Related papers: Random perturbation of low rank matrices: Improvin…
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…
Matrix perturbation bounds (such as Weyl and Davis-Kahan) are used abundantly in many areas of mathematics and data science. Many bounds (such as the above two) involve the spectral norm of the noise matrix and are sharp in worst case…
Perturbation theory is developed to analyze the impact of noise on data and has been an essential part of numerical analysis. Recently, it has played an important role in designing and analyzing matrix algorithms. One of the most useful…
Computing the first few singular vectors of a large matrix is a problem that frequently comes up in statistics and numerical analysis. Given the presence of noise, exact calculation is hard to achieve, and the following problem is of…
The Davis-Kahan-Wedin $\sin \Theta$ theorem describes how the singular subspaces of a matrix change when subjected to a small perturbation. This classic result is sharp in the worst case scenario. In this paper, we prove a stochastic…
Concatenating matrices is a common technique for uncovering shared structures in data through singular value decomposition (SVD) and low-rank approximations. The fundamental question arises: How does the singular value spectrum of the…
The classical Davis-Kahan theorem provides an efficient bound on the perturbation of eigenspaces of a matrix under a large (eigenvalue) gap condition. In this paper, we consider the case when the gap is moderate. Using a bootstrapping…
We extend several relative perturbation bounds to Hermitian matrices that are possibly singular, and also develop a general class of relative bounds for Hermitian matrices. As a result, corresponding relative bounds for singular values of…
We develop deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors, in a spirit similar to classical results for matrices such as those due to Weyl, Davis, Kahan and Wedin. Our bounds…
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…
This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar…
In statistics and machine learning, people are often interested in the eigenvectors (or singular vectors) of certain matrices (e.g. covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or…
We perturb a real matrix $A$ of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our bounds, which extend existing lower-order expressions,…
Perturbation bounds for singular spaces, in particular Wedin's $\sin \Theta$ theorem, are a fundamental tool in many fields including high-dimensional statistics, machine learning, and applied mathematics. In this paper, we establish…
Solutions to scalar theories with derivative self-couplings often have regions where non-linearities are important. Given a classical source, there is usually a region, demarcated by the Vainshtein radius, inside of which the classical…
These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition…
The quasi-variational inequalities play a significant role in analyzing a wide range of real-world problems. However, these problems are more complicated to solve than variational inequalities as the constraint set is based on the current…
A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank $1$ perturbation. Considered in this review are the additive rank $1$ perturbation of the…
This paper derives exponential tail bounds and polynomial moment inequalities for the spectral norm deviation of a random matrix from its mean value. The argument depends on a matrix extension of Stein's method of exchangeable pairs for…
In this note we want to have another look on Schwinger-Dyson equations for the eigenvalue distributions and the fluctuations of classical unitarily invariant random matrix models. We are exclusively dealing with one-matrix models, for which…