On the SoS Certifiability of Log-Concave Distributions

Abstract:For an arbitrary isotropic log-concave distribution $P$ on $\mathbb{R}^d$, we prove that the polynomial $(Cm)^m\|v\|_2^m - \mathbb{E}_{X\sim P}\langle X,v\rangle^m$ is a sum of squares for every even $m\ge2$, where $C>0$ is a universal constant. This removes the dependence on the Poincaré constant in the theorem of Kothari and Steinhardt (arXiv:1711.07465), recovering the optimal moment bounds for log-concave distributions. As an immediate corollary, we obtain computationally efficient algorithms with dimension-free error guarantees for a wide range of high-dimensional statistical estimation problems. Our proof uses stochastic localization to decompose $P$ as an average of random strongly log-concave measures, whose centered moments admit the subgaussian certificates of Diakonikolas, Hopkins, Pensia, and Tiegel (STOC 2025; arXiv:2410.21194). With a covariance-adapted choice of localization, we show that a fourth-moment certificate derived from Letwin's variance inequality for quadratic forms (arXiv:2607.24164) suffices to control this averaging at every even degree.

Machine Learning (cs.LG); Computational Complexity (cs.CC); Probability (math.PR)

arXiv:2609.30105 [cs.LG]

(or arXiv:2609.30105v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.30105 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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