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 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 ≥ 1 and ǫ > 0, draws O d (1/ǫ) (d+5)/2 samples from an unknown target log-concave density on R d , and outputs a hypothesis that (with high probability) is ǫ-close to the target, in total variation distance. Our upper bound on the sample complexity comes close to the known lower bound of Ω d (1/ǫ) (d+1)/2 for this problem.
For example, REF used this approach to bound the sample complexity of learning high-dimensional log-concave distributions.
Learning Multivariate Log-concave Distributions
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Computer Science Mathematics