Let X be a nonsingular simply connected projective variety of dimension m, E a rank n vector bundle on X, and L a line bundle on X. Suppose that S2(E∗)⊗L is an ample vector bundle and that there is a constant even rank r≥2 symmetric bundle map E→E∗⊗L. We prove that m≤n−r. We u…
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…
New framework solves low-rank optimization problems to certifiable optimality.
problem Low-rank optimization problems with certifiable solutions.
method Mixed-Projection Conic Optimization framework using symmetric projection matrices and outer-approximation algorithms.
result Solves low-rank problems to certifiable optimality, outperforming existing methods.
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.
Efficiently approximates eigenspaces for symmetric and general matrices.
problem Fast computation of eigenspaces for large matrices.
method Factor eigenspaces into fundamental components using transformations, solve minimization problems, and iteratively update.
result Improved computational efficiency for eigenspace approximation.
Minimal submanifolds in matrix spaces proven for specific ranks.
problem Minimal submanifolds in matrix spaces.
method Proving semialgebraic sets of matrices are minimal.
result Rectangular, skew-symmetric, and symmetric matrices with prescribed eigenvalues are minimal.
After recalling the notion of caustics of plane curves and basic equations, we first show the birationality of the caustic map for a general source point S in the plane. Then we prove more generally a theorem for curves D in the projective space of 3x3 symmetric matrices B. For a general 3x1 vector S the projection to …
Study isotropy groups for complex orthogonal and skew-symmetric matrices.
problem Understanding isotropy subgroups of orthogonal similarity transformations.
method Analysis of group structure of nonsingular block matrices.
result Group structure of isotropy subgroups related to block Toeplitz matrices.
Algorithm finds isotropy subgroups of orthogonal similarity on symmetric matrices.
problem Computing isotropy subgroups of orthogonal similarity on symmetric matrices.
method Algorithmic procedure solving a Toeplitz matrix equation.
result Structure of isotropy subgroups described.
FedSPDnet improves federated learning for SPD matrices, outperforming existing methods.
problem Federated learning for SPD matrices with orthogonality constraints.
method Two efficient aggregation strategies: ProjAvg and RLAvg, preserving geometric structure.
result FedSPDnet outperforms federated EEGnet in F1 score and robustness to federation and partial participation.
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.
In this paper we extend DDVV-type inequalities involving the Frobenius norm of commutators from real symmetric and skew-symmetric matrices to Hermitian and skew-Hermitian matrices.
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…
New method for calculating HOMFLY polynomials in symmetric representations.
problem Calculating HOMFLY polynomials for symmetric representations.
method Planar decomposition and projection to symmetric representations.
result Restoration of planarity and new insights into HOMFLY polynomials.
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 nodal domain theorems for symmetric matrices via signed graphs.
problem Establish nodal domain theorems for symmetric matrices.
method Explore signed graph structure to define nodal domains for any function.
result Improved lower bound estimates for the number of strong nodal domains.
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.
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.
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.
New algorithm for online optimization over symmetric cones, unifying previous methods.
problem Online convex optimization over symmetric cones.
method Symmetric-Cone Multiplicative Weights Update (SCMWU) algorithm.
result SCMWU is a no-regret algorithm.
In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit f…
Paper introduces skew-symmetric matrices for virtual doodle classification.
problem Classifying virtual doodles and distinguishing non-classical doodles.
method Use skew-symmetric augmented matrices and homology intersection number.
result Characterization of virtualization of classical doodles.
Random projections help in representing sparse graphs efficiently.
problem Efficiently representing sparse graphs of varying sizes and vertex sets.
method Random projection of adjacency matrices to retain graph functionality and properties.
result Random projections can accurately represent graphs of different sizes and vertex sets in the same space.
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.
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
problem Defining projective analogues of Lie bialgebras and Poisson-Lie groups.
method Introducing projective tensor products and adapting classical notions to these structures.
result Every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra.
Paper develops new method for detecting latent structure in large symmetric data matrices.
problem Testing for latent structure in large symmetric data matrices.
method Introduces Wilcoxon--Wigner random matrices based on normalized rank statistics.
result Establishes asymptotic Gaussian fluctuations for leading eigenvalue and eigenvector of Wilcoxon--Wigner 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.
