Sparse Approximation of Triangular Transports, Part I: The Finite-Dimensional Case
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365_2022_Article_9569.pdf
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Author(s)
Zech, Jakob
Marzouk, Youssef
Date Issued
March 19, 2022
Publisher
Springer US
Citation
Zech, Jakob and Marzouk, Youssef. 2022. "Sparse Approximation of Triangular Transports, Part I: The Finite-Dimensional Case."
Version
Final published version
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.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
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DOI of Published Version
https://doi.org/10.1007/s00365-022-09569-2
