# On the SoS Certifiability of Log-Concave Distributions

arXiv: https://arxiv.org/abs/2609.30105 · HTML: https://mlsanity.com/paper/2609.30105/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 22 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 10 of 10)

- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Resilience: A Criterion for Learning in the Presence of Arbitrary Outliers](https://mlsanity.com/paper/1703.04940/) (2017)
- [Robust Linear Regression: Optimal Rates in Polynomial Time](https://mlsanity.com/paper/2007.01394/) (2020)
- [Private Robust Estimation by Stabilizing Convex Relaxations](https://mlsanity.com/paper/2112.03548/) (2021)
- [Outlier-robust moment-estimation via sum-of-squares](https://mlsanity.com/paper/1711.11581/) (2017)
- [High-dimensional estimation via sum-of-squares proofs](https://mlsanity.com/paper/1807.11419/) (2018)
- [Better Agnostic Clustering Via Relaxed Tensor Norms](https://mlsanity.com/paper/1711.07465/) (2017)
- [SoS Certifiability of Subgaussian Distributions and its Algorithmic Applications](https://mlsanity.com/paper/2410.21194/) (2024)
- [Strongly Refuting Random CSPs Below the Spectral Threshold](https://mlsanity.com/paper/1605.00058/) (2016)
- [The Localization Method for High-Dimensional Inequalities](https://mlsanity.com/paper/2512.10848/) (2025)

## Abstract (arXiv)

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.