[Paper Review] A Slightly Improved Bound for the KLS Constant

Arun Jambulapati, Yin Tat Lee+1 more|arXiv (Cornell University)|Aug 24, 2022

Molecular spectroscopy and chirality4 citations

TL;DR

This paper improves the polylogarithmic bound for the KLS constant in high-dimensional convex geometry by refining the stochastic localization technique of Klartag and Lehec. By optimizing the trade-off between mean squared norm growth and covariance control in the Lee-Vempala stochastic process, the authors achieve a tighter bound: $\sigma_n \lesssim \log^{2.2226}n$ and $\psi_n \lesssim \log^{3.2226}n$, representing a significant improvement over the prior $\log^4 n$ and $\log^5 n$ bounds.

ABSTRACT

We refine the recent breakthrough technique of Klartag and Lehec to obtain an improved polylogarithmic bound for the KLS constant.

Motivation & Objective

• To improve the polylogarithmic upper bound on the KLS constant for isotropic log-concave distributions in high dimensions.
• To refine the stochastic localization framework of Klartag and Lehec to achieve a tighter dependence on dimension.
• To optimize the trade-off between the growth of the mean vector norm and the covariance operator in the stochastic process.
• To establish a quantitative improvement over the prior $\log^4 n$ and $\log^5 n$ bounds for the thin-shell and KLS constants, respectively.
• To demonstrate that the KLS conjecture's polylogarithmic bound can be improved through careful analysis of the Lee-Vempala stochastic process and the $T_\mu$ functional.

Proposed method

• Refines the stochastic localization process by analyzing the squared norm of the mean vector $\|a_t\|^2$ as a proxy for the thin-shell constant $\sigma_n^2$, rather than relying solely on $\mathrm{Tr}(A_t^q)$.
• Applies the Lee-Vempala variant of stochastic localization, which simplifies the dynamics by removing the inverse covariance scaling, enabling tighter control of the mean vector growth.
• Uses a new time-dependent optimization over $t_\lambda$ in the integral bound from Lemma 8 to balance the terms $\frac{1}{n\lambda^2}\mathbb{E}\|a_{t_\lambda}\|^2$ and $\frac{1}{\lambda t_\lambda}$.
• Introduces a parameter $\gamma$ to control the decay rate of the mean vector norm, and optimizes over $\gamma$ and the parameter $q$ in the $T_\mu$ functional bound.
• Leverages the improved $T_\mu$ bound from Chen (2022) for $q$-strongly log-concave distributions to control the derivative of $\mathrm{Tr}(A_t^q)$.
• Derives a recursive inequality in terms of $\sigma_n$ and $\psi_n$, and solves for the minimal exponent $\eta$ such that $\sigma_n \lesssim \log^\eta n$, leading to the final bound.

Experimental results

Research questions

• RQ1 Can the polylogarithmic bound for the KLS constant be improved beyond the $\log^5 n$ result of Klartag and Lehec?
• RQ2 Does refining the analysis of the mean vector norm growth in the Lee-Vempala stochastic process yield a better exponent in the logarithmic bound?
• RQ3 What is the optimal trade-off between the time parameter $t_\lambda$ and the decay rate $\gamma$ in the integral bound for $\sigma_n^2$?
• RQ4 Can the $T_\mu$ functional bound be leveraged more effectively in the stochastic localization framework to reduce the dependence on dimension?
• RQ5 What value of $q$ and $\gamma$ minimizes the exponent $\eta$ in the bound $\sigma_n \lesssim \log^\eta n$?

Key findings

• The paper establishes a new upper bound of $\sigma_n \lesssim \log^{2.2226}n$ for the thin-shell constant, improving upon the previous $\log^4 n$ bound.
• The KLS constant is improved to $\psi_n \lesssim \log^{3.2226}n$, a significant tightening from the prior $\log^5 n$ result.
• The improvement is achieved by optimizing the time parameter $t_\lambda$ in the integral bound and refining the analysis of the mean vector norm growth in the Lee-Vempala stochastic process.
• The optimal value of $\gamma = 2\sqrt{2}$ and $q = 3$ yield the minimal exponent $\eta = \frac{63\sqrt{2} - 36}{82} \leq 0.6476$ for the thin-shell constant, leading to the final bound.
• The analysis shows that the KLS conjecture's polylogarithmic bound can be improved by a factor in the exponent, suggesting a path toward a universal constant bound.
• The result demonstrates that the $\psi_n \lesssim \sigma_n$ assumption leads to a tighter bound than the general case, highlighting the role of the KLS conjecture in refining the analysis.