Neural nets learn robust geometric data representations.
problem Ensuring neural networks are robust to adversarial attacks.
method Topological Data Analysis via persistence diagrams, Lipschitz stability.
result Certified ε-robustness on ORBIT5K dataset. GSAN learns adaptive node representations using geometric scattering and attention.
problem Oversmoothing in node representation learning.
method Attention-based architecture integrating geometric scattering and GCN channels.
result GSAN outperforms previous networks in semi-supervised node classification.
This paper analyzes Barlow Twins' representation efficiency using information-geometric methods.
problem Understanding and comparing the efficiency of self-supervised learning methods.
method Introduces an information-geometric framework to quantify representation efficiency and applies it to Barlow Twins.
result Proves that Barlow Twins achieves optimal representation efficiency (η=1).
New method learns disentangled representations using Gromov-Monge maps.
problem Learning disentangled representations from unlabelled data.
method Introduces a novel approach based on Gromov-Monge maps to preserve geometric features while aligning data distributions.
result Demonstrates effectiveness on four benchmarks, outperforming other methods.
Novel representer theorem for metric and preference learning in RKHSs.
problem Metric and preference learning problems in Hilbert spaces.
method Regularization with respect to task structure norm, RKHS representation, and novel algorithm.
result Significant performance improvement over baseline methods in real-world rank inference benchmarks.
GD-VAEs learn dynamics from observations using geometric and topological information.
problem Learning parsimonious representations of nonlinear dynamics from observations.
method Develops data-driven methods incorporating geometric and topological information using Variational Autoencoders (VAEs).
result GD-VAEs provide methods for learning reduced dimensional representations of nonlinear dynamics.
Geometric stability measures neural network robustness, distinguishing from similarity metrics.
problem Lack of robustness in neural network representations.
method Introduces geometric stability, quantified by Shesha metric measuring self-consistency.
result Stability and similarity are uncorrelated, revealing distinct properties of neural network robustness.
This paper reviews discrete curvature models for geometric data analysis.
problem Capturing intrinsic geometric structure in diverse data representations.
method Comprehensive review of discrete curvature models from Riemannian and metric geometry perspectives.
result Systematic pipeline for curvature-driven data analysis and learning.
Proposes a method to learn representations of higher-dimensional simplicial complexes.
problem Lack of methods for representing entire simplicial complexes.
method Geometric message passing schemes for end-to-end learning of simplicial complex representations.
result First method for learning representations of entire simplicial complexes.
GQML uses symmetries from representation theory to improve quantum machine learning.
problem Creating quantum models with symmetries to improve performance.
method Introduction to representation theory for quantum learning, focusing on group actions and symmetries.
result Effective implementation of GQML requires knowledge of group representation theory.
Proposes IIKL for preserving geometric properties of non-Euclidean data.
problem Loss of geometric information in non-Euclidean data representation.
method IIKL method builds Riemannian manifold and isometrically induces metric.
result Preserves geometric structure of original data in 3D and high-dimensional datasets.
Survey on geometric foundations of data reduction methods.
problem High-dimensional data with intrinsic nonlinear structure.
method Spectral manifold learning methods.
result Derivation and convergence analysis of spectral manifold learning.
New framework explains neural network behavior through geometric postulates.
problem Understanding neural network mechanisms and making them more transparent.
method Introducing the Pursuit of Subspaces (PoS) hypothesis as an axiomatic framework.
result Unified geometric perspective on neural network representation, computation, and generalization.
Neural networks can learn distance metrics affecting model performance.
problem Understanding how neural networks learn and represent data.
method Experiments with six MNIST architectures, constrained to learn either distance or intensity representations.
result Distance-based learning affects model performance, validating the geometric framework.
We show that the detection of geometric intersection in an arbitrary representation of the mapping class group of surface implies the injectivity of that representation up to center, and vice versa. As an application, we discuss the geometric intersection in the Johnson filtration. Also, we further consider the problem…
Deep learning models complex multivariate extremes using geometric shapes.
problem Modeling complex extremal dependencies in high-dimensional data.
method Geometric representation and deep learning for flexible semi-parametric models.
result First approach to modeling limit sets using deep learning for high-dimensional data.
