arXiv subjects

Konstantin Tikhomirov

Publications and source records attributed to Konstantin Tikhomirov.

At least 55 records · Page 3Linked to original sources

arXiv2019

On delocalization of eigenvectors of random non-Hermitian matrices

We study delocalization of null vectors and eigenvectors of random matrices with i.i.d entries. Let $A$ be an $n\times n$ random matrix with i.i.d real subgaussian entries of zero mean and unit variance. We show that with probability at least $1-e^{-\log^{2} n}$ $$ \min\limits_{I\subset[n],\,|I|= m}\|{\bf v}_I\| \geq \frac{m^{3/2}}{n^{3/2}\log^Cn}\|{\bf v}\| $$ for any real eigenvector ${\bf v}$ and any $m\in[\log^C n,n]$, where ${\bf v}_I$ denotes the restriction of ${\bf v}$ to $I$. Further, when the entries of $A$ are complex, with i.i.d real and imaginary parts, we show that with probability at least $1-e^{-\log^{2} n}$ all eigenvectors of $A$ are delocalized in the sense that $$ \min\limits_{I\subset[n],\,|I|= m}\|{\bf v}_I\| \geq \frac{m}{n\log^Cn}\|{\bf v}\| $$ for all $m\in[\log^C{n},n]$. Comparing with related results, in the range $m\in[\log^{C'}{n},n/\log^{C'}{n}]$ in the i.i.d setting and with weaker probability estimates, our lower bounds on $\|{\bf v}_I\|$ strengthen an earlier estimate $\min\limits_{|I|= m}\|{\bf v}_I\| \geq c(m/n)^6\|{\bf v}\| $ obtained in [M. Rudelson, R. Vershynin, Geom. Func. Anal., 2016], and bounds $\min\limits_{|I|= m}\|{\bf v}_I\| \geq c(m/n)^2\|{\bf v}\|$ (in the real setting) and $\min\limits_{|I|= m}\|{\bf v}_I\| \geq c(m/n)^{3/2}\|{\bf v}\|$ (in the complex setting) established in [K. Luh, S. O'Rourke, arXiv:1810.00489]. As the case of real and complex Gaussian matrices shows, our bounds are optimal up to the polylogarithmic multiples. We derive stronger estimates without the polylogarithmic error multiples for null vectors of real $(n-1)\times n$ random matrices.

math.PR↗

arXiv2019

Singularity of random Bernoulli matrices

For each $n$, let $M_n$ be an $n\times n$ random matrix with independent $\pm 1$ entries. We show that ${\mathbb P}\{\mbox{$M_n$ is singular}\}=(1/2+o_n(1))^n$, which settles an old problem. Some generalizations are considered.

math.PR↗

arXiv2019

Small ball probability for the condition number of random matrices

Let $A$ be an $n\times n$ random matrix with i.i.d. entries of zero mean, unit variance and a bounded subgaussian moment. We show that the condition number $s_{\max}(A)/s_{\min}(A)$ satisfies the small ball probability estimate $${\mathbb P}\big\{s_{\max}(A)/s_{\min}(A)\leq n/t\big\}\leq 2\exp(-c t^2),\quad t\geq 1,$$ where $c>0$ may only depend on the subgaussian moment. Although the estimate can be obtained as a combination of known results and techniques, it was not noticed in the literature before. As a key step of the proof, we apply estimates for the singular values of $A$, ${\mathbb P}\big\{s_{n-k+1}(A)\leq ck/\sqrt{n}\big\}\leq 2 \exp(-c k^2), \quad 1\leq k\leq n,$ obtained (under some additional assumptions) by Nguyen.

