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

Local Algorithms for Block Models with Side Information

There has been a recent interest in understanding the power of local algorithms for optimization and inference problems on sparse graphs. Gamarnik and Sudan (2014) showed that local algorithms are weaker than global algorithms for finding large independent sets in sparse random regular graphs. Montanari (2015) showed that local algorithms are suboptimal for finding a community with high connectivity in the sparse Erd\H{o}s-R\'enyi random graphs. For the symmetric planted partition problem (also named community detection for the block models) on sparse graphs, a simple observation is that local algorithms cannot have non-trivial performance. In this work we consider the effect of side information on local algorithms for community detection under the binary symmetric stochastic block model. In the block model with side information each of the $n$ vertices is labeled $+$ or $-$ independently and uniformly at random; each pair of vertices is connected independently with probability $a/n$ if both of them have the same label or $b/n$ otherwise. The goal is to estimate the underlying vertex labeling given 1) the graph structure and 2) side information in the form of a vertex labeling positively correlated with the true one. Assuming that the ratio between in and out degree $a/b$ is $\Theta(1)$ and the average degree $(a+b) / 2 = n^{o(1)}$, we characterize three different regimes under which a local algorithm, namely, belief propagation run on the local neighborhoods, maximizes the expected fraction of vertices labeled correctly. Thus, in contrast to the case of symmetric block models without side information, we show that local algorithms can achieve optimal performance for the block model with side information.Comment: Due to the limitation "The abstract field cannot be longer than 1,920 characters", the abstract here is shorter than that in the PDF fil

Moment estimates for convex measures

Let $p\geq 1$, \eps >0, r\geq (1+\eps) p, and $X$ be a $(-1/r)$-concave random vector in $\R^n$ with Euclidean norm $|X|$. We prove that (\E |X|^{p})^{1/{p}}\leq c (C(\eps) \E|X|+\sigma_{p}(X)), where \sigma_{p}(X)=\sup_{|z|\leq 1}(\E||^{p})^{1/p}, C(\eps) depends only on \eps and $c$ is a universal constant. Moreover, if in addition $X$ is centered then (\E |X|^{-p})^{-1/{p}}\geq c(\eps) (\E|X| - C \sigma_{p}(X))

Remarks on the KLS conjecture and Hardy-type inequalities

We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body $\Omega \subset \mathbb{R}^n$, not necessarily vanishing on the boundary $\partial \Omega$. This reduces the study of the Neumann Poincar\'e constant on $\Omega$ to that of the cone and Lebesgue measures on $\partial \Omega$; these may be bounded via the curvature of $\partial \Omega$. A second reduction is obtained to the class of harmonic functions on $\Omega$. We also study the relation between the Poincar\'e constant of a log-concave measure $\mu$ and its associated K. Ball body $K_\mu$. In particular, we obtain a simple proof of a conjecture of Kannan--Lov\'asz--Simonovits for unit-balls of $\ell^n_p$, originally due to Sodin and Lata{\l}a--Wojtaszczyk.Comment: 18 pages. Numbering of propositions, theorems, etc.. as appeared in final form in GAFA seminar note

Estimation in high dimensions: a geometric perspective

This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of asymptotic convex geometry, and demonstrates connections between geometric results and estimation problems. The theory is illustrated with applications to sparse recovery, matrix completion, quantization, linear and logistic regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change

An Auxin Transport-Based Model of Root Branching in Arabidopsis thaliana

Root architecture is a crucial part of plant adaptation to soil heterogeneity and is mainly controlled by root branching. The process of root system development can be divided into two successive steps: lateral root initiation and lateral root development/emergence which are controlled by different fluxes of the plant hormone auxin. While shoot architecture appears to be highly regular, following rules such as the phyllotactic spiral, root architecture appears more chaotic. We used stochastic modeling to extract hidden rules regulating root branching in Arabidopsis thaliana. These rules were used to build an integrative mechanistic model of root ramification based on auxin. This model was experimentally tested using plants with modified rhythm of lateral root initiation or mutants perturbed in auxin transport. Our analysis revealed that lateral root initiation and lateral root development/emergence are interacting with each other to create a global balance between the respective ratio of initiation and emergence. A mechanistic model based on auxin fluxes successfully predicted this property and the phenotype alteration of auxin transport mutants or plants with modified rythms of lateral root initiation. This suggests that root branching is controlled by mechanisms of lateral inhibition due to a competition between initiation and development/emergence for auxin