Pierre Youssef

• PhD
• Professor (Associate) at New York University Abu Dhabi

The chain rule lies at the heart of the powerful Gamma calculus for Markov diffusions on manifolds, providing remarkable connections between several fundamental notions such as Bakry-\'Emery curvature, entropy decay, and hypercontractivity. For Markov chains on finite state spaces, approximate versions of this chain rule have recently been put forw...

We establish a Central Limit Theorem for tensor product random variables $c_k:=a_k \otimes a_k$ , where $(a_k)_{k \in \mathbb {N}}$ is a free family of variables. We show that if the variables $a_k$ are centered, the limiting law is the semi-circle. Otherwise, the limiting law depends on the mean and variance of the variables $a_k$ and corresponds...

We establish a central limit theorem for the sum of $\epsilon$-independent random variables, extending both the classical and free probability setting. Central to our approach is the use of graphon limits to characterize the limiting distribution, which depends on the asymptotic structure of the underlying graphs governing $\epsilon$-independence....

We study the limiting spectral distribution of quantum channels whose Kraus operators sampled as n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n\times n$$\end{doc...

It is well known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko–Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monotoni...

We prove that a wide class of random quantum channels with few Kraus operators, sampled as random matrices with mild moment assumptions, exhibit a large spectral gap, and are therefore optimal quantum expanders. In particular, our result provides a recipe to construct random quantum expanders from their classical (random or deterministic) counterpa...

For reversible Markov chains on finite state spaces, we show that the modified log-Sobolev inequality (MLSI) can be upgraded to a log-Sobolev inequality (LSI) at the surprisingly low cost of degrading the associated constant by $\log (1/p)$, where $p$ is the minimum non-zero transition probability. We illustrate this by providing the first log-Sobo...

It is well-known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko--Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monoton...

Consider the switch chain on the set of d-regular bipartite graphs on n vertices with \(3\le d\le n^{c}\), for a small universal constant \(c>0\). We prove that the chain satisfies a Poincaré inequality with a constant of order O(nd); moreover, when d is fixed, we establish a log-Sobolev inequality for the chain with a constant of order \(O_d(n\log...

In this work, we develop a comparison procedure for the Modified log-Sobolev Inequality (MLSI) constants of two reversible Markov chains on a finite state space. Efficient comparison of the MLSI Dirichlet forms is a well known obstacle in the theory of Markov chains. We approach this problem by introducing a {\it regularized} MLSI constant which, u...

In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let be a sequence of random symmetric matrices such that each Wn is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product , where is a real centered uniformly bounded random variable of unit variance and bn is a...

We introduce the maximal correlation coefficient $R(M_1,M_2)$ between two noncommutative probability subspaces $M_1$ and $M_2$ and show that the maximal correlation coefficient between the sub-algebras generated by $s_n:=x_1+\ldots +x_n$ and $s_m:=x_1+\ldots +x_m$ equals $\sqrt{m/n}$ for $m\le n$, where $(x_i)_{i\in \mathbb{N}}$ is a sequence of fr...

We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carré du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives ris...

Consider the switch chain on the set of $d$-regular bipartite graphs on $n$ vertices with $3\leq d\leq n^{c}$, for a small universal constant $c>0$. We prove that the chain satisfies a Poincar\'e inequality with a constant of order $O(nd)$; moreover, when $d$ is fixed, we establish a log-Sobolev inequality for the chain with a constant of order $O_...

In this note, we show that the norm of an $n\times n$ random jointly exchangeable matrix with zero diagonal can be estimated in terms of the norm of its $n/2\times n/2$ submatrix located in the top right corner. As a consequence, we prove a relation between the second largest singular values of a random matrix with constant row and column sums and...

We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carré du champ operator. This extends to the noncommutative setting a classical phenomenon in the scalar case. Moreover, the proof g...

We show that any probability measure satisfying a Matrix Poincar\'e inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carr\'e du champ operator. This extends to the noncommutative setting a classical phenomenon in the scalar case. Moreover, the pro...

We show that any |$n\times m$| matrix |$A$| can be approximated in operator norm by a submatrix with a number of columns of order the stable rank of |$A$|⁠. This improves on existing results by removing an extra logarithmic factor in the size of the extracted matrix. Our proof uses the recent solution of the Kadison–Singer problem. We also develop...