Robot learns to manipulate objects using multiple geometric representations.
problem Manipulation tasks are poorly represented by Cartesian coordinates.
method Extends Gaussian distributions on Riemannian manifolds to analyze demonstrations, formulating the problem as an optimal control problem.
result Robot can generalize manipulation tasks using multiple geometric representations.
The volume of a credal set correlates with epistemic uncertainty in binary classification but not in multi-class.
problem Representing and quantifying epistemic uncertainty in machine learning.
method Examined the geometric representation of credal sets as d-dimensional polytopes and their volume as a measure of uncertainty. result The volume of a credal set is a meaningful measure of epistemic uncertainty in binary classification but not in multi-class.
Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity. We introduc…
Surveying Hitchin representations of Fuchsian groups.
problem Understanding representations of Fuchsian groups.
method Survey and conjectural geometric description.
result Conjectural geometric picture of an augmented Hitchin component.
We propose a method to learn object representations from 3D point clouds using bundles of geometrically interpretable hidden units, which we call geometric capsules. Each geometric capsule represents a visual entity, such as an object or a part, and consists of two components: a pose and a feature. The pose encodes whe…
Study quandle modules over geometric quandles and their relation to Lie-Yamaguti representations.
problem Understanding quandle modules and their connection to Lie-Yamaguti representations.
method Examine quandle modules over quandle spaces, focusing on geometric structures.
result Modules over quandle spaces are linked to representations of Lie-Yamaguti algebras.
Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.
problem Understanding geometrically finite Fuchsian groups and their representations.
method Theory of Anosov representations, type-preserving deformations, limit maps, relative Anosov and dominated representations.
result Cusped Hitchin representations are Borel Anosov, stable under deformations, and limit maps vary analytically.
This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.
problem Improving topological inference and visualization of large-scale geometric datasets.
method Proposes a method for learning topologically-faithful covers of geometric datasets using optimization.
result Simplicial complexes obtained from learned covers outperform standard methods in terms of size and representation of large-scale topology.
SVarM uses varifold representations for shape classification and regression.
problem Challenges in analyzing geometric data due to non-Euclidean shape spaces.
method Develops a neural network-based framework for varifold representations of shapes.
result Demonstrates strong performance and robustness in shape classification and regression.
Special orthogonal representations from octonions have geometric properties linked to binary cubics.
problem Understanding geometric properties of special orthogonal representations from octonions.
method Using octonions and their derivations, spinors, and covariants to show geometric properties.
result Covariants and Mathews identities of these representations are related to the Fano plane and (Z2)3. Proves EGF representations in specific geometric contexts.
problem Understanding representations of groups with hyperbolic properties.
method Analyzes projectively convex cocompact manifolds and convex projective manifolds with generalized cusps.
result Holonomy representations of specific geometric manifolds are EGF representations.
GAMLA learns manifold structures with auto-encoding for global insights.
problem Limited global insight and lack of interpretable analytical descriptions in manifold learning.
method Two-round auto-encoding process to derive character and complementary representations.
result GAMLA provides global and analytical descriptions of smooth manifolds.
New insights into Anosov representations of hyperbolic groups.
problem Understanding Anosov representations of relatively hyperbolic groups.
method Proving representations can be interpreted as restricted Anosov representations over flow spaces and showing stability under deformations.
result Representations of certain types are divergent, extended geometrically finite and stable under small deformations.
Kernel VICReg improves SSL in RKHS, capturing nonlinear structures.
problem Limited ability of existing SSL methods to handle nonlinear dependencies.
method Kernel VICReg framework in RKHS, kernelizing VICReg objectives.
result Kernel VICReg mitigates representational collapse and improves performance.
New method uses random convex polytopes to measure representation quality.
problem Measuring the quality of deep learning representations.
method Random Polytope Descriptor method based on random convex polytopes.
result Regularization in autoencoders can degrade out-of-distribution detection.
Geometric framework detects concept frustration between human concepts and machine representations.
problem Aligning human concepts with machine learning representations.
method Geometric framework and similarity measures for detecting concept frustration.
result Concept frustration affects machine learning model performance and reorganizes learned concept representations.
