Sparse Approximation of Triangular Transports, Part I: The Finite-Dimensional Case

Abstract

Abstract For two probability measures

$${\rho }$$

ρ

and

$${\pi }$$

π

with analytic densities on the d-dimensional cube

$$[-1,1]^d$$

[ - 1 , 1 ]

d

, we investigate the approximation of the unique triangular monotone Knothe–Rosenblatt transport

$$T:[-1,1]^d\rightarrow [-1,1]^d$$

T :

[ - 1 , 1 ]

d

→

[ - 1 , 1 ]

d

, such that the pushforward

$$T_\sharp {\rho }$$

T ♯

ρ

equals

$${\pi }$$

π

. It is shown that for

$$d\in {{\mathbb {N}}}$$

d ∈ N

there exist approximations

$${\tilde{T}}$$

T ~

of T, based on either sparse polynomial expansions or deep ReLU neural networks, such that the distance between

$${\tilde{T}}_\sharp {\rho }$$

T ~

♯

ρ

and

$${\pi }$$

π

decreases exponentially. More precisely, we prove error bounds of the type

$$\exp (-\beta N^{1/d})$$

exp ( - β

N

1 / d

)

(or

$$\exp (-\beta N^{1/(d+1)})$$

exp ( - β

N

1 / ( d + 1 )

)

for neural networks), where N refers to the dimension of the ansatz space (or the size of the network) containing

$${\tilde{T}}$$

T ~

; the notion of distance comprises the Hellinger distance, the total variation distance, the Wasserstein distance and the Kullback–Leibler divergence. Our construction guarantees

$${\tilde{T}}$$

T ~

to be a monotone triangular bijective transport on the hypercube

$$[-1,1]^d$$

[ - 1 , 1 ]

d

. Analogous results hold for the inverse transport

$$S=T^{-1}$$

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.