Study differential properties of matrix square roots in specific cases.
problem Understanding matrix square roots in semi-simple, symmetric, and orthogonal cases.
method Analysis of differential and metric structures of real square roots of matrices under specific conditions.
result Differential properties of matrix square roots in semi-simple, symmetric, and orthogonal cases.
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.
We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin representations for all series of classical groups (for large values of parameters of re…
We present a formula for the trace of any symmetric power of a n×n matrix (with coefficients in a field) in terms of the ordinary powers of the matrix, an arbitrarily chosen linear function which vanishes on the identity matrix, and n−2 polynomial functions defined recursively.
Adaptive algorithm improves convergence rate of Langevin dynamics.
problem Improving convergence rate of Langevin dynamics.
method Adaptive non-reversible stochastic gradient Langevin dynamics algorithm.
result Improved convergence rate of the algorithm.
Given a symmetric nonnegative matrix A, symmetric nonnegative matrix factorization (symNMF) is the problem of finding a nonnegative matrix H, usually with much fewer columns than A, such that A≈HHT. SymNMF can be used for data analysis and in particular for various clustering tasks. In this paper, we p…
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.
Gradient descent achieves exact linear convergence rate for symmetric matrix completion.
problem Low-rank symmetric matrix completion using gradient descent.
method Local analysis of gradient descent for symmetric matrices without additional assumptions.
result Closed-form expression of exact linear convergence rate matches practice.
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 short note reviews so-called Natural Gradient Descent (NGD) for multivariate Gaussians. The Fisher Information Matrix (FIM) is derived for several different parameterizations of Gaussians. Careful attention is paid to the symmetric nature of the covariance matrix when calculating derivatives. We show that there ar…
New method for mixed memberships using symmetrized Laplacian inverse matrix.
problem Mixed memberships in community detection.
method Spectral clustering on symmetrized Laplacian inverse matrix.
result Mixed-SLIM methods outperform state-of-the-art methods.
This work accelerates constrained sampling using large deviation principles.
problem Sampling constrained probability distributions efficiently.
method Large deviation principles applied to skew-reflected non-reversible Langevin dynamics.
result The skew-symmetric matrix accelerates convergence and reduces asymptotic variance.
The paper is on the vanishing topology of singular Milnor fibres of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices with sufficiently many parameters. We define vanishing cycles on such fibres, prove an extended form of the Damon-Pike μ=τ conjecture about the families of a special type…
A necessary condition for a connection in a vector bundle to be locally metric is for its curvature matrix, which consists of 2 forms, to be skew symmetric with respect to some local frame. In this paper we give a simple algorithm that can be used to decide when a matrix of 2 forms is equivalent to a skew symmetric…
The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.
problem Analyzing Riemannian Gaussian distributions on symmetric spaces.
method Analytical computation of marginals using orthogonal and skew orthogonal polynomials, and diffusion kernels.
result Riemannian Gaussian distributions are random matrix types, and their probability density functions can be computed analytically.
Unimodular classification of symmetric matrix map-germs.
problem Classifying symmetric matrix map-germs under volume-preserving equivalence.
method Introducing symmetrical quasi-homogeneity and volume-preserving equivalence.
result All simple G-equivalence classes coincide with volume-preserving equivalence classes. 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.
Determinants of theta curves and symmetric graphs are studied.
problem Understanding the determinants of theta curves and symmetric graphs.
method Combinatorial approach using Kirchhoff's Matrix Tree Theorem and spanning tree enumeration.
result The determinant of a simple theta curve is the product of the determinants of its constituent knots.
Extends knot invariant computation to symmetrically colored sl_N.
problem Computing quantum knot invariants for slN. method Develops symmetrically colored R matrix for slN. result Defines FKslN,sym for positive braid knots. 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.
Derives a Hamiltonian model for 3D axially symmetric magnetohydrodynamics.
problem Modeling of 3D axially symmetric magnetohydrodynamics.
method Hamiltonian formulation and matrix discretization.
result First discrete model for 3D magnetohydrodynamics compatible with underlying Lie-Poisson structure.
A new method computes link invariants from diagrams.
problem Computing link invariants efficiently.
method Single symmetric matrix from a link diagram.
result Multivariable Alexander polynomial computation.
Symmetric nonnegative matrix factorization (NMF), a special but important class of the general NMF, is demonstrated to be useful for data analysis and in particular for various clustering tasks. Unfortunately, designing fast algorithms for Symmetric NMF is not as easy as for the nonsymmetric counterpart, the latter adm…
Anosov groups study matrix coefficients and orbit counting in symmetric spaces.
problem Anosov groups and their matrix coefficients in symmetric spaces.
method Asymptotic analysis of matrix coefficients and higher rank measures.
result Asymptotic behavior of matrix coefficients and orbit counting results.
We investigate some geometric properties of the real algebraic variety Δ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart-Young-Mirsky-type theorem for the distance function from a generic matrix to points in Δ. We exhibit conne…
The paper analyzes how over-parameterization affects GD convergence in matrix sensing problems.
problem Matrix sensing problem with over-parameterized gradient descent.
method Analyzes symmetric and asymmetric parameterizations, provides lower bounds and convergence rates.
result Over-parameterization slows down GD convergence, but asymmetric parameterization can speed up convergence.
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.
Simple matrix formulas for Grassmannian curvatures.
problem Modeling Grassmannian for curvature calculations.
method Symmetric orthogonal matrices and standard matrix operations.
result Explicit, simple formulas for various curvatures.
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.
Spectral clustering is a standard approach to label nodes on a graph by studying the (largest or lowest) eigenvalues of a symmetric real matrix such as e.g. the adjacency or the Laplacian. Recently, it has been argued that using instead a more complicated, non-symmetric and higher dimensional operator, related to the n…
The SCMU algorithm computes cone factorizations for symmetric cones, improving upon existing methods.
problem Computing cone factorizations for symmetric cones in optimization.
method Introduces and analyzes the symmetric-cone multiplicative update (SCMU) algorithm.
result The SCMU algorithm non-decreases the squared loss objective.
We develop a method to factorize symmetric sparse Boolean matrices efficiently.
problem Finding a symmetric factorization of a given matrix into a sparse, Boolean matrix.
method Polynomial-time algorithm based on bootstrapping higher-order information and tensor decomposition.
result A matrix with full column rank can be recovered with high probability when the matrix size is sufficiently large.
Improves matrix multiplication throughput for asymmetric bit-width operands.
problem Matrix multiplications between asymmetric bit-width operands, especially 8- and 4-bit, are not efficiently handled by existing SIMD instructions.
method Proposes a new SIMD matrix multiplication instruction that uses mixed precision on inputs (8- and 4-bit) and accumulates into 16-bit output, improving throughput.
result Offers 2x improvement in throughput compared to existing symmetric-operand-size instructions, with negligible overflow.
Classification of SL(n) covariant valuations on Orlicz spaces.
problem Classifying continuous SL(n) covariant valuations on Orlicz spaces.
method Complete classification without symmetric assumptions, focusing on moment matrix and a new functional in dimension two.
result The moment matrix is the only SL(n) covariant valuation for n≥3, and a new functional appears in dimension two.
New method for inferring network topology from partial data.
problem Inferring network topology from limited node data.
method Vector autoregressive model and Gaussian mixture algorithm.
result The proposed method converges to the network combination matrix in probability.
Basket links are shown to be isotopic to T(2,n+1).
problem Understanding isotopy of basket links to torus links.
method Using symmetrized Seifert form congruence to An matrix. result Basket links are isotopic to T(2,n+1). A study of proper affine vector fields in plane symmetric static space-times by using the rank of the Rieman matrix and holonomy. Studying proper affine vector fields in each case, It is shown that the special class of the above space-times admit proper affine vector fields.
Estimates the probability of a random symmetric tensor being close to rank-one.
problem Estimating the probability of a random symmetric tensor being close to rank-one.
method Using Weyl's tube formula and techniques from Random Matrix theory, we study metric invariants of the real Veronese variety.
result Explicit formula for the reach and curvature coefficients of the real Veronese variety with respect to the Bombieri-Weyl metric.
Paper shows no spurious local minima in a specific matrix factorization problem.
problem Optimization of ℓ1-norm rank-one symmetric matrix factorization. method Second-order variational analysis to study the landscape of the problem.
result Any second-order stationary point is globally optimal.
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.
Symmetric nonnegative matrix factorization has found abundant applications in various domains by providing a symmetric low-rank decomposition of nonnegative matrices. In this paper we propose a Frank-Wolfe (FW) solver to optimize the symmetric nonnegative matrix factorization problem under a simplicial constraint, whic…
Over the past few years, trace regression models have received considerable attention in the context of matrix completion, quantum state tomography, and compressed sensing. Estimation of the underlying matrix from regularization-based approaches promoting low-rankedness, notably nuclear norm regularization, have enjoye…
Algorithm calculates Jones polynomial from Goeritz matrix.
problem Calculating Jones polynomial from link diagrams.
method Explicit algorithm using Goeritz matrices.
result Jones polynomial can be recovered from orientable checkerboard surfaces.
Olshausen and Field (OF) proposed that neural computations in the primary visual cortex (V1) can be partially modeled by sparse dictionary learning. By minimizing the regularized representation error they derived an online algorithm, which learns Gabor-filter receptive fields from a natural image ensemble in agreement …
New connections on symmetric spaces with invariant properties.
problem Understanding invariant connections on hermitian symmetric spaces.
method Introduced a class of G-invariant connections on homogeneous bundles over hermitian symmetric spaces. result Parameter space of connections is a normal variety with a canonical anti-holomorphic involution.
We analyze the spectral properties of correlation matrices between distinct statistical systems. Such matrices are intrinsically non symmetric, and lend themselves to extend the spectral analyses usually performed on standard Pearson correlation matrices to the realm of complex eigenvalues. We employ some recent random…
We consider height functions on symmetric spaces M≅G/K embedded in the associated matrix Lie group G. In particular we study the relationship between the critical sets of the height function on G and its restriction to M. Also we prove that the gradient flow on M can be integrated by means of a generaliz…
We present a method based on the orthogonal symmetric non-negative matrix tri-factorization of the normalized Laplacian matrix for community detection in complex networks. While the exact factorization of a given order may not exist and is NP hard to compute, we obtain an approximate factorization by solving an optimiz…