Related papers: Rate-Optimal Perturbation Bounds for Singular Subs…
We present a comprehensive analysis of singular vector and singular subspace perturbations in the signal-plus-noise matrix model with random Gaussian noise. Assuming a low-rank signal matrix, we extend the Davis-Kahan-Wedin theorem in a…
Reliable and efficient computation of the pseudospectral abscissa in the large-scale setting is still not settled. Unlike the small-scale setting where there are globally convergent criss-cross algorithms, all algorithms in the large-scale…
Let A be a self-adjoint operator on a Hilbert space H. Assume that {\sigma} is an isolated component of the spectrum of A, i.e. dist({\sigma},{\Sigma})=d>0 where {\Sigma}=spec(A)\{\sigma}. Suppose that V is a bounded self-adjoint operator…
This paper establishes a variant of Stewart's theorem (Theorem~6.4 of Stewart, {\em SIAM Rev.}, 15:727--764, 1973) for the singular subspaces associated with the SVD of a matrix subject to perturbations. Stewart's original version uses both…
Classical matrix perturbation results, such as Weyl's theorem for eigenvalues and the Davis-Kahan theorem for eigenvectors, are general purpose. These classical bounds are tight in the worst case, but in many settings sub-optimal in the…
For a fixed symmetric matrix A and symmetric perturbation E we develop purely deterministic bounds on how invariant subspaces of A and A+E can differ when measured by a suitable "row-wise" metric rather than via traditional measures of…
In this paper, we study the perturbation of the extreme singular values of a matrix in the particular case where it is obtained after appending an arbitrary column vector. Such results have many applications in bifurcation theory, signal…
In this paper, we develop novel perturbation bounds for the high-order orthogonal iteration (HOOI) [DLDMV00b]. Under mild regularity conditions, we establish blockwise tensor perturbation bounds for HOOI with guarantees for both tensor…
The Shannon lower bound has been the subject of several important contributions by Berger. This paper surveys Shannon bounds on rate-distortion problems under mean-squared error distortion with a particular emphasis on Berger's techniques.…
Driven by a wide range of applications, many principal subspace estimation problems have been studied individually under different structural constraints. This paper presents a unified framework for the statistical analysis of a general…
We analyze a class of estimators based on convex relaxation for solving high-dimensional matrix decomposition problems. The observations are noisy realizations of a linear transformation $\mathfrak{X}$ of the sum of an approximately) low…
We establish optimal convergence rates up to a log-factor for a class of deep neural networks in a classification setting under a restraint sometimes referred to as the Tsybakov noise condition. We construct classifiers in a general setting…
An analogue of the Davis-Kahan sin2\Theta\ theorem from [SIAM J. Numer. Anal. 7 (1970), 1--46] is proved under a general spectral separation condition. This extends the generic sin2\theta\ estimates recently shown by Albeverio and Motovilov…
Accurate modeling of boundary conditions is crucial in computational physics. The ever increasing use of neural networks as surrogates for physics-related problems calls for an improved understanding of boundary condition treatment, and its…
Modern applications in statistics, computer science and network science have seen tremendous values of finer matrix spectral perturbation theory. In this paper, we derive a generic $\ell_{2\rightarrow\infty}$ eigenspace perturbation bound…
High-dimensional inference refers to problems of statistical estimation in which the ambient dimension of the data may be comparable to or possibly even larger than the sample size. We study an instance of high-dimensional inference in…
We study $\textit{sparse singular value certificates}$ for random rectangular matrices. If $M$ is an $n \times d$ matrix with independent Gaussian entries, we give a new family of polynomial-time algorithms which can certify upper bounds on…
Physical theories have a limited regime of validity and hence must be accompanied by a breakdown diagnostic to establish when they cease to be valid as parameters are varied. For perturbative theories, estimates of the first neglected order…
The singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and exploiting the geometric…
We consider differentially private approximate singular vector computation. Known worst-case lower bounds show that the error of any differentially private algorithm must scale polynomially with the dimension of the singular vector. We are…