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 $(Cm{)}^m\Vert v{\Vert }_2^m-{\mathbb{E}}_{X\sim P}⟨X,v{⟩}^m$ is a sum of squares for every even $m\ge 2$, 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.