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Konstantin Tikhomirov

Publications and source records attributed to Konstantin Tikhomirov.

63 records · Page 4Linked to original sources
arXiv2016

On the norm of a random jointly exchangeable matrix

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 its top right $n/2\times n/2$ submatrix. The result has an application to estimating the spectral gap of random undirected $d$-regular graphs in terms of the second singular value of {\it directed} random graphs with predefined degree sequences.

math.PR↗
arXiv2016

Adjacency matrices of random digraphs: singularity and anti-concentration

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. To this end, we establish a few properties of $d$-regular directed graphs. One of them, a Littlewood-Offord type anti-concentration property, is of independent interest. Let $J$ be a subset of vertices of $G$ with $|J|\approx n/d$. Let $δ_i$ be the indicator of the event that the vertex $i$ is connected to $J$ and define $δ= (δ_1, δ_2, ..., δ_n)\in \{0, 1\}^n$. Then for every $v\in\{0,1\}^n$ the probability that $δ=v$ is exponentially small. This property holds even if a part of the graph is "frozen".

math.PR↗
arXiv2016

Illumination of convex bodies with many symmetries

Let $n\geq C$ for a large universal constant $C>0$, and let $B$ be a convex body in $R^n$ such that for any $(x_1,x_2,\dots,x_n)\in B$, any choice of signs $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_n\in\{-1,1\}$ and for any permutation $σ$ on $n$ elements we have $(\varepsilon_1x_{σ(1)},\varepsilon_2x_{σ(2)},\dots,\varepsilon_nx_{σ(n)})\in B$. We show that if $B$ is not a cube then $B$ can be illuminated by strictly less than $2^n$ sources of light. This confirms the Hadwiger--Gohberg--Markus illumination conjecture for unit balls of $1$-symmetric norms in $R^n$ for all sufficiently large $n$.

math.MG↗
arXiv2016

Sample covariance matrices of heavy-tailed distributions

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 independent copies of $X$, and $Σ_N:=\frac{1}{N}\sum_{i=1}^N X_i {X_i}^T$ be the sample covariance matrix. We prove that $$K^{-1}\|Σ_N-I_n\|_{2\to 2}\leq\frac{1}{N}\max\limits_{i\leq N}\|X_i\|^2 +\Bigl(\frac{n}{N}\Bigr)^{1-2/p}\log^4\frac{N}{n}+\Bigl(\frac{n}{N}\Bigr)^{1-2/\min(p,4)}$$ with probability at least $1-\frac{1}{n}$, where $K>0$ depends only on $B$ and $p$. In particular, for all $p>4$ we obtain a quantitative Bai-Yin type theorem.

math.PR↗
arXiv2015

When does a discrete-time random walk in $\mathbb{R}^n$ absorb the origin into its convex hull?

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 $\mathbb{S}^{n-1}$. In this way, the case of a discretized Brownian motion is related to Gordon's escape theorem dealing with standard Gaussian matrices. The approach allows us to prove that with high probability, the $π/2$-covering time of certain random walks on $\mathbb{S}^{n-1}$ is of order $n$. For certain spherical simplices on $\mathbb{S}^{n-1}$, we extend the "escape" phenomenon to a broad class of random matrices; as an application, we show that $e^{Cn}$ steps are sufficient for the standard walk on $\mathbb{Z}^n$ to absorb the origin into its convex hull with a high probability.

math.PR↗
arXiv2015

On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence

Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko--Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refinement of the rank one update approach used by Srivastava and Vershynin to produce non-asymptotic quantitative estimates. In other words we provide a new proof of the Bai and Yin theorem using basic tools from probability theory and linear algebra, together with a new extension of this theorem to random matrices with dependent entries.

math.PR↗
arXiv2015

Minimax of an n-dimensional Brownian motion

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 $\{ \langle \bar{u},BM_n(t)\rangle,\, \bar{u}\in S^{n-1},\, t\in [1,2^{cn}]\}$.

math.PR↗
arXiv2014

The limit of the smallest singular value of random matrices with i.i.d. entries

Let $\{a_{ij}\}$ $(1\le i,j<\infty)$ be i.i.d. real valued random variables with zero mean and unit variance and let an integer sequence $(N_m)_{m=1}^\infty$ satisfy $m/N_m\longrightarrow z$ for some $z\in(0,1)$. For each $m\in{\mathbb N}$ denote by $A_m$ the $N_m\times m$ random matrix $(a_{ij})$ $(1\le i\le N_m,1\le j\le m)$ and let $s_{m}(A_m)$ be its smallest singular value. We prove that the sequence $\bigl({N_m}^{-1/2} s_{m}(A_m)\bigr)_{m=1}^\infty$ converges to $1-\sqrt{z}$ almost surely. Our result does not require boundedness of any moments of $a_{ij}$'s higher than the $2$-nd and resolves a long standing question regarding the weakest moment assumptions on the distribution of the entries sufficient for the convergence to hold.

math.PR↗