This paper derives radial fields on manifolds of symmetric positive definite matrices.
problem Lack of an expression for radial fields on manifolds of symmetric positive definite matrices.
method Derives an expression for radial fields on manifolds of symmetric positive definite matrices.
result Derives an expression for radial fields on manifolds of symmetric positive definite matrices.
New method classifies manifold-valued data using Riemannian geometry.
problem Classifying data on curved Riemannian manifolds.
method Probabilistic Learning Vector Quantization on Symmetric Positive Definite Matrices.
result The method outperforms traditional Euclidean methods on manifold-valued data.
Paper proposes a novel R-JDRDL method for SPD manifolds.
problem High-dimensional noisy signals analysis.
method Riemannian optimization framework for joint DR and DL.
result R-JDRDL outperforms existing algorithms in image classification.
Paper introduces a new distance measure for Gaussian Mixture Models.
problem Developing a new distance measure for Gaussian Mixture Models.
method Embedding K-component Gaussian Mixture Models into the manifold of symmetric positive definite matrices and calculating a lower bound for the Fisher-Rao metric.
result Demonstrated effectiveness through experiments on standard datasets.
Characterizes real left symmetric algebras with positive definite Koszul form and related Kähler-Einstein structures.
problem Characterizing real left symmetric algebras with positive definite Koszul form.
method Analyzes the properties of left multiplication operators and symmetric bilinear forms.
result Provides a complete characterization of real left symmetric algebras with positive definite Koszul form.
This work improves understanding of symmetrizing Bregman divergences on positive definite matrices.
problem Understanding which mean to use for symmetrizing Bregman divergences on positive definite matrices.
method Axiomatic definition of mean functionals and variational principles over the cone of positive definite matrices.
result The arithmetic mean is canonical for forward symmetrization, and the arithmetic, log-Euclidean, and harmonic means for reverse symmetrization.
Improved method for computing Fréchet means on SPD matrices.
problem Computing Fréchet means on the manifold of SPD matrices.
method Random matrix theory-based approach for estimating Fréchet means.
result Significantly outperforms state-of-the-art methods in experiments.
Knots and 4-manifolds linked via matrix kinking.
problem Understanding equivalence of symmetric matrices and their implications.
method Isotopy and kinking moves on Goeritz matrices.
result Every nonsingular symmetric integer matrix is kink-equivalent to positive or negative-definite matrices.
We study the differential geometric properties of the manifold of non-singular symmetric real matrices endowed with the trace metric; in case of positive definite matrices we describe the full group of isometries
Researchers found a way to measure energy in black hole perturbations.
problem Lack of positive-definite and conserved energy in black hole stability.
method Dimensional reduction and construction of a positive-definite energy functional.
result Conserved Hamiltonian energy for axially symmetric perturbations of Kerr black holes.
New balanced metrics introduced for SPD matrices, improving metric choice.
problem Lack of principles for choosing SPD matrix metrics.
method Introducing balanced metrics that relate existing metrics.
result Two new balanced metric families introduced: mixed-power-Euclidean and mixed-power-affine.
The study extends kernel universality to Riemannian symmetric spaces.
problem Understanding kernel universality in non-Euclidean domains.
method Harmonic analysis on Riemannian symmetric spaces.
result Proves universality of recent kernels on Riemannian symmetric spaces.
Proves inequality for tensor fields on curved spaces.
problem Generalizing inequality for tensor fields on curved spaces.
method Alexandrov-Bakelman-Pucci (ABP) method
result Proves Michael-Simon-Sobolev inequality for tensor fields.
Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
problem Analyzing locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
method Examines Riemannian and Finslerian surfaces, providing necessary and sufficient conditions for locally symmetric fourth root metrics in 2D and more complex conditions for higher dimensions.
result Formulates conditions for positive definiteness of locally symmetric polynomial metrics in Finslerian surfaces and provides explicit examples.
Researchers construct a gauge-invariant energy functional for axially symmetric perturbations around Kerr black holes.
problem Understanding energy of axially symmetric perturbations around Kerr black holes.
method Hamiltonian dimensional reduction to a 2+1 Einstein-wave map system, constructing a positive-definite, gauge-invariant energy functional. result The energy functional serves as a Hamiltonian for the constrained evolution of linear perturbations.
