Eldan's Stochastic Localization and the KLS Hyperplane Conjecture: An Improved Lower Bound for Expansion
2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) (2017)
Berkeley, California, USA
Oct. 15, 2017 to Oct. 17, 2017
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2017.96
We show that the KLS constant for n-dimensional isotropic logconcave measures is O(n1/4), improving on the current best bound of O(n1/3√log n). As corollaries we obtain the same improved bound on the thin-shell estimate, Poincaréconstant and Lipschitz concentration constant and an alternative proof of this bound for the isotropic constant; it also follows that the ball walk for sampling from an isotropic logconcave density in ℝn converges in O*(n2.5) steps from a warm start.
computational complexity, random processes, statistical distributions, stochastic processes
Y. T. Lee and S. S. Vempala, "Eldan's Stochastic Localization and the KLS Hyperplane Conjecture: An Improved Lower Bound for Expansion," 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), Berkeley, California, USA, 2017, pp. 998-1007.