In this paper, we study the effect of sparsity on the appearance of outliers in the semi-circular law. Let $(W_n)_{n=1}^\infty$ be a sequence of random symmetric matrices such that each $W_n$ is $n\times n$ with i.i.d entries above and on the main diagonal equidistributed with the product $b_n\xi$, where $\xi$ is a real centered uniformly bounded r...

We derive a lower bound on the smallest singular value of a random $d$-regular matrix, that is, the adjacency matrix of a random $d$-regular directed graph. More precisely, let $C_1<d< c_1 n/\log^2 n$ and let $\mathcal{M}_{n,d}$ be the set of all $0/1$-valued square $n\times n$ matrices such that each row and each column of a matrix $M\in \mathcal{...

For any α ∈ (0, 1) and any n α ≤ d ≤ n/2, we show that λ(G) ≤ C α √d with probability at least 1 - 1/n, where G is the uniform random undirected d-regular graph on n vertices, λ(G) denotes its second largest eigenvalue (in absolute value) and C α is a constant depending only on α. Combined with earlier results in this direction covering the case of...

Let $X$ be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that $$ \mathbf{E}\|X\|_{S_p} \asymp \mathbf{E}\Bigg[ \Bigg(\sum_i\Bigg(\sum_j X_{ij}^2\Bigg)^{p/2}\Bigg)^{1/p} \Bigg] $$ for any $2\le p\le\infty$, where $S_p$ denotes the $p$-Schatten class and the constants are universal. The...

Let $n$ be a large integer, let $d$ satisfy $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in {\mathcal C}$. Further, denote by $M$ the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. In this paper, we study structure of the kernel of submatrices of $M-z\,{\rm Id}$, formed by removing a...

Let $d$ be a fixed large integer. For any $n$ larger than $d$, let $A_n$ be the adjacency matrix of the random directed $d$-regular graph on $n$ vertices, with the uniform distribution. We show that $A_n$ has rank at least $n-1$ with probability going to one as $n$ goes to infinity. The proof combines the method of simple switchings and a recent re...

Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices. We show that, as long as $d\to\infty$ with $n$, the empirical spectral distribution of appropriately rescaled matrix $A_n$ converges weakly in probabili...

Let $d$ be a fixed large integer. For any $n$ larger than $d$, let $A_n$ be the adjacency matrix of the random directed $d$-regular graph on $n$ vertices, with the uniform distribution. We show that $A_n$ has rank at least $n-1$ with probability going to one as $n$ goes to infinity. The proof combines the method of simple switchings and a recent re...

Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices. We show that, as long as $d\to\infty$ with $n$, the empirical spectral distribution of appropriately rescaled matrix $A_n$ converges weakly in probabili...

Let $d$ and $n$ be integers satisfying $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in \mathbb{C}$. Denote by $M$ the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. In this paper, we study the structure of the kernel of submatrices of $M-z\,{\rm Id}$, formed by removing a subset of ro...

Let $X$ be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that $$ \mathbf{E}\|X\|_{S_p} \asymp \mathbf{E}\Bigg[ \Bigg(\sum_i\Bigg(\sum_j X_{ij}^2\Bigg)^{p/2}\Bigg)^{1/p} \Bigg] $$ for any $2\le p\le\infty$, where $S_p$ denotes the $p$-Schatten class and the constants are universal. The...

Suppose that \(m,n \in \mathbb{N}\) and that \(A: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}\) is a linear operator. It is shown here that if \(k,r \in \mathbb{N}\) satisfy \(k <r\leqslant \mathbf{rank}(A)\) then there exists a subset σ ⊆ {1, …, m} with | σ | = k such that the restriction of A to \(\mathbb{R}^{\sigma } \subseteq \mathbb{R}^{m}\) is...

We derive a lower bound on the smallest singular value of a random $d$-regular matrix, that is, the adjacency matrix of a random $d$-regular directed graph. More precisely, let $C_1<d< c_1 n/\log^2 n$ and let $\mathcal{M}_{n,d}$ be the set of all $0/1$-valued square $n\times n$ matrices such that each row and each column of a matrix $M\in \mathcal{...

We connect this question to a problem of estimating the probability that the image of certain random matrices does not intersect with a subset of the unit sphere Sⁿ⁻¹. In this way, the case of a discretized Brownian motion is related to Gordon's escape theorem dealing with standard Gaussian matrices. We show that for the random walk BMn(i), i ∈ N,...