The Gaussian kernel is never positive-definite on Riemannian symmetric spaces.
problem Proving the non-positive-definiteness of the Gaussian kernel on non-Euclidean symmetric spaces.
method Developed new geometric and analytical arguments to rigorously characterize the positive-definiteness of the Gaussian kernel.
result Lp-Godement theorems provide necessary and sufficient conditions for positive-definiteness. Paper proves MS convergence for radially symmetric kernels with large bandwidths.
problem Proving convergence of mean shift algorithm with radially symmetric kernels.
method Analyzes convergence of mean shift algorithm with radially symmetric, positive definite kernels.
result Guaranteed convergence for sufficiently large bandwidth in any dimension.
New Sliced-Wasserstein distances for non-Euclidean data.
problem Computational burden of Wasserstein distance on non-Euclidean manifolds.
method Derive Sliced-Wasserstein distances and flows on Cartan-Hadamard manifolds.
result General constructions and non-parametric schemes for minimizing new distances.
Two criteria for a closed connected definite 4-manifold with infinite cyclic fundamental group to be TOP-split are given. One criterion extends a sufficient condition made in a previous paper. The result is equivalent to a purely algebraic result on the question asking when a positive definite Hermitian form over the r…
Introduces matrix MLP for learning symmetric positive definite matrices.
problem Learning structured parameters like symmetric positive definite matrices.
method Develops matrix multilayer perceptron (matrix MLP) for structured parameter learning.
result Extends variational autoencoder (VAE) for dense covariance matrices.
Paper proposes a new covariance estimator ensuring positive semi-definite matrices.
problem Estimating spot covariance matrices while maintaining positive semi-definiteness.
method Modification of the Fourier covariance estimator with a symmetric positive semi-definite constraint.
result The estimator is consistent and produces accurate positive semi-definite matrices.
Extends differential privacy to Riemannian manifolds, improving utility.
problem Releasing private statistical summaries on Riemannian manifolds.
method Extended Laplace or K-norm mechanism using intrinsic distances and volumes.
result Demonstrates rate optimality and utility improvement over ambient spaces.
Estimates Laplace eigenvalues and diameter for Lie group metrics.
problem Estimating Laplace eigenvalues and diameter for left-invariant metrics on compact Lie groups.
method Relates left-invariant metrics to positive definite matrices and uses eigenvalue properties.
result Partial answers to Eldredge's conjecture on Laplace eigenvalues and diameter.
New Riemannian metric for SPD matrices avoids swelling effect.
problem Efficiency and stability in computing with SPD matrices.
method Log-Cholesky decomposition and Lie group structure.
result Log-Cholesky average maintains determinant bounds.
Study of strictly accretive matrices using Finsler geometry.
problem Characterize the set of strictly accretive matrices.
method Introduced Finsler metrics and characterized geodesics and distance.
result Geodesic distance applied to matrix approximation problem.
Proposes a normalization technique for manifold valued data.
problem Instability in optimization for manifold valued data.
method Develops a general normalization technique for manifold valued data.
result Demonstrates performance gain in synthetic and real datasets.
Recent advances suggest that a wide range of computer vision problems can be addressed more appropriately by considering non-Euclidean geometry. This paper tackles the problem of sparse coding and dictionary learning in the space of symmetric positive definite matrices, which form a Riemannian manifold. With the aid of…
As is well known, a metric on a manifold determines a unique symmetric connection for which the metric is parallel: the Levi-Civita connection. In this paper we investigate the inverse problem: to what extent is the metric of a Riemannian manifold determined by its Levi-Civita connection? It is shown that for a generic…
Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.
problem Establishing inequalities for tensor fields on curved manifolds.
method Applying the ABP method to symmetric tensor fields on manifolds with nonnegative sectional curvature.
result Log Sobolev and Michael Simon inequalities for tensor fields.
New method for symmetric matrix completion using ReLU sampling.
problem Symmetric positive semi-definite low-rank matrix completion with deterministic entry-dependent sampling.
method ReLU sampling, gradient descent with tailored initialization.
result Gradient descent with tailored initialization achieves global minima.
This work presents a parametrized family of divergences, namely Alpha-Beta Log- Determinant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Alpha-Beta Log-Determinant divergences between symmetric, positive definite matrices to the infi…
New geometric structures defined on SPD matrices for better understanding.
problem Understanding SPD matrices and their geometric properties.
method Introducing Finslerian and dual information-geometric structures on James' bicone domain.
result Geodesics correspond to straight lines in coordinate systems, and new dissimilarities generalize existing ones.
