Related papers: Four Deviations Suffice for Rank 1 Matrices

The aim of this paper is to study the matrix discrepancy problem. Assume that $\xi_1,\ldots,\xi_n$ are independent scalar random variables with finite support and $\mathbf{u}_1,\ldots,\mathbf{u}_n\in \mathbb{C}^d$. Let $\mathcal{C}_0$ be…

Akemann and Weaver showed Lyapunov-type theorem for rank one positive semidefinite matrices which is an extension of Weaver's KS$_2$ conjecture that was proven by Marcus, Spielman, and Srivastava in their breakthrough solution of the…

In 2013, Marcus, Spielman, and Srivastava resolved the famous Kadison-Singer conjecture. It states that for $n$ independent random vectors $v_1,\cdots, v_n$ that have expected squared norm bounded by $\epsilon$ and are in the isotropic…

Recently Marcus, Spielman and Srivastava proved Weaver's ${\rm{KS}}_r$ conjecture, which gives a positive solution to the Kadison-Singer problem. Cohen and Br\"and\'en independently extended this result to obtain the arbitrary-rank version…

Consider the matrix $\Sigma_n = n^{-1/2} X_n D_n^{1/2} + P_n$ where the matrix $X_n \in \C^{N\times n}$ has Gaussian standard independent elements, $D_n$ is a deterministic diagonal nonnegative matrix, and $P_n$ is a deterministic matrix…

Marcus, Spielman, and Srivastava recently solved the Kadison-Singer problem by showing that if u_1, ..., u_m are column vectors in C^d such that \sum u_iu_i^* = I, then a set of indices S \subseteq {1, ..., m} can be chosen so that \sum_{i…

A well-known discovery of Feige's is the following: Let $X_1, \ldots, X_n$ be nonnegative independent random variables, with $\mathbb{E}[X_i] \leq 1 \;\forall i$, and let $X = \sum_{i=1}^n X_i$. Then for any $n$, \[\Pr[X < \mathbb{E}[X] +…

Marcus, Spielman and Srivastava (Annals of Mathematics 2014) solved the Kadison--Singer Problem by proving a strong form of Weaver's conjecture: they showed that for all $\alpha > 0$ and all lists of vectors of norm at most $\sqrt{\alpha}$…

Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We…

In 2001 Heinrich, Novak, Wasilkowski and Wo\'zniakowski proved that the inverse of the star discrepancy satisfies $n(d,\varepsilon)\leq c_{\abs}d \varepsilon^{-2}$ by showing that there exists a set of points in $[0,1)^d$ whose…

We sharpen the constant in the $KS_2$ conjecture of Weaver \cite{We}, which was validated by Marcus, Spielman, and Srivastava \cite{MSS} in their solution of the Kadison--Singer problem. We then apply this result to prove optimal asymptotic…

For a fixed unit vector a=(a_1,a_2,...,a_n) in S^{n-1}, i.e. sum_{i=1}^n a_i^2=1, we consider the 2^n sign vectors epsilon=(epsilon_1,epsilon_2,...,epsilon_n) in {-1,1}^n and the corresponding scalar products a.epsilon=sum_{i=1}^n a_i…

We adapt the arguments of Marcus, Spielman and Srivastava in their proof of the Kadison-Singer problem to prove improved paving estimates. Working with Anderson's paving formulation of Kadison-Singer instead of Weaver's vector balancing…

We prove deviation inequalities for sums of high-dimensional random matrices and operators with dependence and {\rc heavy tails}. Estimation of high-dimensional matrices is a concern for numerous modern applications. However, most results…

Marcus, Spielman, and Srivastava in their seminal work \cite{MSS13} resolved the Kadison-Singer conjecture by proving that for any set of finitely supported independently distributed random vectors $v_1,\dots, v_n$ which have "small"…

Let $p>2$, $B\geq 1$, $N\geq n$ and let $X$ be a centered $n$-dimensional random vector with the identity covariance matrix such that $\sup\limits_{a\in S^{n-1}}{\mathrm E}|\langle X,a\rangle|^p\leq B$. Further, let $X_1,X_2,\dots,X_N$ be…

We establish a large deviation principle for the empirical spectral measure of a sample covariance matrix with sub-Gaussian entries, which extends Bordenave and Caputo's result for Wigner matrices having the same type of entries [7]. To…

Let $A$ be an $n\times n$ random matrix with independent, identically distributed mean 0, variance 1 subgaussian entries. We prove that $$ \mathbb{P}(A\text{ has distinct singular values})\geq 1-e^{-cn} $$ for some $c>0$, confirming a…

We prove deviation bounds for the random variable $\sum_{i=1}^{n} f_i(Y_i)$ in which $\{Y_i\}_{i=1}^{\infty}$ is a Markov chain with stationary distribution and state space $[N]$, and $f_i: [N] \rightarrow [-a_i, a_i]$. Our bound improves…

Let $a_1, \dots, a_n \in \mathbb{R}$ satisfy $\sum_i a_i^2 = 1$, and let $\varepsilon_1, \ldots, \varepsilon_n$ be uniformly random $\pm 1$ signs and $X = \sum_{i=1}^{n} a_i \varepsilon_i$. It is conjectured that $X = \sum_{i=1}^{n} a_i…