## Adaptation in multivariate log-concave density estimation

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##### ISSN

0090-5364

##### Publisher

Institute of Mathematical Statistics

##### Type

Article

##### This Version

AM

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Feng, O., Guntuboyina, A., Kim, A. K., & Samworth, R. Adaptation in multivariate log-concave density estimation. https://doi.org/10.17863/CAM.48299

##### 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 H\"older 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 $\beta$-H\"older log-concave densities with $\beta\in (1,3]$, again up to a polylogarithmic factor.

##### Keywords

math.ST, math.ST, stat.TH, 62G07, 62G20

##### Sponsorship

EPSRC Fellowship EP/P031447/1
Leverhulme Trust grant RG81761

##### Funder references

Leverhulme Trust (PLP-2014-353)

EPSRC (EP/P031447/1)

##### Embargo Lift Date

2023-01-23

##### Identifiers

This record's DOI: https://doi.org/10.17863/CAM.48299

This record's URL: https://www.repository.cam.ac.uk/handle/1810/301217

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All rights reserved