New algorithm accelerates optimization on Riemannian manifolds, including Wasserstein space.
problem Accelerating optimization methods in Riemannian geometry.
method Dynamic stepsize algorithms on Riemannian manifolds with specific vector transport.
result First provable accelerated gradient method in Wasserstein space.
This research simplifies Riemannian LBFGS for SPD manifolds.
problem Optimization on Riemannian manifolds, especially SPD.
method Two mappings for tangent space, making vector transports and adjoint vector transports identity.
result RLBFGS becomes less computationally expensive and easier to analyze.
Mathematical foundation for phylogenetic tree uncertainty quantification.
problem Uncertainty in evolutionary relationships between species.
method Introducing the Wald space as a subset of symmetric positive definite matrices, studying its topology and structure, and proposing a new numerical method for geodesics and curvature.
result Wald space has a topology of disjoint open cubes, is contractible, and is a Whitney stratified space of type (A).
Paper introduces a generalized Bures-Wasserstein geometry for SPD matrices.
problem Understanding the geometry of SPD matrices for machine learning.
method Proposes a generalized Bures-Wasserstein geometry parameterized by a symmetric positive definite matrix.
result The GBW geometry outperforms the BW geometry in machine learning applications.
On a Hermitian manifold we construct a symmetric (1,1)- tensor H using the torsion and the curvature of the Chern connection. On a compact balanced Hermitian manifold we find necessary and sufficient conditions in terms of the tensor H for a harmonic 1-form to be analytic and for an analytic 1-form to be harm…
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.
Geometric analysis of normal distributions using Fisher and Killing metrics.
problem Quantifying the difference between Fisher and Killing metrics on the space of normal distributions.
method Riemannian geometry, Fisher information metric, Killing metric, asymptotic geodesics.
result Approximation of Fisher metric by Killing metric for long distances is justified.
The present paper proposes generalized Gaussian kernel adaptive filtering, where the kernel parameters are adaptive and data-driven. The Gaussian kernel is parametrized by a center vector and a symmetric positive definite (SPD) precision matrix, which is regarded as a generalization of the scalar width parameter. These…
New k-means method clusters radar image sequences using SPD matrices.
problem Clustering radar image sequences efficiently.
method Developed k-means on SPD matrices for non-Euclidean data. result Effective clustering of radar image sequences via SPD matrices.
Efficiently clusters data on manifolds using Fréchet maps.
problem Clustering on high-dimensional, non-Euclidean manifolds is computationally challenging.
method Introduces p-Fréchet map to embed manifold data into Euclidean space for k-means clustering. result Significant performance gains in runtime and accuracy compared to existing methods.
Proposes a new Sliced-Wasserstein distance for covariance matrices in M/EEG signals.
problem Efficiently dealing with distributions of covariance matrices in M/EEG multivariate time series.
method Defines a Sliced-Wasserstein distance for symmetric positive definite matrices and applies it to brain-age prediction and Brain Computer Interface applications.
result Demonstrates computational efficiency and strong theoretical guarantees for the proposed distance.
Recent advances suggest that encoding images through Symmetric Positive Definite (SPD) matrices and then interpreting such matrices as points on Riemannian manifolds can lead to increased classification performance. Taking into account manifold geometry is typically done via (1) embedding the manifolds in tangent space…
Following Cao-Hamilton-Ilmanen, in this paper we study the linear stability of Perelman's ν-entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with resp…
We introduce a wrapped Gaussian for SPD matrices, enhancing data analysis.
problem Handling circular and non-flat data distributions on SPD manifolds.
method Introduced a non-isotropic wrapped Gaussian using the exponential map, derived theoretical properties, and proposed a maximum likelihood framework.
result Demonstrated the robustness and flexibility of the wrapped Gaussian model on synthetic and real-world datasets.
This work proves Kerr black holes are dynamically stable under certain perturbations.
problem Dynamical stability of Kerr black holes under axially symmetric perturbations.
method Dimensional reduction to 2+1 Einstein-wave map system, construction of positive-definite energy functional, proving boundary terms vanish.
result Strictly conserved positive energy for axially symmetric linear perturbations of Kerr black holes.
Study proves Kato inequality leads to definite 4-manifolds with positive curvature.
problem Positive sectional curvature and Kato type inequalities on 4-manifolds.
method Analyzes Kato type inequalities on compact Riemannian 4-manifolds with positive sectional curvature.
result Proves compact Riemannian 4-manifolds satisfying a Kato type inequality are definite.