math.PR↗

arXiv2019

Outliers in spectrum of sparse Wigner matrices

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 $b_nξ$, where $ξ$ is a real centered uniformly bounded random variable of unit variance and $b_n$ is an independent Bernoulli random variable with a probability of success $p_n$. Assuming that $\lim\limits_{n\to\infty}n p_n=\infty$, we show that for the random sequence $(ρ_n)_{n=1}^\infty$ given by $$ρ_n:=θ_n+\frac{n p_n}{θ_n},\quad θ_n:=\sqrt{\max\big(\max\limits_{i\leq n}\|{\rm Row_i}(W_n)\|_2^2-np_n,n p_n\big)},$$ the ratio $\frac{\|W_n\|}{ρ_n}$ converges to one in probability. A non-centered counterpart of the theorem allows to obtain asymptotic expressions for eigenvalues of the Erdős--Renyi graphs, which were unknown in the regime $n p_n=Θ(\log n)$. In particular, denoting by $A_n$ the adjacency matrix of $\mathcal{G}(n,p_n)$ and by $λ_{|k|}(A_n)$ its $k$-th largest (by the absolute value) eigenvalue, under the assumptions $\lim\limits_{n\to\infty }n p_n=\infty$ and $\lim\limits_{n\to\infty}p_n=0$ we have: -(No non-trivial outliers) If $\liminf\frac{n p_n}{\log n}\geq\frac{1}{\log (4/e)}$ then for any fixed $k\geq2$, $\frac{|λ_{|k|}(A_n)|}{2\sqrt{n p_n}}$ converges to $1$ in probability. -(Outliers) If $\limsup\frac{n p_n}{\log n}<\frac{1}{\log (4/e)}$ then there is $\varepsilon>0$ such that for any $k\in\mathbb{N}$, we have $\lim\limits_{n\to\infty}\mathbb{P}\Big\{\frac{|λ_{|k|}(A_n)|}{2\sqrt{n p_n}}>1+\varepsilon\Big\}=1$. On a conceptual level, our result highlights similarities in appearance of outliers in spectrum of sparse matrices and the so-called BBP phase transition phenomenon in deformed Wigner matrices.

math.PR↗

arXiv2019

The sparse circular law under minimal assumptions

The circular law asserts that the empirical distribution of eigenvalues of appropriately normalized $n\times n$ matrix with i.i.d. entries converges to the uniform measure on the unit disc as the dimension $n$ grows to infinity. Consider an $n\times n$ matrix $A_n=(δ_{ij}^{(n)}ξ_{ij}^{(n)})$, where $ξ_{ij}^{(n)}$ are copies of a real random variable of unit variance, variables $δ_{ij}^{(n)}$ are Bernoulli ($0/1$) with ${\mathbb P}\{δ_{ij}^{(n)}=1\}=p_n$, and $δ_{ij}^{(n)}$ and $ξ_{ij}^{(n)}$, $i,j\in[n]$, are jointly independent. In order for the circular law to hold for the sequence $\big(\frac{1}{\sqrt{p_n n}}A_n\big)$, one has to assume that $p_n n\to \infty$. We derive the circular law under this minimal assumption.

math.PR↗

arXiv2019

Estimates for order statistics in terms of quantiles

Let $X_1, \ldots, X_n$ be independent non-negative random variables with cumulative distribution functions $F_1,F_2,\ldots,F_n$, each satisfying certain (rather mild) conditions. We show that the median of $k$-th smallest order statistic of the vector $(X_1, \ldots, X_n)$ is equivalent to the quantile of order $(k-1/2)/n$ with respect to the averaged distribution $F=\frac{1}{n}\sum_{i=1}^n F_i$.

math.PR↗

arXiv2018

Structure of eigenvectors of random regular digraphs

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 rows. We show that with large probability the kernel consists of two non-intersecting types of vectors, which we call very steep and gradual with many levels. As a corollary, we show, in particular, that every eigenvector of $M$, except for constant multiples of $(1,1,\dots,1)$, possesses a weak delocalization property: its level sets have cardinality less than $Cn\ln^2 d/\ln n$. For a large constant $d$ this provides a principally new structural information on eigenvectors, implying that the number of their level sets grows to infinity with $n$. As a key technical ingredient of our proofs we introduce a decomposition of $\mathbb{C}^n$ into vectors of different degrees of `structuredness', which is an alternative to the decomposition based on the least common denominator in the regime when the underlying random matrix is very sparse.

math.PR↗

arXiv2018

The rank of random regular digraphs of constant degree

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 result of the authors on delocalization of eigenvectors of $A_n$.

math.PR↗

arXiv2018

The smallest singular value of a shifted $d$-regular random square matrix

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 $M\in \mathcal{M}_{n,d}$ has exactly $d$ ones. Let $M$ be uniformly distributed on $\mathcal{M}_{n,d}$. Then the smallest singular value $s_{n} (M)$ of $M$ is greater than $c_2 n^{-6}$ with probability at least $1-C_2\log^2 d/\sqrt{d}$, where $c_1$, $c_2$, $C_1$, and $C_2$ are absolute positive constants independent of any other parameters.

math.PR↗

arXiv2018

On the spectral radius of a random matrix: an upper bound without fourth moment

Consider a square matrix with independent and identically distributed entries of zero mean and unit variance. It is well known that if the entries have a finite fourth moment, then, in high dimension, with high probability, the spectral radius is close to the square root of the dimension. We conjecture that this holds true under the sole assumption of zero mean and unit variance, in other words that there are no outliers in the circular law. In this work we establish the conjecture in the case of symmetrically distributed entries with a finite moment of order larger than two. The proof uses the method of moments combined with a novel truncation technique for cycle weights that might be of independent interest.

math.CO↗

arXiv2018

On the Banach-Mazur distance to cross-polytope

Let $n\geq 3$, and let $B_1^n$ be the standard $n$-dimensional cross-polytope (i.e. the convex hull of standard coordinate vectors and their negatives). We show that there exists a symmetric convex body $\mathcal G_m$ in ${\mathbb R}^n$ such that the Banach--Mazur distance $d(B_1^n,\mathcal G_m)$ satisfies $d(B_1^n,\mathcal G_m)\geq n^{5/9}\log^{-C}n$, where $C>0$ is a universal constant. The body $\mathcal G_m$ is obtained as a typical realization of a random polytope in ${\mathbb R}^n$ with $2m:=2n^C$ vertices (for a large constant $C$). The result improves upon an earlier estimate of S.Szarek which gives $d(B_1^n,\mathcal G_m)\geq c n^{1/2}\log n$ (with a different choice of $m$). This shows in a strong sense that the cross-polytope (or the cube $[-1,1]^n$) cannot be an "approximate" center of the Minkowski compactum.

math.MG↗

arXiv2018

Circular law for sparse random regular digraphs

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 probability to the circular law. This result, together with an earlier work of Cook, completely settles the problem of weak convergence of the empirical distribution in directed $d$-regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of $A_n$ based on studying random normals to rowspaces and on constructing a product structure to deal with the lack of independence between the matrix entries.

math.PR↗

arXiv2017

Cube is a strict local maximizer for the illumination number

It was conjectured by Levi, Hadwiger, Gohberg and Markus that the boundary of any convex body in ${\mathbb R}^n$ can be illuminated by at most $2^n$ light sources, and, moreover, $2^n-1$ light sources suffice unless the body is a parallelotope. We show that if a convex body is close to the cube in the Banach-Mazur metric, and it is not a parallelotope, then indeed $2^n-1$ light sources suffice to illuminate its boundary. Equivalently, any convex body sufficiently close to the cube, but not isometric to it, can be covered by $2^n-1$ smaller homothetic copies of itself.

math.MG↗

arXiv2017

The variance of the $\ell_p^n$-norm of the Gaussian vector, and Dvoretzky's theorem

Let $n$ be a large integer, and let $G$ be the standard Gaussian vector in $R^n$. Paouris, Valettas and Zinn (2015) showed that for all $p\in[1,c\log n]$, the variance of the $\ell_p^n$--norm of $G$ is equivalent, up to a constant multiple, to $\frac{2^p}{p}n^{2/p-1}$, and for $p\in[C\log n,\infty]$, $\mathbb{Var}\|G\|_p\simeq (\log n)^{-1}$. Here, $C,c>0$ are universal constants. That result left open the question of estimating the variance for $p$ logarithmic in $n$. In this note, we resolve the question by providing a complete characterization of $\mathbb{Var}\|G\|_p$ for all $p$. We show that there exist two transition points (windows) in which behavior of $\mathbb{Var}\|G\|_p$, viewed as a function of $p$, significantly changes. We also discuss some implications of our result in context of random Dvoretzky's theorem for $\ell_p^n$.

math.FA↗

arXiv2017

Order statistics of vectors with dependent coordinates, and the Karhunen-Loève basis

Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not identically distributed coordinates and let $T$ be an orthogonal trasformation of $\mathbb R^n$. We show that the random vector $Y=T(X)$ satisfies $$\mathbb E\sum\limits_{j=1}^k j\mbox{-}\min_{i\leq n}{X_{i}}^2 \leq C\mathbb E\sum\limits_{j=1}^k j\mbox{-}\min_{i\leq n}{Y_{i}}^2$$ for all $k 0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni regarding optimality of the Karhunen-Loeve basis for the nonlinear signal approximation. As a by-product we obtain some relations for order statistics of random vectors (not only Gaussian) which are of independent interest.

math.PR↗

arXiv2017

Superconcentration, and randomized Dvoretzky's theorem for spaces with 1-unconditional bases

Let $n$ be a sufficiently large natural number and let $B$ be an origin-symmetric convex body in $R^n$ in the $\ell$-position, and such that the normed space $(R^n,\|\cdot\|_B)$ admits a $1$-unconditional basis. Then for any $\varepsilon\in(0,1/2]$, and for random $c\varepsilon\log n/\log\frac{1}{\varepsilon}$-dimensional subspace $E$ distributed according to the rotation-invariant (Haar) measure, the section $B\cap E$ is $(1+\varepsilon)$-Euclidean with probability close to one. This shows that the "worst-case" dependence on $\varepsilon$ in the randomized Dvoretzky theorem in the $\ell$-position is significantly better than in John's position. It is a previously unexplored feature, which has strong connections with the concept of superconcentration introduced by S. Chatterjee. In fact, our main result follows from the next theorem: Let $B$ be as before and assume additionally that $B$ has a smooth boundary and ${\mathbb E}_{γ_n}\|\cdot\|_B\leq n^c\,{\mathbb E}_{γ_n}\big\|{\rm grad}_B(\cdot)\big\|_2$ for a small universal constant $c>0$, where ${\rm grad}_B(\cdot)$ is the gradient of $\|\cdot\|_B$ and $γ_n$ is the standard Gaussian measure in $R^n$. Then for any $p\in[1,c\log n]$ the $p$-th power of the norm $\|\cdot\|_B^p$ is $\frac{C}{\log n}$--superconcentrated in the Gauss space.

math.MG↗

arXiv2017

Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries

Let $A=(a_{ij})$ be an $n\times n$ random matrix with i.i.d. entries such that $\mathbb{E} a_{11} = 0$ and $\mathbb{E} {a_{11}}^2 = 1$. We prove that for any $δ>0$ there is $L>0$ depending only on $δ$, and a subset $\mathcal{N}$ of $B_2^n$ of cardinality at most $\exp(δn)$ such that with probability very close to one we have $$A(B_2^n)\subset \bigcup_{y\in A(\mathcal{N})}\bigl(y+L\sqrt{n}B_2^n\bigr).$$ As an application, we show that for some $L'>0$ and $u\in[0,1)$ depending only on the distribution law of $a_{11}$, the smallest singular value $s_n$ of the matrix $A$ satisfies $\mathbb{P}\{s_n(A)\le \varepsilon n^{-1/2}\}\le L'\varepsilon+u^n$ for all $\varepsilon>0$. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.

math.PR↗

arXiv2016

The spectral gap of dense random regular graphs

For any $α\in (0,1)$ and any $n^α\leq d\leq n/2$, we show that $λ(G)\leq C_α\sqrt{d}$ with probability at least $1-\frac{1}{n}$, where $G$ is the uniform random $d$-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 sparse random graphs, this completely settles the problem of estimating the magnitude of $λ(G)$, up to a multiplicative constant, for all values of $n$ and $d$, confirming a conjecture of Vu. The result is obtained as a consequence of an estimate for the second largest singular value of adjacency matrices of random {\it directed} graphs with predefined degree sequences. As the main technical tool, we prove a concentration inequality for arbitrary linear forms on the space of matrices, where the probability measure is induced by the adjacency matrix of a random directed graph with prescribed degree sequences. The proof is a non-trivial application of the Freedman inequality for martingales, combined with boots-trapping and tensorization arguments. Our method bears considerable differences compared to the approach used by Broder, Frieze, Suen and Upfal (1999) who established the upper bound for $λ(G)$ for $d=o(\sqrt{n})$, and to the argument of Cook, Goldstein and Johnson (2015) who derived a concentration inequality for linear forms and estimated $λ(G)$ in the range $d= O(n^{2/3})$ using size-biased couplings.

math.PR↗