On the SoS Certifiability of Log-Concave Distributions

Abstract: For an arbitrary isotropic log-concave distribution $P$ on $\mathbb{R}<sup>d$, we prove that the polynomial $$(Cm)^{m|v|_2^m} - \mathbb{E}_{X\sim P}\langle X,v\rangle<sup>m$$ is a sum of squares for every even $m\ge2$, where $C&gt;0$ 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.