Related papers: On the Upper Bounds for the Matrix Spectral Norm
In this paper, we give estimates for both upper and lower bounds of eigenvalues of a simple matrix. The estimates are shaper than the known results.
Every sufficiently big matrix with small spectral norm has a nearby low-rank matrix if the distance is measured in the maximum norm (Udell & Townsend, SIAM J Math Data Sci, 2019). We use the Hanson--Wright inequality to improve the estimate…
This paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and Bregman divergence losses. A major focus is on the derivation of a rate sharp minimax…
Computing eigenvalues of very large matrices is a critical task in many machine learning applications, including the evaluation of log-determinants, the trace of matrix functions, and other important metrics. As datasets continue to grow in…
An important yet challenging problem in numerical linear algebra is finding a principal submatrix with maximum determinant from a given symmetric positive semidefinite matrix. This problem arises in experimental design, statistics, and…
This paper proposes lower bounds on a quantity called $L^p$-norm joint spectral radius, or in short, $p$-radius, of a finite set of matrices. Despite its wide range of applications to, for example, stability analysis of switched linear…
While Spectral Methods have long been used for Principal Component Analysis, this survey focusses on work over the last 15 years with three salient features: (i) Spectral methods are useful not only for numerical problems, but also discrete…
Spectral estimators are fundamental in lowrank matrix models and arise throughout machine learning and statistics, with applications including network analysis, matrix completion and PCA. These estimators aim to recover the leading…
Matrix concentration inequalities provide a direct way to bound the typical spectral norm of a random matrix. The methods for establishing these results often parallel classical arguments, such as the Laplace transform method. This work…
Practical model building processes are often time-consuming because many different models must be trained and validated. In this paper, we introduce a novel algorithm that can be used for computing the lower and the upper bounds of model…
We focus on \emph{row sampling} based approximations for matrix algorithms, in particular matrix multipication, sparse matrix reconstruction, and \math{\ell_2} regression. For \math{\matA\in\R^{m\times d}} (\math{m} points in \math{d\ll m}…
We consider robust low rank matrix estimation as a trace regression when outputs are contaminated by adversaries. The adversaries are allowed to add arbitrary values to arbitrary outputs. Such values can depend on any samples. We deal with…
The matrix spectral and nuclear norms appear in enormous applications. The generalizations of these norms to higher-order tensors is becoming increasingly important but unfortunately they are NP-hard to compute or even approximate. Although…
We study subset selection for matrices defined as follows: given a matrix $\matX \in \R^{n \times m}$ ($m > n$) and an oversampling parameter $k$ ($n \le k \le m$), select a subset of $k$ columns from $\matX$ such that the pseudo-inverse of…
In distributed optimization or Nash-equilibrium seeking over directed graphs, it is crucial to find a matrix norm under which the disagreement of individual agents' states contracts. In existing results, the matrix norm is usually defined…
Low-rank matrix approximations are often used to help scale standard machine learning algorithms to large-scale problems. Recently, matrix coherence has been used to characterize the ability to extract global information from a subset of…
We propose a new pivotal method for estimating high-dimensional matrices. Assume that we observe a small set of entries or linear combinations of entries of an unknown matrix $A\_0$ corrupted by noise. We propose a new method for estimating…
For real symmetric matrices that are accessible only through matrix vector products, we present Monte Carlo estimators for computing the diagonal elements. Our probabilistic bounds for normwise absolute and relative errors apply to Monte…
First, we extend the results of approximate matrix multiplication from the Frobenius norm to the spectral norm. Second, We develop a class of fast approximate generalized linear regression algorithms with respect to the spectral norm.…
In this paper we present a new bound obtained with the probabilistic method for the solution of the Set Covering problem with unit costs. The bound is valid for problems of fixed dimension, thus extending previous similar asymptotic…