Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
Geometric Algebra Transformer (GATr) handles various geometric data types efficiently.
problem Lack of a single architecture for diverse geometric data types.
method GATr uses projective geometric algebra, equivariant to E(3), and is a Transformer architecture.
result GATr outperforms non-geometric and equivariant baselines in various geometric tasks.
CupNet prunes neural nets for cup-shaped data.
problem Pruning neural networks for cup-shaped data.
method Used simulated cup drawing data to prune a neural network.
result Pruning effectively reduces network size for cup-shaped data.
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.
A new geometric method for clustering SPD data improves upon Euclidean and Riemannian approaches.
problem Skewed interpretations of SPD data in Euclidean analysis and computational inefficiency of Riemannian methods.
method Proposes a geometric method based on the Thompson metric for unsupervised clustering of SPD data.
result Demonstrates improved clustering results using inductive midrange centroid computation.
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.
Machine learning methods struggle with geometric data, but shape space analysis provides a framework for studying and analyzing geometric variability.
problem Machine learning methods struggle with geometric data
method Shape space analysis provides a mathematical and computational framework
result Characterizes shape variability, compares geometric objects, and analyzes structural trajectories
Survey of de Casteljau's algorithm's applications in geometric data analysis.
problem No specific problem stated; focuses on algorithm applications.
method Constructive approach to generalize parametric smooth curves to manifolds.
result Algorithm provides principled way to analyze geometric data.
SRCA reduces high-dimensional data to lower dimensions while preserving geometric structures.
problem High-dimensional datasets with underlying geometric structures.
method Spherical Rotation Component Analysis (SRCA) incorporating geometric loss functions.
result SRCA provides a low-rank spherical representation of data with general theoretic guarantees.
Quantum dynamics reveals hidden geometric structure in data.
problem Understanding complex, high-dimensional datasets through geometric structure.
method Introducing semiclassical and microlocal analysis to data analysis.
result First tractable algorithm for approximating wave dynamics and geodesics on data manifolds.
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.
Several data analysis techniques employ similarity relationships between data points to uncover the intrinsic dimension and geometric structure of the underlying data-generating mechanism. In this paper we work under the model assumption that the data is made of random perturbations of feature vectors lying on a low-di…
Geometric Graph Alignment enhances IoT intrusion detection using NID data.
problem Data scarcity hinders IoT intrusion detection accuracy.
method Geometric Graph Alignment (GGA) approach to transfer knowledge between network intrusion detection and IoT intrusion detection domains.
result GGA approach boosts IoT intrusion detection performance on multiple datasets.
We construct transformations which take asymptotically AdS hyperbolic initial data into asymptotically flat initial data, and which preserve relevant physical quantities. This is used to derive geometric inequalities in the asymptotically AdS hyperbolic setting from counterparts in the asymptotically flat realm, whenev…
Normal-bundle bootstrap generates new data preserving geometric structure.
problem Probabilistic models often exhibit salient geometric structure.
method NBB method decomposes probability measure into manifold and normal spaces, estimates manifold as density ridge, and generates new data by bootstrapping projection vectors.
result NBB generates new data that preserves the geometric structure of a given data set.
CP2 uses geometric information to improve conformal prediction robustness.
problem CP fails under geometric data shifts, losing coverage guarantees.
method Integrates geometric pose information into CP via canonicalization.
result Integrating geometric information with CP ensures robustness under geometric shifts.
We develop a theory of parametrized geometric cobordism by introducing smooth Thom stacks. This requires identifying and constructing a smooth representative of the Thom functor acting on vector bundles equipped with extra geometric data, leading to a geometric refinement of the the Pontrjagin-Thom construction in stac…
Equivariant networks improve geometric prediction without scalar approximations.
problem Efficiently predicting geometric tensors in real-world scenarios.
method Equivariant networks for geometric prediction.
result Equivariant networks can generalize to unseen systems for geometric prediction.
A new method for generative modeling of discrete data using geometric latent subspaces.
problem Learning generative models for discrete data with statistical dependencies.
method Geometric latent-subspace framework in exponential parameter space of product manifolds of categorical distributions.
result Low-dimensional latent space encodes statistical dependencies and accurately models high-dimensional discrete data.
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. 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.
Quantum neural networks need both data-dependent and trainable unitaries for effective geometric deformation.
problem Quantum neural networks lack the geometric flexibility of classical networks due to limitations in state reachability.
method Viewing quantum states as embedded manifolds, we analyze infinitesimal unitary actions and introduce the CLA maps and aCLS criterion.
result Geometric flexibility in quantum neural networks requires a joint dependence on data and trainable weights.
Researchers geometrically define asymptotic coordinates in General Relativity.
problem Understanding the asymptotic behavior of relativistic initial data sets.
method Geometrization of asymptotic flatness and analysis of geometric invariants.
result Geometrically defined asymptotic coordinates for mass, energy, momentum, and angular momentum.
MCBP detects boundaries in high-dimensional data using curvature.
problem Boundary detection in high-dimensional data.
method MCBP uses mean curvature to model data manifold curvature.
result MCBP improves clustering performance in complex scenarios.
The method of geometric harmonics is adapted to the situation of incomplete data by means of the iterated geometric harmonics (IGH) scheme. The method is tested on natural and synthetic data sets with 50--500 data points and dimensionality of 400--10,000. Experiments suggest that the algorithm converges to a near optim…
New approach combines geometric and probabilistic methods to estimate manifold dimension in high-dimensional data.
problem Estimating the dimension of manifolds in high-dimensional data.
method Combines a modified box-counting algorithm (geometric) and a new probabilistic method (nearest neighbor distance analysis).
result The combined method is robust, fast, and effective in estimating manifold dimension.
