An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture

By Yuansi Chen. 2020

145 citations recorded by the source corpus.

We prove an almost constant lower bound of the isoperimetric coefficient in the KLS conjecture. The lower bound has the dimension dependency $$d^{-o_d(1)}$$ 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 $$d^{-1/4}$$ 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.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5a2edbb507d8be4bea6cdc70c63905cab3d244ac

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In Applied Mathematics

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Bourgain’s slicing problem and KLS isoperimetry up to polylog (2022, methodology): “(10) In [15], Chen uses Eldan’s stochastic localization [21] and the analysis of Lee and Vempala [37] in order to show that ψn ≤ C1 exp (”
Provable Guarantees for Self-Supervised Deep Learning with Spectral Contrastive Loss (2021, background): “In many cases, the Cheeger constant is at least inverse polynomial in the data dimension (Chen, 2021, Lee and Vempala, 2016).”
Logarithmic bounds for isoperimetry and slices of convex sets (2023, background): “The existence and uniqueness of a strong solution to (24) are proven via a standard argument (see, e.g. Chen [14]).”