Researchers develop geodesics for a new metric on correlation matrices.
problem Lack of intrinsic tools for statistical analyses of correlation matrices.
method Developed geodesics for the quotient-affine metric on full-rank correlation matrices.
result Provided fundamental Riemannian operations for the quotient-affine metric.
Investigates O(n)-invariant metrics on SPD matrices, extending kernel metrics.
problem Limited coverage of O(n)-invariant metrics by kernel metrics.
method Characterization of O(n)-invariant metrics, intermediate classes construction.
result Introduction of cometric-stability as a key property for geodesics.
New metrics defined for full-rank correlation matrices, ensuring unique operations.
problem No suitable problem statement as the abstract does not describe a problem to be solved.
method New Riemannian metrics defined on full-rank correlation matrices, providing unique operations.
result Unique Riemannian logarithm and Fréchet mean defined for full-rank correlation matrices.
We present a new Riemannian metric, termed Log-Cholesky metric, on the manifold of symmetric positive definite (SPD) matrices via Cholesky decomposition. We first construct a Lie group structure and a bi-invariant metric on Cholesky space, the collection of lower triangular matrices whose diagonal elements are all posi…
Researchers construct explicit bundles for ALF metrics, revealing rational patching matrices for gravitational instantons.
problem Constructing explicit toric Ricci-flat metrics and their associated bundles.
method Explicit construction of patching matrices for ALF metrics and gravitational instantons.
result Rational form of patching matrices for gravitational instantons in the Chen--Teo family.
We introduce an universum of the Polish (=complete separable metric) space - the convex cone of distance matrices and study its geometry. It happened that the generic Polish spaces in this sense of this universum is so called Urysohn spaces defined by P.S.Urysohn in 20-th, and generic metric triple (= metric space with…
The paper explores totally geodesic submanifolds in SPD matrices and their properties.
problem Characterizing and understanding totally geodesic submanifolds in SPD matrices.
method Detailed geometric analysis and projection properties of SPD matrices.
result A non-linear projection on totally geodesic submanifolds has the minimizing property.
We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.
The paper presents two schemes for sampling matrices from specific distributions on a manifold.
problem Sampling matrices from Gibbs distributions on the manifold of positive semi-definite matrices with fixed rank.
method Two explicit schemes based on Euler-Maruyama discretization of the Riemannian Langevin equation with Brownian motion on the manifold.
result Numerical validation of the schemes using specific energy functions and metrics.
The paper explores Cholesky decompositions for symmetric matrices and their geometric properties.
problem Understanding the structure and properties of symmetric matrices through Cholesky decompositions.
method Introducing cones of symmetric matrices, proving Cholesky-type factorizations, and showing geometric properties.
result Each symmetric matrix admits an uncountable family of Cholesky-type factorizations, and these cones are isometric Riemannian manifolds.
Develops log-Euclidean Lie groups for SPD and correlation matrices.
problem Unifies various log-Euclidean constructions for SPD and correlation matrices.
method Theory and explicit isometries linking different log-Euclidean metrics.
result Explicit log-Euclidean metrics on SPD and correlation matrices.
New metrics defined on SPD matrices link to divergences and curvature.
problem Defining and characterizing metrics on SPD matrices.
method Developed a principle of deformed metrics and introduced balanced bilinear forms.
result Introduce Mixed-Euclidean metrics with negative sectional curvature.
Inverse metric matrices on Siegel-Jacobi spaces are calculated for Berezin quantization.
problem Calculating inverse metric matrices on Siegel-Jacobi spaces.
method Inversion of metric matrices on XnJ and ildeXnJ. result Explicit calculations of inverse metric matrices for n=2. WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particulary universal) space to the properties of distance …
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
New metric tensor field on symmetric matrices simplifies eigenvector computation.
problem Complex eigenvector computation for 2x2 symmetric matrices.
method Introducing a metric tensor field on the space of symmetric matrices, resulting in a curved manifold.
result Parallel transport simplifies eigenvector computation for one-parameter families of matrices.
Defines cross product for m vectors in n-dimensional spaces.
problem No universal definition for cross product in high-dimensional spaces.
method Defines cross product for m vectors in n-dimensional spaces with any metric matrices.
result Cross product length represents m-dimensional volume, components represent volume directions.
Study linear perturbations of Spin(7) metrics, finding only rank one nilpotent matrices.
problem Linear perturbations of Spin(7) metrics.
method Applying the method of linear perturbations to Spin(7)-structures.
result Only rank one nilpotent matrices determine nontrivial perturbations.
New methods link Calabi-Yau metrics to random matrices.
problem Lack of explicit metrics on Calabi-Yau manifolds hinders particle physics computations.
method Numerical approximations of the Laplacian spectrum on Calabi-Yau spaces.
result Surprising link found between Calabi-Yau metrics and random matrix theory.
This paper studies geodesics between covariance matrices of different ranks using the Bures-Wasserstein metric.
problem Geodesics between covariance matrices of varying ranks.
method Analyzes the Bures-Wasserstein distance on covariance matrices, completing previous work on geodesics and providing explicit formulas.
result The set of all minimizing geodesics between two covariance matrices is parametrized by a closed unit ball in R(k−r)imes(l−r). We consider three different approaches to define natural Riemannian metrics on polytopes of stochastic matrices. First, we define a natural class of stochastic maps between these polytopes and give a metric characterization of Chentsov type in terms of invariance with respect to these maps. Second, we consider the Fish…
The paper solves PDEs from matrices with orthogonal columns, linking them to Hessian metrics and symmetric spaces.
problem Solving third order PDEs for strictly convex smooth functions.
method Geometric methods using Hessian metrics and symmetric spaces.
result Explicit solutions and a family of non-generic solutions with applications in Poisson geometry and Kahler structures.