We explore the generalization of scattering transforms from traditional (e.g., image or audio) signals to graph data, analogous to the generalization of ConvNets in geometric deep learning, and the utility of extracted graph features in graph data analysis. In particular, we focus on the capacity of these features to r…
Refines geometric center of mass analysis for Einstein field equations.
problem Analyzing the geometric center of mass of Willmore surfaces in initial data for Einstein field equations.
method Refined Lyapunov-Schmidt analysis to study geometric center of mass of area-constrained Willmore surfaces.
result The geometric center of mass agrees with the Hamiltonian center of mass under specific conditions.
L-GATr transforms high-energy physics data using geometric algebra and Lorentz symmetry.
problem Extracting scientific understanding from particle-physics experiments with high precision and efficiency.
method L-GATr, a geometric algebra Transformer, representing data in 4D space-time and being equivariant under Lorentz transformations.
result L-GATr achieves performance comparable to or better than domain-specific baselines on regression, classification, and generative tasks.
GIBLy adds geometric priors to 3D segmentation models, improving performance with minimal overhead.
problem Lack of explicit geometric information in 3D semantic segmentation models.
method Introduces GIBLy, a lightweight geometric inductive bias layer that integrates learnable geometric priors into existing 3D segmentation pipelines.
result Consistent performance gains across multiple benchmarks, including up to +11.5% mIoU on TS40K with PTV3.
This paper introduces online algorithms to estimate robust geometric median in large data streams.
problem Detecting outliers in large data sets using robust statistical measures.
method Online stochastic Newton methods for estimating the geometric median.
result Rates of convergence for online estimation of the geometric median.
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.
Geometric deep learning predicts knot invariants.
problem Predicting knot invariants from knot data.
method Constructing a functor from knots to graphs and using graph neural networks.
result High generalization capabilities demonstrated.
We present a deformable generator model to disentangle the appearance and geometric information for both image and video data in a purely unsupervised manner. The appearance generator network models the information related to appearance, including color, illumination, identity or category, while the geometric generator…
Paper defines and proves geometric uniqueness of Einstein field equations.
problem Einstein field equations characteristic Cauchy problem
method Covariant definition of double null data, proving geometric uniqueness
result Double null data fully covariant and geometrically unique
We investigate learning of the differential geometric structure of a data manifold embedded in a high-dimensional Euclidean space. We first analyze kernel-based algorithms and show that under the usual regularizations, non-probabilistic methods cannot recover the differential geometric structure, but instead find mostl…
In this paper we investigate the life-span of classical solutions to the hyperbolic geometric flow in two space variables with slow decay initial data. By establishing some new estimates on the solutions of linear wave equations in two space variables, we give a lower bound of the life-span of classical solutions to th…
We introduce PyTorch Geometric, a library for deep learning on irregularly structured input data such as graphs, point clouds and manifolds, built upon PyTorch. In addition to general graph data structures and processing methods, it contains a variety of recently published methods from the domains of relational learnin…
The paper geometrically characterizes graded manifolds and proves the Frobenius theorem.
problem Understanding and characterizing graded manifolds.
method Geometric characterization and Frobenius theorem proof.
result Frobenius theorem proven for graded distributions.
Geometric approach improves functional outlier detection.
problem Detecting outliers in functional data sets.
method Developed a geometric perspective on functional manifold for outlier detection.
result Improved understanding and differentiation of outliers.
Generative diffusion models gradually memorize training data, losing independent dimensions.
problem Understanding how generative diffusion models memorize training data, especially on low-dimensional manifolds.
method Measuring latent dimensionality via the learned score field, proposing a geometric memorization theory.
result Generative diffusion models experience a smooth collapse of their capacity to vary across independent directions as data become scarce, leading to near point-wise replication of salient features.
GCML preserves geometric structure in manifold clustering for diverse data types.
problem Loss functions in manifold clustering can corrupt latent space structure.
method GCML framework with isometric and ranking losses for geometric structure preservation.
result GCML outperforms other methods in latent space structure preservation and performance metrics.
With the renewed and growing interest in geometric continuity in mind, this article gives a general definition of geometrically continuous polygonal surfaces and geometrically continuous spline functions on them. Polynomial splines defined by G1 gluing data in terms of rational functions are analyzed further. A general…
New framework robustly handles outliers in Wasserstein DRO for better decision-making.
problem Non-geometric perturbations like adversarial outliers distort Wasserstein distance.
method Proposes an outlier-robust WDRO framework using a robust Wasserstein ball.
result Derives minimax optimal excess risk bounds for robust WDRO.
Smooth bundles with rough data maintain Hodge kernel isomorphism.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
Develops geometric causal models for causal inference from dependent data.
problem Causal inference from structured, dependent data (e.g., spatial, network, molecular).
method Geometric causal models (GCMs) exploiting symmetries of data generating process, combining group theory, ergodic theory, and Bayesian inference.
result Establishes identification and estimation of causal effects from dependent data.
The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of the success of convolutional neural networks (ConvNets) in image data analysis and other tasks. Inspired by recent interest in geometric deep learning, which aims to generalize ConvNets to manifold and gra…
Study describes limits of non-collapsing K3 surfaces using algebraic data.
problem Understanding limits of non-collapsing polarized K3 surfaces.
method Explicit description via period mapping and algebro-geometric data.
result Bubbling limits depend solely on algebro-geometric data.