In this paper, we address the problem of estimating the spectral gap of random $d$-regular graphs distributed according to the uniform model. Namely, let $\alpha>0$ be any positive number and let integers $d$ and $n$ satisfy $n^\alpha\leq d\leq n/2$. Let $G$ be uniformly distributed on the set of all $d$-regular simple undirected graphs on $n$ vert...

In this note, we show that the norm of an $n\times n$ random jointly exchangeable matrix with zero diagonal can be estimated in terms of the norm of its $n/2\times n/2$ submatrix located in the top right corner. As a consequence, we prove a relation between the second largest singular values of a random matrix with constant row and column sums and...

We exploit the recent solution of the Kadison-Singer problem to show that any $n\times m$ matrix $A$ can be approximated in operator norm by a submatrix with a number of columns of order the stable rank of $A$. This improves on existing results by removing an extra logarithmic factor in the size of the extracted matrix. We develop a sort of tensori...

Let be the set of all directed d-regular graphs on n vertices. Let G be a graph chosen uniformly at random from and M be its adjacency matrix. We show that M is invertible with probability at least for , where are positive absolute constants. To this end, we establish a few properties of directed d-regular graphs. One of them, a Littlewood–Offord-t...

Suppose that $m,n\in \mathbb{N}$ and that $A:\mathbb{R}^m\to \mathbb{R}^n$ is a linear operator. It is shown here that if $k,r\in \mathbb{N}$ satisfy $k<r\le \mathrm{\bf rank(A)}$ then there exists a subset $\sigma\subseteq \{1,\ldots,m\}$ with $|\sigma|=k$ such that the restriction of $A$ to $\mathbb{R}^{\sigma}\subseteq \mathbb{R}^m$ is invertibl...

Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability at least $1-C\ln^{3} d/\sqrt{d}$ for $C\leq d\leq cn/\ln^2 n$, where $c, C$ are positive absolute constants....

In this paper we obtain a Bernstein type inequality for the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality can be viewed as an extension to the matrix setting of the Bernstein-type inequality obtained by Merlev\`ede et al. (2009) in the context of real-valued bounde...

For some absolute constants $c$, $n_0$ and any $n\geq n_0$, we show that with probability close to one the convex hull of the $n$-dimensional Brownian motion ${\rm conv}\{BM_n(t):\, t\in[1,2^{cn}]\}$ does not contain the origin. The result can be interpreted as an estimate of the minimax of the Gaussian process $\{ \langle \bar{u},BM_n(t)\rangle,\,...

Motivated by a question of Eldan, we prove that for certain random walks in $\mathbb{R}^n$, the number of steps required to get the origin in the interior of their convex hull is less than $\exp(Cn)$ with high probability. These estimates are consequences of some general statements about random matrices, closely related to Gordon's escape theorem.

Given $U$ an $n\times m$ matrix of rank $n$ and $V$ block of columns inside $U$, we consider the problem of extracting a block of columns of rank $n$ which minimize the Hilbert-Schmidt norm of the inverse while preserving the block $V$. This generalizes a previous result of Gluskin-Olevskii, and improves the estimates when given a "good" block $V$.

In this thesis, we address three themes : columns subset selection in a matrix, the Banach-Mazur distance to the cube and the estimation of the covariance of random matrices. Although the three themes seem distant, the techniques used are similar throughout the thesis. In the first place, we generalize the restricted invertibility principle of Boug...

We extend to the matrix setting a recent result of Srivastava-Vershynin about estimating the covariance matrix of a random vector. The result can be in- terpreted as a quantified version of the law of large numbers for positive semi-definite matrices which verify some regularity assumption. Beside giving examples, we dis- cuss the notion of log-con...

Given a matrix U, using a deterministic method, we extract a "large" submatrix of U'(whose columns are obtained by normalizing those of U) and estimate its smallest and largest singular value. We apply this result to the study of contact points of the unit ball with its maximal volume ellipsoid. We consider also the paving problem and give a determ...

We prove a normalized version of the restricted invertibility principle obtained by Spielman-Srivastava. Applying this result, we get a new proof of the proportional Dvoretzky-Rogers factorization theorem recovering the best current estimate. As a consequence, we also recover the best known estimate for the Banach-Mazur distance to the cube: the di...