Study relates Finsler structures to Clifford bundles for flat metrics.
problem Relating Finsler structures to Clifford bundles for flat metrics.
method Examines extensions of Clifford bundles and Finsler type structures for flat metrics.
result Triangle map exists between Finsler structures constructed from metrics and 1-forms.
We propose a fast general projection-free metric learning framework, where the minimization objective minM∈SQ(M) is a convex differentiable function of the metric matrix M, and M resides in the set S of generalized graph Laplacian matrices for con…
The space of matrices of positive determinant GL^+_n inherits an extrinsic metric space structure from R^{n^2}. On the other hand, taking the infimum of the lengths of all paths connecting two points in GL^+_n gives an intrinsic metric. We prove bilipschitz equivalence for intrinsic and extrinsic metrics on GL^+_n, exp…
We study the differential-geometric properties of the loci of fixed points of the elliptic isometries of the manifold of definite positive real matrices with the trace metric. We also give an explicit description of such loci and in particular we find their De Rham decomposition.
We investigate the connections between the differential-geometric properties of the exponential map from the space of real skew symmetric matrices onto the group of real special orthogonal matrices and the manifold of real orthogonal matrices equipped with the Riemannian structure induced by the Frobenius metric.
Study of metrics on positive-definite matrices from power potential, linking to power means.
problem Understanding metrics on positive-definite matrices derived from power potential.
method Explicit expressions for geodesics and distance function derived from Hessian of power potential.
result Geodesics and distance function converge to weighted matrix geometric mean as β tends to zero.
DiffeoCFM efficiently generates realistic brain connectivity matrices using pullback metrics.
problem Generating realistic brain connectivity matrices for population heterogeneity analysis.
method Conditional flow matching on matrix manifolds via pullback metrics induced by global diffeomorphisms.
result DiffeoCFM achieves state-of-the-art performance on large-scale fMRI and EEG datasets.
Researchers found sub-Lorentzian geodesics on a specific Lie subgroup.
problem Finding geodesics on a specific Lie subgroup with a sub-Lorentzian metric.
method Formulated a time-anti-optimal control problem, applied Pontryagin's minimum principle, and used geodesics and shortest arcs of a sub-Riemannian metric.
result Discovered sub-Lorentzian nonspacelike geodesics and longest arcs.
Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle …
The curvature tensor and the scalar curvature are computed in the space of positive definite real matrices endowed by the Kubo-Mori inner product as a Riemannian metric.
Study evaluates thresholds for removing noise from DNN weights using random matrix theory.
problem Removing noise from deep neural network weights for better approximation.
method Model weights as signal + noise, use random matrix theory to estimate thresholds, evaluate using cosine similarity.
result Proposed threshold estimation method improves approximation quality.
The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Here we d…
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.
Symmetric Positive Definite (SPD) matrices have been widely used in medical data analysis and a number of different Riemannian met-rics were proposed to compute with them. However, there are very few methodological principles guiding the choice of one particular metric for a given application. Invariance under the acti…
It is the aim of this article to determine curvature quantities of an arbitrary Riemannian monotone metric on the space of positive matrices resp. nonsingular density matrices. Special interest is focused on the scalar curvature due to its expected quantum statistical meaning. The scalar curvature is explained in more …
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.
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.
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.
Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability distributions, namely distributions whose densities have elliptical level sets. We endow thes…
Extends metrics for SPD matrices to infinite dimensions.
problem Lack of generalized forms for Riemannian metrics.
method Unitized Hilbert-Schmidt operators and extended Mahalanobis norm.
result Improved performance in high-dimensional comparisons.
The abstract introduces a new concept called flagfolds to model multi-dimensional shapes.
problem Modeling multi-dimensional shapes in a way that avoids going through higher dimensional spaces.
method Interpreting covariance matrices as nested subspaces and defining a Riemannian metric on the highest dimensional stratum.
result A Riemannian metric on the highest dimensional stratum allows for geodesics between subspaces of different dimensions.
This work tackles regression on non-Euclidean spaces, specifically positive-definite matrices with the Bures-Wasserstein metric.
problem Regression on non-Euclidean spaces, specifically positive-definite matrices with the Bures-Wasserstein metric.
method Developed a sufficient condition for the existence of a minimizer of the conditional barycenter problem, characterized the optimization landscape, and developed a projection-free algorithm for approximate computation of first-order stationary points.
result The objective is free of local maxima under the sufficient condition, and the algorithm enables the use of stochastic Riemannian optimization methods for large-scale setups.
This thesis consists of two independent parts: random matrices, which form the first one-third of this thesis, and machine learning, which constitutes the remaining part. The main results of this thesis are as follows: a necessary and sufficient condition for the inverse moments of (m,n,β)-Laguerre matrices and compo…
E2M predicts metric space outputs using deep learning.
problem Predicting non-Euclidean outputs like distributions and matrices.
method Weighted Fréchet means over learned weights.
result E2M achieves state-of-the-art performance across various outputs.
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.
Python package for SPD matrix distances, reproducible and extensible.
problem Computing distances between SPD matrices for various applications.
method Unified, extensible framework supporting multiple SPD metrics.
result Reproducible and accessible SPD matrix comparison tool.