Related papers: Lower bounds on the smallest eigenvalue of a sampl…

We provide non-asymptotic, relative deviation bounds for the eigenvalues of empirical covariance and Gram matrices in general settings. Unlike typical uniform bounds, which may fail to capture the behavior of smaller eigenvalues, our…

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.

Given $X$ a random vector in ${\mathbb{R}}^n$, set $X_1,...,X_N$ to be independent copies of $X$ and let $\Gamma=\frac{1}{\sqrt{N}}\sum_{i=1}^N <X_i,\cdot>e_i$ be the matrix whose rows are $\frac{X_1}{\sqrt{N}},\dots, \frac{X_N}{\sqrt{N}}$.…

Some monotone increasing sequences of the lower bounds for the minimum eigenvalue of $M$-matrices are given. It is proved that these sequences are convergent and improve some existing results. Numerical examples show that these sequences…

How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite…

We show that the empirical distribution of the eigenvalues of the sample covariance matrix of certain random vectors (not necessarily independent entries) with bounded marginal $L^{4}$ norms converges weakly to a compound free Poisson…

We prove lower bounds on the number of samples needed to privately estimate the covariance matrix of a Gaussian distribution. Our bounds match existing upper bounds in the widest known setting of parameters. Our analysis relies on the…

This paper develops nonasymptotic information inequalities for the estimation of the eigenspaces of a covariance operator. These results generalize previous lower bounds for the spiked covariance model, and they show that recent upper…

Let $A$ be an $N\times n$ random matrix whose entries are coordinates of an isotropic log-concave random vector in $\mathbb{R}^{Nn}$. We prove sharp lower tail estimates for the smallest singular value of $A$ in the following cases: (1)…

We consider estimation of a sparse parameter vector that determines the covariance matrix of a Gaussian random vector via a sparse expansion into known "basis matrices". Using the theory of reproducing kernel Hilbert spaces, we derive lower…

We obtain a lower bound on each entry of the principal eigenvector of a non-regular connected graph.

We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form \[ M = A\circ X + B…

In this paper, we consider the interference rejection combining (IRC) receiver, which improves the cell-edge user throughput via suppressing inter-cell interference and requires estimating the covariance matrix including the inter-cell…

This paper considers the problem of matrix-variate logistic regression. It derives the fundamental error threshold on estimating low-rank coefficient matrices in the logistic regression problem by obtaining a lower bound on the minimax…

We develop new techniques for proving lower bounds on the least singular value of random matrices with limited randomness. The matrices we consider have entries that are given by polynomials of a few underlying base random variables. This…

This paper establishes sharp dimension-free concentration and expectation bounds for the deviation of a sample cross-covariance matrix from its mean. For sub-Gaussian random vectors, we prove a high-probability operator-norm bound governed…

Finite sample properties of random covariance-type matrices have been the subject of much research. In this paper we focus on the "lower tail" of such a matrix, and prove that it is subgaussian under a simple fourth moment assumption on the…

This work is concerned with finite range bounds on the variance of individual eigenvalues of random covariance matrices, both in the bulk and at the edge of the spectrum. In a preceding paper, the author established analogous results for…

We consider an $n\times n$ matrix of independent real Gaussian random variables and determine the asymptotic distribution of the smallest gaps between complex eigenvalues.

We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the…