Adaptation in multivariate log-concave density estimation
Adaptation in multivariate log-concave density estimation
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DOI：
10.17863/cam.48299
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发表时间：
2018-12
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影响因子：
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通讯作者：
Oliver Y. Feng;Adityanand Guntuboyina;Arlene K. H. Kim;R. Samworth
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文献类型：
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作者：
Oliver Y. Feng;Adityanand Guntuboyina;Arlene K. H. Kim;R. Samworth
研究了多元对数凹极大似然估计在对数凹密度的两个子类上的适应性。第一类是具有多面体支撑的密度，其对数是分段仿射的。这种密度的复杂性可以用由$f$诱导的支承的多面体剖分中的子域的面数之和$\Gamma(F)$来度量。给出了d维对数凹密度的n个独立观测值，证明了一个尖锐的预言不等式，特别地，它意味着这种密度的对数凹极大似然估计的Kullback-Leibler风险在$Gamma(F)/n$以上有界，直到一个多对数因子.因此，即使在这种多变量设置中，速率基本上也可以是参数的。第二类子类由轮廓分离良好的密度组成；这些新的子类被构造为仿射不变的，并且证明包含各种各样的密度，包括那些满足Holder正则性条件的密度。在这里，我们证明了另一个尖锐的预言不等式，它特别地揭示了当$d=3$时，对数凹极大似然估计在$\beta$-Holder对数凹密度类上达到$n^{-\min\bigl(FRAC{\beta+3}{\beta+7}，\FRAC{4}{7}\BiGR)}$的Kullback-Leibler风险界，同样也是多对数因子.
We study the adaptation properties of the multivariate log-concave maximum likelihood estimator over two 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. The second 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 Holder regularity conditions. Here, we prove another sharp oracle inequality, which reveals in particular that the log-concave maximum likelihood estimator attains a Kullback--Leibler risk bound of order $n^{-\min\bigl(\frac{\beta+3}{\beta+7},\frac{4}{7}\bigr)}$ when $d=3$ over the class of $\beta$-Holder log-concave densities with $\beta > 1$, again up to a polylogarithmic factor.