[Paper Review] Concentration phenomena in high dimensional geometry

Olivier Guédon|arXiv (Cornell University)|Jan 1, 2014

Point processes and geometric inequalities85 references22 citations

TL;DR

This paper investigates concentration phenomena in high-dimensional geometry, focusing on log-concave and s-concave probability measures. It establishes sharp bounds on the isotropic constant of convex bodies and random vectors, proving that (−1/r)-concave isotropic vectors exhibit strong concentration of measure, particularly for the ℓ² norm, with explicit tail bounds that improve covariance matrix estimation and volume computation algorithms in high dimensions.

ABSTRACT

The purpose of this note is to present several aspects of concentration phenomena in high dimensional geometry. At the heart of the study is a geometric analysis point of view coming from the theory of high dimensional convex bodies. The topic has a broad audience going from algorithmic convex geometry to random matrices. We have tried to emphasize different problems relating these areas of research. Another connected area is the study of probability in Banach spaces where some concentration phenomena are related with good comparisons between the weak and the strong moments of a random vector.

Motivation & Objective

• To understand the geometric and probabilistic mechanisms behind concentration of measure in high-dimensional spaces.
• To establish uniform bounds on the isotropic constant LK for convex bodies and log-concave measures, linking it to the hyperplane (slicing) conjecture.
• To develop new tools for analyzing the moments and concentration of random vectors in Banach spaces, particularly for s-concave and ψ₂-type vectors.
• To apply these results to algorithmic problems such as volume computation and covariance matrix estimation in high dimensions.
• To extend classical moment comparison inequalities (weak vs. strong moments) to the setting of log-concave and s-concave measures.

Proposed method

• Uses the Prékopa–Leindler inequality and geometric analysis to characterize log-concave measures and their stability under linear transformations and convolution.
• Introduces the family of sets Kp(f) for a log-concave function f, proving they are convex and linking their isotropic constant to that of f.
• Applies Gaussian concentration and Gordon’s min-max theorem to derive moment comparison inequalities via the H(p, λ) assumption on random vectors.
• Employs the Zp body duality and norm approximation via linear forms to control the p-th moment of linear functionals.
• Uses the inertia ellipsoid and randomized rounding to place convex bodies in nearly isotropic position, enabling efficient volume computation.
• Applies multiphase Monte Carlo algorithms with bounds on mixing time derived from isoperimetric inequalities and spectral gap estimates.

Experimental results

Research questions

• RQ1 What is the relationship between the isotropic constant of a convex body and the concentration of measure of its associated log-concave measure?
• RQ2 Can the weak and strong moments of a log-concave or s-concave random vector be compared with uniform constants independent of dimension?
• RQ3 To what extent do (−1/r)-concave random vectors satisfy the H(p, λ) assumption, and what are the implications for moment and tail bounds?
• RQ4 How can the concentration properties of the ℓ² norm of isotropic s-concave vectors be leveraged to improve covariance matrix estimation?
• RQ5 What is the optimal rate of convergence for the empirical covariance matrix of s-concave vectors, and how does it depend on r and n?

Key findings

• The hyperplane conjecture is equivalent to the uniform boundedness of the isotropic constant LK over all dimensions and convex bodies.
• For any log-concave function f, the isotropic constant satisfies Lf ≤ c n^{1/4}, improving previous bounds.
• Any (−1/r)-concave isotropic random vector X in R^n satisfies the H(p, C) assumption for all 0 < p < r/2 with a universal constant C.
• For such vectors, the ℓ² norm satisfies the tail bound P(|X|₂ > t√n) ≤ (c max(1, r/√n)/t)^{r/2}, showing strong concentration.
• When r ≥ log n and N ≥ C(ε)n, the empirical covariance matrix of N i.i.d. copies of an isotropic (−1/r)-concave vector approximates the identity within ε with high probability.
• The results imply improved bounds on the mixing time of Markov chains used in randomized volume computation algorithms, particularly via the inertia ellipsoid approximation.