High-dimensional nonparametric density estimation via symmetry and shape constraints

Understand

We tackle the problem of high-dimensional nonparametric density estimation by taking the class of log-concave densities on $\mathbb{R}^p$ and incorporating within it symmetry assumptions, which facilitate scalable estimation algorithms and can mitigate the curse of dimensionality.

• Our main symmetry assumption is that the super-level sets of the density are $K$-homothetic (i.e.
• scalar multiples of a convex body $K \subseteq \mathbb{R}^p$).
• When $K$ is known, we prove that the $K$-homothetic log-concave maximum likelihood estimator based on $n$ independent observations from such a density has a worst-case risk bound with respect to, e.g., squared Hellinger loss, of $O(n^{-4/5})$, independent of $p$.