Alain Pajor
The Gelfand widths of $\ell_p$-balls for $0<p\leq 1$.
Thu, 02/04/2010 - 16:01 — SomNambulAuthors: Alain Pajor, Holger Rauhut, Simon Foucart, Tino Ullrich
Subjects: Functional Analysis
AbstractWe provide sharp lower and upper bounds for the Gelfand widths of
$\ell_p$-balls in the $N$-dimensional $\ell_q^N$-space for $0<p\leq 1$ and $p<q
\leq 2$. Such estimates are highly relevant to the novel theory of compressive
sensing, and our proofs rely on methods from this area.Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles.
Fri, 08/28/2009 - 23:56 — SomNambulAuthors: Radosław Adamczak, Alexander E. Litvak, Alain Pajor, Nicole Tomczak-Jaegermann
Subjects: Probability
AbstractLet $K$ 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.
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