On the SoS Certifiability of Log-Concave Distributions
Abstract
We prove that for every isotropic log-concave distribution P on ℝd and every even m≥2, the polynomial (Cm)m|v|2m − 𝔼X∼ P⟨ X,v⟩m is a sum of squares, where C>0 is a universal constant. This improves on the Poincaré-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.
Related
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- High-Dimensional Ultra-Log-Concave Distributions
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- An invariant variational proof for canonical processes with independent symmetric log-concave tails
- Concave transforms and volumes of adelic line bundles
BibTeX
@misc{storozhenko2026certifiability,
title = {{On the SoS Certifiability of Log-Concave Distributions}},
author = {Aleksandr Storozhenko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.30105},
eprint = {2609.30105},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30105},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the SoS Certifiability of Log-Concave Distributions
AU - Aleksandr Storozhenko
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.30105
UR - https://arxiv.org/abs/2609.30105
AB - We prove that for every isotropic log-concave distribution P on ℝ^d and every even m≥2, the polynomial (Cm)^m|v|_2^m − 𝔼_X∼ P⟨ X,v⟩^m is a sum of squares, where C>0 is a universal constant. This improves on the Poincaré-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.
ER -