Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures

By Olivier Guédon, Emanuel Milman. 2010

142 citations recorded by the source corpus.

Given an isotropic random vector X with log-concave density in Euclidean space $${\mathbb{R}^n}$$ , we study the concentration properties of |X| on all scales, both above and below its expectation. We show in particular that$$\begin{array}{l} \mathbb{P}\left ( \left | |X| - \sqrt{n} \right | \geq t\sqrt{n} \right ) \leq C \, {\rm exp} \left ( -cn^{1/2} {\rm min}(t^{3}, t) \right) \; \forall t \geq 0, \end{array}$$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 $${\sqrt{\mathbb{V}{\rm ar}(|X|)}}$$ we obtain is of the order of n1/3, and improves down to n1/4 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.Lovász-Simonovits is deduced.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7c6794a8c0b5dd431b1f75b15e38cda39fbf416f

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Log-Concavity and Strong Log-Concavity: a review (2014, background): “Related papers include Guédon and Milman (2011) and Fradelizi, Guédon and Pajor (2013).”
Invariances in variance estimates (2011, methodology): “This is applied to spectral gap estimates for log-concave measures with many symmetries and to non-interacting conservative spin systems.”
A probabilistic take on isoperimetric-type inequalities (2012, unknown): “, [28], [21], [17], [19].”
Cone-volume measure of general centered convex bodies (2016, unknown): “Milman [20], and B.”
An Analysis of the t-SNE Algorithm for Data Visualization (2018, background): “We ﬁrst invoke the “thin-shell” property of log-concave distributions: Lemma A.6 (Gu´edon and Milman (2011)) .”