attributions using straight paths from Integrated Gradients (IG).
method Introduces a model-induced Riemannian metric and computes attributions along geodesics.
result GIG produces more faithful attributions than IG on benchmarks.
New method uses Riemannian geometry to improve neural network adversarial attacks.
problem Improving robustness of neural networks against adversarial attacks.
method Proposes a new adversarial attack using Riemannian foliation theory and curvature of data space.
result The new attack is more efficient and accurate compared to existing methods.
This paper reconstructs equivalence classes in 1D neural networks.
problem Reconstructing equivalence classes in neural networks.
method Singular Riemannian geometry approach.
result Algorithm to build the set of points on the same equivalence class.
In a number of disciplines, the data (e.g., graphs, manifolds) to be analyzed are non-Euclidean in nature. Geometric deep learning corresponds to techniques that generalize deep neural network models to such non-Euclidean spaces. Several recent papers have shown how convolutional neural networks (CNNs) can be extended …
Proposes graph neural network layers for manifold-valued graphs.
problem Graphs with features in a Riemannian manifold.
method Diffusion layer and tangent multilayer perceptron.
result Outperforms state-of-the-art networks on Alzheimer's classification.
Unified geometric flows improve deep learning efficiency and simplify neural network topologies.
problem Improving deep learning performance and simplifying neural network structures.
method Proposes a thermodynamically coupled Ricci flow that dynamically adapts parameter space geometry to loss landscape topology, enabling automated singularity resolution and providing entanglement entropy bounds.
result Demonstrates 2.1× convergence acceleration and 63% topological simplification while maintaining complexity, outperforming Riemannian baselines by 15.2% in few-shot accuracy.
Why do deep neural networks (DNNs) benefit from very high dimensional parameter spaces? Their huge parameter complexities vs stunning performance in practice is all the more intriguing and not explainable using the standard theory of model selection for regular models. In this work, we propose a geometrically flavored …
Generative models can approximate high-dimensional data from lower dimensions without needing a latent dimension equal to or greater than the data's intrinsic dimension.
problem Theoretical limitations on the latent dimension required for generative models to approximate high-dimensional data distributions.
method Inspired by space-filling curves, the work demonstrates that generative networks can approximate distributions on -dimensional manifolds from inputs of any arbitrary dimension, even lower than .
result Generative models can approximate high-dimensional data distributions from lower-dimensional inputs without needing a latent dimension equal to or greater than the data's intrinsic dimension.
Simplified optimization for structured matrices in deep learning.
problem Computational challenges in Riemannian submanifold optimization for structured symmetric positive-definite matrices.
method Proposed a generalized Riemannian normal coordinates that dynamically orthonormalizes the metric and converts the problem into an unconstrained Euclidean space problem.
result Simplified existing approaches for structured covariances and developed matrix-inverse-free 2nd-order optimizers for deep learning with low precision.
New bounds for neural networks on curved manifolds improve generalization.
problem Existing generalization theories fail to account for non-Euclidean manifold structures.
method Derive covering number bounds incorporating manifold-specific properties like curvature.
result Sharp Rademacher complexity bounds for neural networks on compact manifolds.
The Normalizing Flow (NF) models a general probability density by estimating an invertible transformation applied on samples drawn from a known distribution. We introduce a new type of NF, called Deep Diffeomorphic Normalizing Flow (DDNF). A diffeomorphic flow is an invertible function where both the function and its i…
The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric…
Deep Gaussian processes on manifolds improve performance on complex data.
problem Complex data on manifolds that shallow models struggle with.
method Residual deep Gaussian processes on Riemannian manifolds.
result Significant improvement in prediction quality and uncertainty calibration.
Adapts IG for better feature attributions and robustness.
problem Reliability concerns in feature attributions for deep learning models.
method Adaptation of path-based feature attribution to Riemannian geometry of data manifolds.
result IG along geodesics generates more intuitive and robust explanations.
The article generalizes Pearson correlation to Riemannian manifolds.
problem Analyzing statistical models on non-linear manifolds.
method Reconstitutes Pearson correlation properties and derives a nonlinear generalization.
result Developed the Riemann-Pearson Correlation for manifold analysis.
Neural samplers such as variational autoencoders (VAEs) or generative adversarial networks (GANs) approximate distributions by transforming samples from a simple random source---the latent space---to samples from a more complex distribution represented by a dataset. While the manifold hypothesis implies that the densit…
On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operatorsmath.DG
The paper studies geometric properties of group equivariant operators and their Riemannian structure.
problem Understanding the geometric structure of group equivariant operators.
method Endowing the space of group equivariant non-expansive operators with a Riemannian manifold structure and using gradient descent methods.
result Gradient descent methods can be applied to minimize cost functions on the space of group equivariant non-expansive operators.