# Learning Multivariate Log-concave Distributions

arXiv: https://arxiv.org/abs/1605.08188 · HTML: https://mlsanity.com/paper/1605.08188/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 May 2016.

## Built on by (most cited first, 4 of 4)

- [Efficient Certificates of Anti-Concentration Beyond Gaussians](https://mlsanity.com/paper/2405.15084/) (2024)
- [Discrete distributions are learnable from metastable samples](https://mlsanity.com/paper/2410.13800/) (2024)
- [Beyond Universal Approximation Theorems: Algorithmic Uniform Approximation by Neural Networks Trained with Noisy Data](https://mlsanity.com/paper/2509.00924/) (2025)
- [Density estimation for Hellinger via minimum-distance estimators: mixtures of Gaussians, log-concave, and more](https://mlsanity.com/paper/2606.11469/) (2026)

## Abstract (arXiv)

We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on ℝd, for all d ≥ 1. Prior to our work, no upper bound on the sample complexity of this learning problem was known for the case of d>3. In more detail, we give an estimator that, for any d ≥ 1 and ε>0, draws tildeOd left( (1/ε)(d+5)/2 right) samples from an unknown target log-concave density on ℝd, and outputs a hypothesis that (with high probability) is ε-close to the target, in total variation distance. Our upper bound on the sample complexity comes close to the known lower bound of Omegad left( (1/ε)(d+1)/2 right) for this problem.