On the SoS Certifiability of Log-Concave Distributions

We prove that for every isotropic log-concave distribution P on ℝd and every even m≥2, the polynomial (Cm)m|v|2m - mathbbEX~ Plangle X,vranglem is a sum of squares, where C>0 is a universal constant. This improves on the Poincar'e-dependent bounds (Kothari and Steinhardt, 2017), 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 statistical estimation problems. Our proof proceeds by using stochastic localization to decompose P as an average of strongly log-concave measures, whose centered moments admit the subgaussian certificates (Diakonikolas et al., 2025). With a covariance-adapted choice of localization, we show that a fourth-moment certificate derived from the variance inequality for quadratic forms (Letwin, 2026) suffices to control this averaging at every even degree.