Learning Multivariate Logconcave Distributions
[edit]
Proceedings of the 2017 Conference on Learning Theory, PMLR 65:711727, 2017.
Abstract
We study the problem of estimating multivariate logconcave probability density functions. We prove the first sample complexity upper bound for learning logconcave densities on $\mathbb{R}^d$, 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 $d \ge 1$ and $ε>0$, draws $\tilde{O}_d \left( (1/ε)^(d+5)/2 \right)$ samples from an unknown target logconcave density on $R^d$, 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 $\Omega_d \left( (1/ε)^(d+1)/2 \right)$ for this problem.
Related Material


