On the SoS Certifiability of Log-Concave Distributions
The paper claims sum-of-squares certificates can match optimal log-concave moment bounds without a Poincaré-constant loss.
Aleksandr Storozhenko proves that a standard even-moment polynomial for any isotropic log-concave distribution is a sum of squares with only a universal constant. The result removes a dependence in Kothari and Steinhardt’s earlier theorem and yields dimension-free error guarantees for efficient high-dimensional estimation algorithms. The proof decomposes the distribution through stochastic localization into random strongly log-concave measures, then controls the averaging with a fourth-moment certificate tied to Letwin’s variance inequality. ArXiv · AI/CL/LG's note
Aleksandr Storozhenko proves that a standard even-moment polynomial for any isotropic log-concave distribution is a sum of squares with only a universal constant. The result removes a dependence in Kothari and Steinhardt’s earlier theorem and yields dimension-free error guarantees for efficient high-dimensional estimation algorithms. The proof decomposes the distribution through stochastic localization into random strongly log-concave measures, then controls the averaging with a fourth-moment certificate tied to Letwin’s variance inequality. ArXiv · AI/CL/LG's note
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