Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sparse approximation of triangular transports. Part I: the finite
  dimensional case   For two probability measures $\rho$ and $\pi$ with analytic densities on the
$d$-dimensional cube $[-1,1]^d$, we investigate the approximation of the unique
triangular monotone Knothe-Rosenblatt transport $T:[-1,1]^d\to [-1,1]^d$, such
that the pushforward $T_\sharp\rho$ equals $\pi$. It is shown that for
$d\in\mathbb{N}$ there exist approximations $\tilde T$ of $T$, based on either
sparse polynomial expansions or deep ReLU neural networks, such that the
distance between $\tilde T_\sharp\rho$ and $\pi$ decreases exponentially. More
precisely, we prove error bounds of the type $\exp(-\beta N^{1/d})$ (or
$\exp(-\beta N^{1/(d+1)})$ for neural networks), where $N$ refers to the
dimension of the ansatz space (or the size of the network) containing $\tilde
T$; the notion of distance comprises the Hellinger distance, the total
variation distance, the Wasserstein distance and the Kullback-Leibler
divergence. Our construction guarantees $\tilde T$ to be a monotone triangular
bijective transport on the hypercube $[-1,1]^d$. Analogous results hold for the
inverse transport $S=T^{-1}$. The proofs are constructive, and we give an
explicit a priori description of the ansatz space, which can be used for
numerical implementations.

30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
12
The categories of this paper are: nucl-th
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc