On the SoS Certifiability of Log-Concave Distributions

arXiv:2609.30105v1 Announce Type: new Abstract: For an arbitrary isotropic log-concave distribution P on R^d, we prove that the polynomial (Cm)^m|v|2^m - E{Xsim P}langle X,vrangle^m is a sum of squares for every even mge2, where C>0 is a universal constant. This removes the dependence on the Poincare 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.

Source: arXiv cs.LG | 2026-09-25