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.
We introduce Clique Matrices as an alternative representation of undirected graphs, being a generalisation of the incidence matrix representation. Here we use clique matrices to decompose a graph into a set of possibly overlapping clusters, de ned as well-connected subsets of vertices. The decomposition is based on a s…
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.
Study of J-Hermitian matrices and geometric mean definition.
problem Understanding the cone of J-Hermitian matrices and its geometric mean.
method Analysis of the cone structure, Riemannian structure, and definition of J-geometric mean.
result Uniquely characterized J-geometric mean defined as a solution to a Riccati-type equation.
We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.
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.
Study elliptic isometries on a matrix manifold with specific metrics.
problem Differential-geometric properties of fixed point loci.
method Explicit description and De Rham decomposition of fixed point loci.
result Explicit description and De Rham decomposition of fixed point loci.
New geometric framework for positive semidefinite matrices of fixed rank.
problem Statistical analysis of positive semidefinite matrices of fixed rank.
method Introducing a manifold S(n,p)∗ with Riemannian geometry and Lie group structure. result Analytical closed forms for geodesics and Fréchet means.
Paper derives explicit expression of Alekseev-Meinrenken diffeomorphism.
problem Understanding the Alekseev-Meinrenken diffeomorphism.
method Via the Stokes phenomenon of meromorphic linear systems of ODEs with Poncaré rank 1.
result Explicit expression of the Alekseev-Meinrenken diffeomorphism.
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.
Bootstrapping regularizes singular correlation matrices, reducing the need for complex regularization.
problem Singular correlation matrices in large datasets.
method Averaging bootstrapped correlation matrices to ensure positive-definiteness.
result The averaged correlation matrix is almost surely positive-definite with a sufficient number of bootstraps.
The crossing matrix of a braid on N strands is the N×N integer matrix with zero diagonal whose i,j entry is the algebraic number (positive minus negative) of crossings by strand i over strand j . When restricted to the subgroup of pure braids, this defines a homomorphism onto the additive subgroup of $N…
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.
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
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.
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.
A new matrix concentration inequality for random products of matrices.
problem Understanding the behavior of random matrix products under bounded independent positive semidefinite matrices.
method Developed a non-asymptotic concentration inequality for the product of matrices.
result The inequality provides a bound on the deviation of the matrix product from its expected value.
The study sets limits on heat equation solutions' Hessians on curved spaces.
problem Bounding Hessians of positive solutions to heat equations on Kähler manifolds.
method Global and local upper bounds for Hessian matrices under curvature constraints.
result Improved bounds on Hessians for Riemannian manifolds with lower sectional curvature.
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…
A new way to describe correlation matrices makes modeling easier.
problem Describing correlation matrices in a flexible and positive-definite way.
method Introduces a novel parametrization that allows unrestricted vectors for correlation matrices.
result The new parametrization ensures positive definiteness without additional constraints.
Method estimates M-matrices in graphical models with improved accuracy.
problem Estimating M-matrices as precision matrices in Gaussian graphical models.
method Adaptive multiple-stage estimation method solving weighted ℓ1-regularized problems.
result Method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification.
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.
Denise learns a function to quickly decompose covariance matrices robustly.
problem Robustly decomposing covariance matrices for feature extraction.
method Deep learning for symmetric positive semidefinite matrices.
result Denise achieves state-of-the-art performance in decomposition quality and speed.
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.
Positive definite matrices abound in a dazzling variety of applications. This ubiquity can be in part attributed to their rich geometric structure: positive definite matrices form a self-dual convex cone whose strict interior is a Riemannian manifold. The manifold view is endowed with a "natural" distance function whil…
Paper develops Riemannian geometry for SPSD matrices with DA applications.
problem Riemannian geometry of SPSD matrices for DA.
method Closed-form expressions, approximations of geodesic path, PT, canonical representation.
result Proposes an algorithm for DA with improved performance.
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.
This article provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. This analysis has been motivated by a large and growing use of matrix-valued affine processes in finance, including multi-asset option pricing with stochastic volatil…
In 1981 Masur proved the existence of a dense geodesic in the moduli space for a Teichmüller space. We prove an analogue theorem for reduced Outer Space endowed with the Lipschitz metric. We also prove two results possibly of independent interest: we show Brun's unordered algorithm weakly converges and from this prove …
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.
Paper proposes a deep learning method for better covariance matrix forecasting.
problem Suboptimal predictive performance in traditional matrix volatility forecasting.
method Riemannian-geometry-aware deep learning framework for symmetric positive definite matrices.
result Our method outperforms traditional approaches in predictive accuracy.
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.
We study functions whose truncations are convex or quasiconvex.
problem Understanding functions with specific truncation properties.
method Analyzing C2-smooth functions with positive definite Hessians. result Injectivity of restricted gradient in positive definite region.
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.
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.
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.
New inequalities for matrix supermartingales converge under various conditions.
problem Convergence and maximal inequalities of supermartingales in positive semidefinite matrices.
method Developed new concentration inequalities for matrix supermartingales.
result New inequalities for matrix supermartingales under different tail conditions.
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.
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.
SpodNet learns SPD matrices with structural constraints.
problem Estimating SPD matrices with additional structural constraints.
method Introduces SpodNet, a neural network module that guarantees SPD outputs and supports structural constraints.
result SpodNet learns SPD and sparse matrices effectively.
Paper proves Fujimoto's conjecture for even m ≥ 4.
problem Proving Fujimoto's conjecture for even dimensions.
method Using a special planar network in the theory of positive matrices.
result Best possible number for even m ≥ 4 is proven.
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.
We introduce new partial orders on the set Sn+ of positive-definite matrices of dimension n derived from the homogeneous geometry of Sn+ induced by the natural transitive action of the general linear group GL(n). The orders are induced by affine-invariant cone fields, which arise naturally from a local anal…
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.
Study extends bounds on sample covariance matrices with general dependence.
problem Quantitative bounds on sample covariance matrices with i.i.d. columns.
method Extends previous work on deterministic equivalent to rectangular random matrices with general dependence structure.
result Proves quantitative bounds involving dimensions and spectral parameter, including closer proximity to real positive semi-line.
Introduces a new model for mapping matrices to matrices, subsuming linear regression.
problem Learning matrix-to-matrix mappings from data.
method Partial trace regression model, leveraging quantum information theory.
result Relevance demonstrated in matrix-to-matrix regression and positive semidefinite matrix completion.
Signed networks allow to model positive and negative relationships. We analyze existing extensions of spectral clustering to signed networks. It turns out that existing approaches do not recover the ground truth clustering in several situations where either the positive or the negative network structures contain no noi…
A new method for deep Wishart processes improves kernel-based models.
problem Inference in deep Wishart processes is challenging due to the need for flexible distributions over positive semi-definite matrices.
method Developed a novel approach to flexible distributions over positive semi-definite matrices using the Bartlett decomposition of the Wishart probability density. Used this to create an approximate posterior for the DWP.
result Improved performance of inference in the DWP compared to DGP with equivalent prior.