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Radosław Adamczak

Publications and source records attributed to Radosław Adamczak.

46 records · Page 3Linked to original sources
arXiv2011

Chevet type inequality and norms of submatrices

We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of expectation of the supremum of "symmetric exponential" processes compared to the Gaussian ones in the Chevet inequality. This is used to give sharp upper estimate for a quantity $Γ_{k,m}$ that controls uniformly the Euclidean operator norm of the sub-matrices with $k$ rows and $m$ columns of an isotropic log-concave unconditional random matrix. We apply these estimates to give a sharp bound for the Restricted Isometry Constant of a random matrix with independent log-concave unconditional rows. We show also that our Chevet type inequality does not extend to general isotropic log-concave random matrices.

math.PR↗
arXiv2011

Geometry of log-concave Ensembles of random matrices and approximate reconstruction

We study the Restricted Isometry Property of a random matrix $Γ$ with independent isotropic log-concave rows. To this end, we introduce a parameter $Γ_{k,m}$ that controls uniformly the operator norm of sub-matrices with $k$ rows and $m$ columns. This parameter is estimated by means of new tail estimates of order statistics and deviation inequalities for norms of projections of an isotropic log-concave vector.

math.PR↗
arXiv2010

Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails

We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by independent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.

math.PR↗
arXiv2009

Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles

Let $K$ be an isotropic convex body in $\R^n$. Given $\eps>0$, how many independent points $X_i$ uniformly distributed on $K$ are needed for the empirical covariance matrix to approximate the identity up to $\eps$ with overwhelming probability? Our paper answers this question posed by Kannan, Lovasz and Simonovits. More precisely, let $X\in\R^n$ be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector $X$ is a random point in an isotropic convex body. We show that for any $\eps>0$, there exists $C(\eps)>0$, such that if $N\sim C(\eps) n$ and $(X_i)_{i\le N}$ are i.i.d. copies of $X$, then $ \Big\|\frac{1}{N}\sum_{i=1}^N X_i\otimes X_i - \Id\Big\| \le ε, $ with probability larger than $1-\exp(-c\sqrt n)$.

math.PR↗
arXiv2009

Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling

This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors $\pm X_1,...,\pm X_N\in\R^n$, ($N\ge n$). We introduce a class of random sampling matrices and show that they satisfy a restricted isometry property (RIP) with overwhelming probability. In particular, we prove that matrices with i.i.d. centered and variance 1 entries that satisfy uniformly a sub-exponential tail inequality possess this property RIP with overwhelming probability. We show that such "sensing" matrices are valid for the exact reconstruction process of $m$-sparse vectors via $\ell_1$ minimization with $m\le Cn/\log^2 (cN/n)$. The class of sampling matrices we study includes the case of matrices with columns that are independent isotropic vectors with log-concave densities. We deduce that if $K\subset \R^n$ is a convex body and $X_1,..., X_N\in K$ are i.i.d. random vectors uniformly distributed on $K$, then, with overwhelming probability, the symmetric convex hull of these points is an $m$-centrally-neighborly polytope with $m\sim n/\log^2 (cN/n)$.

math.PR↗
arXiv2008

The LIL for canonical $U$-statistics

We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for canonical $U$-statistics of arbitrary order $d$, extending the previously known results for $d=2$. The nasc's are expressed as growth conditions on a parameterized family of norms associated with the $U$-statistics kernel.

math.PR↗
arXiv2008

A few remarks on the operator norm of random Toeplitz matrices

We present some results concerning the almost sure behaviour of the operator norm or random Toeplitz matrices, including the law of large numbers for the norm, normalized by its expectation (in the i.i.d. case). As tools we present some concentration inequalities for suprema of empirical processes, which are refinements of recent results by Einmahl and Li.

math.PR↗
arXiv2007

The LIL for $U$-statistics in Hilbert spaces

We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for $U$-statistics in Hilbert spaces. As a tool we also develop moment and tail estimates for canonical Hilbert-space valued $U$-statistics of arbitrary order, which are of independent interest.

math.PR↗
arXiv2007

Moment inequalities for U-statistics

We present moment inequalities for completely degenerate Banach space valued (generalized) U-statistics of arbitrary order. The estimates involve suprema of empirical processes which, in the real-valued case, can be replaced by simpler norms of the kernel matrix (i.e., norms of some multilinear operators associated with the kernel matrix). As a corollary, we derive tail inequalities for U-statistics with bounded kernels and for some multiple stochastic integrals.

math.PR↗