Feature extraction and dimension reduction for networks is critical in a wide variety of domains. Efficiently and accurately learning features for multiple graphs has important applications in statistical inference on graphs. We propose a method to jointly embed multiple undirected graphs. Given a set of graphs, the jo…
Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
problem Characterizing the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
method Introduce the diffeomorphic logarithm of special orthogonal matrices and an efficient algorithm.
result The region containing the principal logarithm has a special multiplicity structure.
Study of a 32D Rosenfeld projective plane, a symmetric space.
problem Understanding the properties of a specific symmetric space.
method Detailed study and analysis of the geometric structure.
result Comprehensive understanding of the 32D Rosenfeld projective plane.
Study inequalities for singular values of rectangular matrices.
problem Inequalities for singular values of rectangular matrices.
method Study convex cones associated to isotropic representations of symmetric spaces.
result Describe inequalities by cohomological conditions.
Researchers approximate partition functions on Riemannian spaces in the large N limit.
problem Computing normalization factors (partition functions) on Riemannian symmetric spaces is challenging.
method Approximation techniques in the large N limit, including saddle-point equations.
result Formulas for leading order terms in the large N limit of SPD matrices and related spaces.
Structured regularizers enable faster optimization on SPD manifolds with constraints.
problem Optimizing SPD matrices with additional constraints.
method Structured regularizers based on symmetric gauge functions.
result Structured regularizers can preserve or induce desirable structure like convexity.
The paper establishes a majorization result for symmetric matrices.
problem Majorization of symmetric matrices under certain conditions.
method Using Garding-Dirichlet operator and properties of I-central operators. result A definitive inequality for symmetric matrices.
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 studies binary random projections with controllable sparsity patterns for computational and accuracy advantages.
problem Improving computational efficiency and accuracy in random projections.
method Proposes two sparse binary projection models with controllable sparsity patterns.
result Significant computational advantages and improved accuracies in empirical evaluations.
A new method for efficiently computing derivatives of skew-symmetric matrix exponentials.
problem Efficient computation of derivatives for skew-symmetric matrices.
method Characterization of invertibility, construction of nearby logarithm, and efficient implementation.
result Explicit formulae for differentiation and its inverse of skew-symmetric matrix exponentials.
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.
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.
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.
New method reduces computational cost for nonnegative low rank matrix approximation.
problem Efficiently compute nonnegative low rank matrix approximation for nonnegative matrices.
method Alternating projections onto tangent spaces of fixed rank matrices manifold and nonnegative matrix manifold.
result Sequence converges linearly to optimal solutions, showing better performance in terms of computational time and accuracy.
Formula establishes determinant majorization for symmetric matrices.
problem Determining determinant majorization for symmetric matrices.
method Establishes determinant majorization formula for symmetric matrices using invariant Garding-Dirichlet polynomials.
result Formula F(A)N1≥det(A)n1 for symmetric matrices. 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.
The paper calculates R and Racah matrices for SO(5) and finds Kauffman polynomials.
problem Generalizing Reshetikhin-Turaev approach to SO(2n+1) case.
method Provided R and Racah matrices for SO(5) symmetric representation.
result Found Kauffman polynomials for SO(5) symmetric representation.
Study cylindrical symmetric Finsler metrics that are projectively flat.
problem Characterize Finsler metrics that are projectively flat.
method Solve the system of differential equations for cylindrical symmetric Finsler metrics.
result Provide a family of solutions for the projectively flat Finsler metrics.
New estimator learns symmetric dynamics from few observations.
problem Learning parameters of stochastic linear dynamics from limited data.
method Method of moments estimator using T=O(logN) observations. result Achieves small maximum element-wise error on symmetric matrices.
Paper tackles fairness in CCA by minimizing correlation disparity error.
problem Fairness issues in CCA.
method Framework to minimize correlation disparity error in CCA.
result Reduces correlation disparity error without sacrificing CCA accuracy.
The class of spherically symmetric Finsler metrics is studied and locally dually flat and projectively flat spherically symmetric Finsler metrics is classified.