三角形输运的稀疏近似。第一部分：有限维情形
Sparse approximation of triangular transports. Part I: the finite dimensional case
摘要：
对于$d$维立方体$[-1,1]^d$上具有解析密度的两个概率测度$ρ$和$π$，研究了唯一三角单调Knothe-Rosenblatt输运$T:[-1,1]^d\to [-1,1]^d$的近似，使得前推$T_\sharpρ$等于$π$ .结果表明，对于$d\in\mathbb{N}$，存在$T$的近似$\tilde T$，基于稀疏多项式展开或深度ReLU神经网络，使得$\tilde T_\sharpρ$和$π$之间的距离呈指数下降。更确切地说，我们证明了$\exp(-βN^{1/d})$类型（或神经网络的$\exp(-βN^{1/(d+1)})$）的误差界，其中$N$指的是包含$\tilde T$的拓扑空间的维数（或网络的大小）;距离的概念包括Hellinger距离，总变差距离，Wasserstein距离和Kullback-Leibler散度。我们的构造保证了$\tilde T$是超立方体$[-1,1]^d$上的单调三角双射输运。对于逆输运$S=T^{-1}$，类似的结果也成立。的证明是建设性的，我们给出了一个明确的先验描述的空间，它可以用于数值实现。
英文摘要：
For two probability measures $ρ$ and $π$ 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 $T_\sharpρ$ equals $π$. 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 $\tilde T_\sharpρ$ and $π$ decreases exponentially. More precisely, we prove error bounds of the type $\exp(-βN^{1/d})$ (or $\exp(-βN^{1/(d+1)})$ 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.
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