Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: High-dimensional nonparametric density estimation via symmetry and shape
  constraints   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
$p$. Moreover, we show that the estimator is adaptive in the sense that if the
data generating density admits a special form, then a nearly parametric rate
may be attained. We also provide worst-case and adaptive risk bounds in cases
where $K$ is only known up to a positive definite transformation, and where it
is completely unknown and must be estimated nonparametrically. Our estimation
algorithms are fast even when $n$ and $p$ are on the order of hundreds of
thousands, and we illustrate the strong finite-sample performance of our
methods on simulated data.

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The categories of this paper are: math-ph
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat