Sharp inequalities for matrix means with unknown variance.
problem Estimating matrix means with unknown variance.
method Empirical Bernstein inequalities for symmetric random matrices.
result Adapts to unknown variance with tight deviation bounds.
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48 results for “matrix inequalities”
Graph Hörmander Systemsmath-ph
The paper extends log-Sobolev inequalities to matrix-valued settings using combinatorial methods.
problem Log-Sobolev inequalities for matrix-valued settings.
method Combining noncommutative geometry tools and combinatorial methods.
result Combinatorial methods yield computable lower bounds for matrix-valued log-Sobolev inequalities.
We derive exponential tail inequalities for sums of random matrices with no dependence on the explicit matrix dimensions. These are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit matrix dimensions replaced by a trace quantity that can be small even when the dimens…
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.
In this paper, we present a new framework to obtain tail inequalities for sums of random matrices. Compared with existing works, our tail inequalities have the following characteristics: 1) high feasibility--they can be used to study the tail behavior of various matrix functions, e.g., arbitrary matrix norms, the absol…
Derives matrix Harnack inequalities for semilinear heat equations on manifolds.
problem Bounding solutions of semilinear heat equations on manifolds with geometric constraints.
method Applies Li-Yau estimates to derive Harnack inequalities for positive solutions.
result Derives matrix Harnack inequalities for positive solutions of semilinear heat equations.
Paper proves inequality for Green function on Kähler manifolds.
problem Estimating Green function on Kähler manifolds.
method Matrix Li-Yau-Hamilton inequality for Green function.
result Elliptic analogue of heat equation estimate for Kähler manifolds.
This note gives a simple analysis of a randomized approximation scheme for matrix multiplication proposed by Sarlos (2006) based on a random rotation followed by uniform column sampling. The result follows from a matrix version of Bernstein's inequality and a tail inequality for quadratic forms in subgaussian random ve…
Nonnegative sectional curvature linked to matrix displacement convexity.
problem Nonnegative sectional curvature in Riemannian manifolds.
method Matrix displacement convexity as a criterion for nonnegative sectional curvature.
result Entropy functional matrix displacement convexity implies nonnegative sectional curvature.
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.
We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation on closed manifolds. We also derive a new interpolated Harnack inequality for the equation on closed surfaces under the -Ricci flow. Finally we prove…
We use topological methods to prove a semicontinuity property of the Hodge spectra for analytic germs defined on an isolated surface singularity. For this we introduce an analogue of the Seifert matrix (the fractured Seifert matrix), and of the Levine--Tristram signatures associated with it, defined for null-homologous…
From concentration inequalities for the suprema of Gaussian or Rademacher processes an inequality is derived. It is applied to sharpen existing and to derive novel bounds on the empirical Rademacher complexities of unit balls in various norms appearing in the context of structured sparsity and multitask dictionary lear…
This work establishes always-valid risk bounds for online matrix completion.
problem Challenges in establishing always-valid concentration inequalities for online matrix completion.
method Combines non-asymptotic martingale concentration and regularized low-rank matrix regression.
result Establishes always-valid risk bound process for online matrix completion.
The paper develops concentration inequalities for structured random data, extending beyond independent terms.
problem Developing concentration inequalities for structured weighted sums of random data, including tensors and matrix-valued data.
method The paper develops Hoeffding and Bernstein bounds for structured weighted sums under exchangeability, extending beyond the classical framework of independent terms.
result The paper develops a sharper concentration bound for combinatorial sums of matrix arrays.
Study noncommutative Sobolev inequalities using quantum state metrics.
problem Establishing Sobolev inequalities in noncommutative settings.
method Generalizing monotone metrics in quantum states.
result Developed new matrix-valued Beckner inequalities.
The aim of this paper is to provide some theoretical understanding of quasi-Bayesian aggregation methods non-negative matrix factorization. We derive an oracle inequality for an aggregated estimator. This result holds for a very general class of prior distributions and shows how the prior affects the rate of convergenc…
In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix concentration inequalities, research has advanced to the point where we can conquer many (…
We give a geometric interpretation of Hamilton's matrix Harnack inequality for the Ricci flow as the curvature of a connection on space-time.
Concentration for matrix martingales in continuous time and microscopic activity of social networksmath.PR
This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to the class of continuous time processes. Our analysis relies on a new supermarti…
Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci…
Develops inequalities for high-dimensional linear processes with dependent innovations.
problem Estimating high-dimensional VAR(p) systems and HAC covariance estimation.
method Concentration inequalities for norm of vector linear processes with sub-Weibull, mixingale innovations.
result Obtained concentration bounds for the maximum entrywise norm of lag- autocovariance matrices.
Based on a suggestion of Richard Hamilton, we give an alternate proof of his matrix Harnack inequality for solutions of the Ricci flow with positive curvature operator. This Harnack inequality says that a certain endomorphism, consisting of an expression in the curvature and its first two covariant derivatives, of the …
To estimate the conditional probability functions based on the direct problem setting, V-matrix based method was proposed. We construct V-matrix based constrained quadratic programming problems for which the inequality constraints are inconsistent. In particular, we would like to present that the constrained quadratic …
New Hessian estimates for heat equations on manifolds.
problem Estimating Hessian matrices for heat-type equations on Riemannian manifolds.
method Using Bismut-Stroock Hessian formula, with explicit coefficients and delay/growth rate functions.
result Novel backward weak Harnack inequality and precise pointwise Hessian estimates for eigenfunctions.
Paper proposes a transfer learning method for improving matrix completion.
problem Improving estimation of a low-rank target matrix using auxiliary data.
method Transfer learning procedure leveraging prior information on favorable source datasets.
result Method outperforms traditional methods when source datasets are close to the target matrix.
PAC-Bayesian matrix completion with a spectral scaled Student prior offers efficient inference.
problem Matrix completion with underlying low-rank structure.
method Spectral scaled Student prior and PAC-Bayesian bounds.
result Minimax-optimal oracle inequality for model misspecification and general sampling distribution.
Study generalizes matrix completion with side info in low noise settings.
problem Matrix completion with side information in low noise conditions.
method Inductive matrix completion with i.i.d. subgaussian noise, uniform sampling, and side information.
result Generalization bounds with noise scaling, convergence to zero, and logarithmic dependence on matrix size.
At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coefficients of this polynomial, called mixed volumes. Among the deepest results of this theory is the Alexandrov-Fenchel inequality, which subsumes…
Derives a sharp inequality for trace-free matrices with applications to hypersurfaces.
problem Classifying conformally flat hypersurfaces and characterizing rotational hypersurfaces.
method Derives a sharp inequality relating eigenvalues of trace-free matrices and applies it to hypersurfaces.
result New proof of the classification of conformally flat hypersurfaces and construction of a functional for rotational hypersurfaces.
New technique stabilizes singular values in concatenated matrices.
problem How singular values of concatenated matrices relate to individual components.
method Developed perturbation technique extending classical results to concatenated matrices.
result Dominant singular values remain stable under small perturbations in submatrices.
The paper derives inequalities for Riemannian maps and submersions involving quaternionic space forms.
problem Establishing optimal inequalities for Riemannian maps and submersions involving quaternionic space forms.
method Deriving Casorati inequalities for Riemannian maps and submersions involving quaternionic space forms.
result Geometric characterizations of equality cases for Riemannian maps and submersions involving quaternionic space forms.
We solve robust regression and matrix completion problems with sparse and low-rank models.
problem Adversarial contamination and noisy matrix completion in high-dimensional settings.
method Subgaussian statistical learning framework, trace-regression with matrix decomposition, novel Huber-type loss.
result Near-optimal estimation rates for robust regression and matrix completion.
Low-rank matrix recovery has found many applications in science and engineering such as machine learning, signal processing, collaborative filtering, system identification, and Euclidean embedding. But the low-rank matrix recovery problem is an NP hard problem and thus challenging. A commonly used heuristic approach is…
New algorithms reduce variance in solving complex mathematical problems.
problem Solving convex-concave saddle point problems, variational inequalities, and inclusions.
method Stochastic variance reduction for extragradient, forward-backward-forward, and forward-reflected-backward methods.
result All proposed methods converge with complexities matching or improving deterministic counterparts.
Various Alexandrov-Fenchel type inequalities have appeared and played important roles in convex geometry, matrix theory and complex algebraic geometry. It has been noticed for some time that they share some striking analogies and have intimate relationships. The purpose of this article is to shed new light on this by c…
A society or country with income equally distributed among its people is truly a fiction! The phenomena of socioeconomic inequalities have been plaguing mankind from times immemorial. We are interested in gaining an insight about the co-evolution of the countries in the inequality space, from a data science perspective…
AGD converges in polynomial iterations to optimal matrix factorization.
problem Matrix factorization optimization with alternating gradient descent.
method Alternating gradient descent with fixed step size, proving convergence in polynomial iterations.
result AGD reaches ε-optimal factorization in T iterations with high probability.
The abundance of high-dimensional data in the modern sciences has generated tremendous interest in penalized estimators such as the lasso, scaled lasso, square-root lasso, elastic net, and many others. In this paper, we establish a general oracle inequality for prediction in high-dimensional linear regression with such…
Paper sets fundamental limits for distributed covariance estimation with constrained communication.
problem Estimating high-dimensional covariance matrices in a feature-split setting with limited communication.
method Developed a Conditional Strong Data Processing Inequality (C-SDPI) to establish minimax lower bounds and an optimal estimation protocol.
result Achieved nearly optimal estimation protocol with sample and communication requirements matching lower bounds up to logarithmic factors.
The paper generalizes product inequalities for random vectors and their applications.
problem Understanding concentration of measure for products of random vectors.
method Develops expressions for the concentration of functionals of random vectors based on product norms.
result Provides generalized Hanson-Wright inequalities and applications to random matrices.
The paper derives inequalities for contact CR-warped product submanifolds in cosymplectic space forms.
problem Establishing inequalities for contact CR-warped product submanifolds in cosymplectic space forms.
method Using the Gauss equation and hypotheses for cosymplectic and nearly cosymplectic manifolds, the paper derives inequalities for the norm of the second fundamental form and the shape operator.
result The contact warped product submanifolds in cosymplectic manifolds exhibit a geometric property called -minimality, leading to an optimal general inequality.
Short proof shows how ridge regression works with random data.
problem Understanding prediction error in ridge regression with random design.
method Combination of exchangeability arguments, matrix perturbation, and operator convexity.
result Elementary proof of prediction error without complex inequalities.
Improved MMWU algorithm achieves instance-optimal regret bound for matrix LEA.
problem Matrix Learning from Expert Advice problem.
method Developed a general potential-based framework for matrix LEA, using a new Jensen's trace inequality.
result Achieved instance-optimal regret bound of .
The paper sets bounds on how much regret is unavoidable in adaptive LQR with unknown B-matrix.
problem Understanding the limits of adaptive LQR with unknown B-matrix.
method Local asymptotic minimax regret lower bounds using van Trees' inequality and Bellman error representation.
result Logarithmic regret is impossible if the parametrization induces an uninformative optimal policy.
We consider the problem of finding anomalies in high-dimensional data using popular PCA based anomaly scores. The naive algorithms for computing these scores explicitly compute the PCA of the covariance matrix which uses space quadratic in the dimensionality of the data. We give the first streaming algorithms that use …
Unified analysis of MPLE for Ising models with bounded operator norm or infinity norm.
problem Estimating Ising models in Total Variation distance with limited samples.
method Maximum Pseudo-Likelihood Estimator (MPLE) for two general classes of Ising models.
result Unified framework for polynomial-time estimation in TV distance for two general classes of Ising models.
We prove a generalization of the Li-Yau estimate for a board class of second order linear parabolic equations. As a consequence, we obtain a new Cheeger-Yau inequality and a new Harnack inequality for these equations. We also prove a Hamilton-Li-Yau estimate, which is a matrix version of the Li-Yau estimate, for these …