Concentration of mass on convex bodies

By Grigoris Paouris. 2006

282 citations recorded by the source corpus.

Subfields: Applied Mathematics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/37497f34470fb16acf86c7e7d0c8e529e37968e9

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In Applied Mathematics

In Molecular Biology

Citations in the corpus

A Berry-Esseen type inequality for convex bodies with an unconditional basis (2007, background): “Paouris [31, 32] was remarkably able to generalize inequali ty (30) to the class of all isotropically-normalized random vectors with a logconcave density inRn.”
Rogers–Shephard inequality for log-concave functions (2014, background): “The far reaching influence of this inequality becomes evident as it can be found as an ingredient not only in many important works in classical and asymptotic convex geometry [11,23,21,26] but also in many others with a”
An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture (2020, methodology): “We invoke the log-concave concentration bound by Paouris [18] (Theorem 1) as follows: there exists a constant c > 0 such that for X a random vector in Rd with an isotropic log-concave density, then P ( ‖X‖2 ≥ c √ d ) ≤ exp ( − √ d )”
Power-law estimates for the central limit theorem for convex sets (2006, background): “3 with the sharp large-deviation estimate of Paouris [13,14]: Paouris showed that under the assumptions of Theorem 1.”
On the infimum convolution inequality (2008, background): “3 we show how IC implies a comparison of weak and strong moments and the results of [12] and [18].”
The log-Brunn-Minkowski inequality (2012, unknown): “[5,8–12,16,18,22–30,32–55,58,62–65,67–69]).”
Introduction to the non-asymptotic analysis of random matrices (2010, background): “Isotropic distributions on convex bodies, and more generally isotropic log-concave distributions, are central to asymptotic convex geometry (see [31, 57]) and computational geometry [78].”
Orlicz–John ellipsoids (2014, background): “For more information about the importance of isotropy, one can refer to [6,16,17,41,44].”
How Close is the Sample Covariance Matrix to the Actual Covariance Matrix? (2010, unknown): “Paouris [20] which implies that ‖X‖2 = O( √ n) with probability at least 1− e− √ .”
A central limit theorem for convex sets (2006, background): “A significantly superior estimate to that of Theorem 1.4, for the case where e is a certain universal constant greater than one, is given by Paouris [39, 40 ].”
Covariance estimation for distributions with 2+ε moments (2011, background): “Paouris’s concentration inequality [11] implies that regularity assumption (SR) holds for X .”
Orlicz centroid bodies (2010, background): “, [10], [11], [24], [49], [50], [51], [52]) and even the theory of stable distributions (see, e.”
Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles (2009, background): “This estimate was further improved to N = C(ε, δ)n log(2) n in [23] and to N = C(ε, δ)n logn in [9] and [22]; the former paper treated the case when K is invariant under every reflection with respect to coordinate subspaces, and the latter proved the estimate in full generality.”
Minkowski valuations intertwining the special linear group (2012, unknown): “, [18, 19, 53]), information theory (see, e.”
ℓ1-regularized linear regression: persistence and oracle inequalities (2011, methodology): “To bound Z in a sharp way, we will use a deep result of Paouris [32]: Theorem 4.9 There are absolute constants c1 and c2 for which the following holds.”
Small ball probability estimates, ψ2-behavior and the hyperplane conjecture (2010, background): “The main results of [22] and [23] show that there exists a parameter q∗ := q∗(K) (related to the Lq–centroid bodies of K) with the following properties: (i) q∗(K) > c √ n, (ii) q−c(K, ξ) > q∗(K) for some absolute constant ξ > 1, and hence, I2(K) 6 ξI−q∗(K).”