We examine geometric representability results for various classes of equiaffine curvature operators. We show every Ricci flat algebraic curvature operator is geometrically realizable by a Ricci flat torsion free connection on the tangent bundle of some smooth manifold.
New representation theory for closed geodesic subflows.
problem Classifying representations with good geometric properties.
method Restricting to invariant closed geodesic subflows.
result Equivalent characterizations and properties of new representations.
ICLR 2021 challenge in computational geometry and topology attracted 16 teams.
problem Designing and evaluating computational methods in differential geometry and topology.
method Designing and hosting an open-source competition with repositories Geomstats and Giotto-TDA.
result 16 teams participated in the challenge, showcasing innovative contributions to computational geometry and topology.
GNPs learn operators on non-Euclidean geometries using neural networks.
problem Learning operators on complex geometries like manifolds.
method Geometric Neural Operators (GNPs) that incorporate geometric properties.
result GNPs can estimate metrics, solve PDEs, and learn LB operators on manifolds.
The paper introduces metrics for robust unsupervised learning of vehicle interactions.
problem Robust representation learning of temporal dynamic interactions in robotics.
method Geometric approach using Procrustes distance and optimal transport for comparing interaction distributions.
result Metrics for assessing stability and comparing interaction learning algorithms.
Proof shows volumes of certain geometric representations are always integers.
problem Integrality of volumes of specific geometric representations.
method Elementary, combinatorial-geometrical proof.
result Volumes of representations are integers when n≥2. Defines new representations for hyperbolic groups, unifying existing definitions.
problem Geometrically finite behavior in higher rank groups.
method Introduces a new family of discrete representations for relatively hyperbolic groups.
result Stability of these representations under certain deformations.
We propose a new perspective on representation learning in reinforcement learning based on geometric properties of the space of value functions. We leverage this perspective to provide formal evidence regarding the usefulness of value functions as auxiliary tasks. Our formulation considers adapting the representation t…
New construction provides non-trivial representations for geometric quantisation.
problem Geometric quantisation of non-integral symplectic structures.
method Construction from Noncommutative Differential Geometry adapted to diffeology.
result The construction provides non-trivial representations.
Foundation models fail to preserve continuous geometry, identified as the Geometric Alignment Tax.
problem Continuous geometry is lost in foundation models due to discrete categorical bottlenecks.
method Controlled ablations on synthetic systems and evaluation of 14 biological models using rate-distortion theory and MINE.
result Replacing cross-entropy with a continuous head reduces geometric distortion by up to 8.5x.
Interventional data helps identify latent factors without distributional assumptions.
problem Identifying latent factors from interventional data without distributional assumptions.
method Leveraging geometric signatures of latent factors' support from interventional data.
result Latent causal factors can be identified up to permutation and scaling given data from perfect do-interventions.
Enhances deep neural networks for MRI reconstruction by increasing expressivity.
problem Balancing network complexity and performance in deep learning MRI reconstruction.
method Geometric approach using bootstrapping and subnetwork aggregation with attention module.
result Significant improvement in MRI reconstruction performance with minimal complexity increase.
A new geometric metric identifies true data changes from parametrization artifacts in high-dimensional representations.
problem Quantifying representation drift in high-dimensional data using Euclidean or cosine distances can misattribute changes due to arbitrary parametrizations.
method Introducing the Fubini Study metric to identify representations that differ only by gauge transformations.
result The Fubini Study metric isolates intrinsic evolution by remaining invariant under gauge-induced fluctuations, providing a diagnostic for meaningful structural changes.
We describe recent links between two topics: geometric structures on manifolds in the sense of Ehresmann and Thurston, and dynamics "at infinity" for representations of discrete groups into Lie groups.
Drug repositioning is an attractive cost-efficient strategy for the development of treatments for human diseases. Here, we propose an interpretable model that learns disease self-representations for drug repositioning. Our self-representation model represents each disease as a linear combination of a few other diseases…
Develops VAEs for learning complex physical systems from data.
problem Learning low-dimensional representations of nonlinear physical systems.
method Variational Autoencoders with manifold latent spaces.
result Effective in learning nonlinear Burgers equation and constrained mechanical systems.