Abstract
We prove an almost constant lower bound of the isoperimetric coefficient in the KLS conjecture. The lower bound has the dimension dependency d^{-o_d(1)}. When the dimension is large enough, our lower bound is tighter than the previous best bound which has the dimension dependency d^{-1/4}. Improving the current best lower bound of the isoperimetric coefficient in the KLS conjecture has many implications, including improvements of the current best bounds in Bourgain’s slicing conjecture and in the thin-shell conjecture, better concentration inequalities for Lipschitz functions of log-concave measures and better mixing time bounds for MCMC sampling algorithms on log-concave measures.
Highlights
Given a distribution, the isoperimetric coefficient of a subset is the ratio of the measure of the subset boundary to the minimum of the measures of the subset and its complement
Taking the minimum of such ratios over all subsets defines the isoperimetric coefficient of the distribution, called the Cheeger isoperimetric coefficient of the distribution
If the conjecture is true, the Cheeger isoperimetric coefficient can be determined by going through all the half-spaces instead of all subsets
Summary
The isoperimetric coefficient of a subset is the ratio of the measure of the subset boundary to the minimum of the measures of the subset and its complement. Lovasz and Simonovits (KLS) [12] conjecture that for any distribution that is log-concave, the Cheeger isoperimetric coefficient equals to that achieved by half-spaces up to a universal constant factor. The current best bound is shown in Lee and Vempala [17], where they show that there exists a universal constant c such that for any log-concave density p with covariance matrix A, c ψ(p) ≥ (Tr (A2))1/4.
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