Adaptation in multivariate log-concave density estimation
Abstract
We study the adaptation properties of the multivariate log-concave maximum likelihood estimator over three subclasses of log-concave densities. The first consists of densities with polyhedral support whose logarithms are piecewise affine. The complexity of such densities~$f$ can be measured in terms of the sum $\Gamma(f)$ of the numbers of facets of the subdomains in the polyhedral subdivision of the support induced by $f$. Given $n$ independent observations from a $d$-dimensional log-concave density with $d \in \{2,3\}$, we prove a sharp oracle inequality, which in particular implies that the Kullback--Leibler risk of the log-concave maximum likelihood estimator for such densities is bounded above by $\Gamma(f)/n$, up to a polylogarithmic factor. Thus, the rate can be essentially parametric, even in this multivariate setting. For the second type of adaptation, we consider densities that are bounded away from zero on a polytopal support; we show that up to polylogarithmic factors, the log-concave maximum likelihood estimator attains the rate $n^{-4/7}$ when $d=3$, which is faster than the worst-case rate of $n^{-1/2}$. Finally, our third type of subclass consists of densities whose contours are well-separated; these new classes are constructed to be affine invariant and turn out to contain a wide variety of densities, including those that satisfy Hölder regularity conditions. Here, we prove another sharp oracle inequality, which reveals in particular that the log-concave maximum likelihood estimator attains a risk bound of order $n^{-\min\bigl(\frac{\beta+3}{\beta+7},\frac{4}{7}\bigr)}$ when $d=3$ over the class of $\beta$-Hölder log-concave densities with $\beta\in (1,3]$, again up to a polylogarithmic factor.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2018
- DOI:
- 10.48550/arXiv.1812.11634
- arXiv:
- arXiv:1812.11634
- Bibcode:
- 2018arXiv181211634F
- Keywords:
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- Mathematics - Statistics Theory;
- 62G07;
- 62G20
- E-Print:
- 97 pages, 6 figures