Related papers: Sharp invertibility of random Bernoulli matrices

Let $A(p,n,k)$ be the number of $p$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We formulate the conjecture that, for every fixed $p$ and $n$, the…

A long-standing conjecture asserts that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has nonnegative coefficients whenever $m$ is a positive integer and $A$ and $B$ are any two $n \times n$ positive semidefinite Hermitian matrices. The…

The Littlewood--Richardson process is a discrete random point process arising from the isotypic decomposition of tensor products of irreducible representations of $\operatorname{GL}_N(\mathbb{C})$. Biane--Perelomov--Popov matrices are…

We consider the classical last-success problem for sequential Bernoulli trials in the homogeneous setting where $X_1,\ldots,X_n$ are i.i.d. $\mathrm{Bernoulli}(p)$ but the success probability $p\in(0,1)$ is unknown to the decision maker.…

This work examines various statistical distributions in connection with random Vandermonde matrices and their extension to $d$--dimensional phase distributions. Upper and lower bound asymptotics for the maximum singular value are found to…

It is known that the Entropy Power Inequality (EPI) always holds if the random variables have density. Not much work has been done to identify discrete distributions for which the inequality holds with the differential entropy replaced by…

Let ${\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p\ge 5$ and $r\ge 2$, we prove that \begin{equation} \sum\limits_{\begin{smallmatrix}…

Let $A$ be an $n \times n$ random matrix with iid entries over a finite field of order $q$. Suppose that the entries do not take values in any additive coset of the field with probability greater than $1 - \alpha$ for some fixed $0 < \alpha…

Let $A$ be an $n \times n$ matrix, $X$ be an $n \times p$ matrix and $Y = AX$. A challenging and important problem in data analysis, motivated by dictionary learning and other practical problems, is to recover both $A$ and $X$, given $Y$.…

Let $p_n$ be the characteristic polynomial of an $n \times n$ random matrix drawn from one of the compact classical matrix groups. We show that the critical points of $p_n$ converge to the uniform distribution on the unit circle as $n$…

We establish a stability result for the Shannon-McMillan-Breiman theorem on the one-sided finite shift space. For any shift-invariant probability measure P and any data-dependent parsing whose number of blocks is sublinear in N almost…

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. The authors of [2] showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ was $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$,…

We consider the set Bp of parametric block correlation matrices with p blocks of various (and possibly different) sizes, whose diagonal blocks are compound symmetry (CS) correlation matrices and off-diagonal blocks are constant matrices.…

Let $ \bbB_n =\frac{1}{n}(\bbR_n + \bbT^{1/2}_n \bbX_n)(\bbR_n + \bbT^{1/2}_n \bbX_n)^* $, where $ \bbX_n $ is a $ p \times n $ matrix with independent standardized random variables, $ \bbR_n $ is a $ p \times n $ non-random matrix and $…

Let $X$ be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that $$ \mathbf{E}\|X\|_{S_p} \asymp \mathbf{E}\Bigg[ \Bigg(\sum_i\Bigg(\sum_j X_{ij}^2\Bigg)^{p/2}\Bigg)^{1/p} \Bigg]…

Let $O(2n+\ell)$ be the group of orthogonal matrices of size $\left(2n+\ell\right)\times \left(2n+\ell\right)$ equipped with the probability distribution given by normalized Haar measure. We study the probability \begin{equation*}…

Scale invariant scattering suggests that all Bernoulli numbers B_{2n} can be naturally partitioned, i.e., written as particular finite sums of same-signed, monotonic, rational numbers. Some properties of these rational numbers are discussed…

Let $x_i$, $i\in\mathbb{Z}$ be a sequence of i.i.d. standard normal random variables. Consider rectangular Toeplitz $\mathbf{X}=\left(x_{j-i}\right)_{1\leq i\leq p,1\leq j\leq n}$ and circulant $\mathbf{X}=\left(x_{(j-i)\mod…

Let $n \ge 3$ be an integer. Let $P_n = \{1, 2, 3, ..., n-1, n \}$ and let $S_n$ be the symmetric group of permutations on $P_n$. Motivated by the theory of discrete dynamical systems on the interval, we associate each permutation $\si_n$…

We show a number of fine-grained hardness results for the Closest Vector Problem in the $\ell_p$ norm ($\mathrm{CVP}_p$), and its approximate and non-uniform variants. First, we show that $\mathrm{CVP}_p$ cannot be solved in…