Interpolating Thin-Shell and Sharp Large-Deviation Estimates For Isotropic Log-Concave Measures

Given an isotropic random vector $X$ with log-concave density in Euclidean space \Real^n, we study the concentration properties of $|X|$ on all scales, both above and below its expectation. We show in particular that: \P(\abs{|X| -\sqrt{n}} \geq t \sqrt{n}) \leq C \exp(-c n^{1/2} \min(t^3,t)) \;\;\; \forall t \geq 0 ~, for some universal constants $c,C>0$. This improves the best known deviation results on the thin-shell and mesoscopic scales due to Fleury and Klartag, respectively, and recovers the sharp large-deviation estimate of Paouris. Another new feature of our estimate is that it improves when $X$ is $\psi_\alpha$ ($\alpha \in (1,2]$), in precise agreement with Paouris' estimates. The upper bound on the thin-shell width \sqrt{\Var(|X|)} we obtain is of the order of $n^{1/3}$, and improves down to $n^{1/4}$ when $X$ is $\psi_2$. Our estimates thus continuously interpolate between a new best known thin-shell estimate and the sharp large-deviation estimate of Paouris. As a consequence, a new best known bound on the Cheeger isoperimetric constant appearing in a conjecture of Kannan--Lov\'asz--Simonovits is deduced.Comment: 29 pages - formulation is now general, estimating deviation of a linear image of X, and dependence on the \psi_\alpha constant is explicit. Corrected typos and refined explanations. To appear in GAF

Thin-shell concentration for convex measures

We prove that for $s<0$, $s$-concave measures on ${\mathbb R}^n$ satisfy a thin shell concentration similar to the log-concave one. It leads to a Berry-Esseen type estimate for their one dimensional marginal distributions. We also establish sharp reverse H\"older inequalities for $s$-concave measures

Community detection in sparse networks via Grothendieck's inequality

We present a simple and flexible method to prove consistency of semidefinite optimization problems on random graphs. The method is based on Grothendieck's inequality. Unlike the previous uses of this inequality that lead to constant relative accuracy, we achieve any given relative accuracy by leveraging randomness. We illustrate the method with the problem of community detection in sparse networks, those with bounded average degrees. We demonstrate that even in this regime, various simple and natural semidefinite programs can be used to recover the community structure up to an arbitrarily small fraction of misclassified vertices. The method is general; it can be applied to a variety of stochastic models of networks and semidefinite programs.Comment: This is the final version, incorporating the referee's comment

Random polytopes obtained by matrices with heavy tailed entries

Let $\Gamma$ be an $N\times n$ random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope $\Gamma^* B_1^N$ in $\mathbb{R}$ (the absolute convex hull of rows of $\Gamma$). In particular, we show that $$\Gamma B_1^N \supset b^{-1} \left( B_{\infty}^n \cap \sqrt{\ln (N/n)}\, B_2^n \right).$$ where $b$ depends only on parameters in small ball inequality. This extends results of \cite{LPRT} and recent results of \cite{KKR}. This inclusion is equivalent to so-called $\ell_1$-quotient property and plays an important role in compressive sensing (see \cite{KKR} and references therein).Comment: Last version, to appear in Communications in Contemporary Mathematic

Random polytopes obtained by matrices with heavy tailed entries

Let $\Gamma$ be an $N\times n$ random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope $\Gamma^* B_1^N$ in $\mathbb{R}$ (the absolute convex hull of rows of $\Gamma$). In particular, we show that $$\Gamma B_1^N \supset b^{-1} \left( B_{\infty}^n \cap \sqrt{\ln (N/n)}\, B_2^n \right).$$ where $b$ depends only on parameters in small ball inequality. This extends results of \cite{LPRT} and recent results of \cite{KKR}. This inclusion is equivalent to so-called $\ell_1$-quotient property and plays an important role in compressive sensing (see \cite{KKR} and references therein).Comment: Last version, to appear in Communications in Contemporary Mathematic

Characterizing the balance between ontogeny and environmental constraints in forest tree development using growth phase duration distributions

ISBN 978-951-651-408-9International audienceWe built segmentation models to identify tree growth phases on the basis of retrospective measurement of annual shoot characteristics along the main stem. Growth phase duration distributions estimated within these models characterize the balance between ontogeny and environmental constraints in tree development at the population scale. These distributions have very contrasted characteristics in terms of shape and relative dispersion between ontogeny-driven and environment-driven tree development