Sparse Approximation of Triangular Transports, Part II: The Infinite-Dimensional Case
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365_2022_Article_9570.pdf
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Author(s)
Zech, Jakob
Marzouk, Youssef
Date Issued
March 17, 2022
Publisher
Springer US
Citation
Zech, Jakob and Marzouk, Youssef. 2022. "Sparse Approximation of Triangular Transports, Part II: The Infinite-Dimensional Case."
Version
Final published version
Abstract
Abstract
For two probability measures
$${\rho }$$
ρ
and
$${\pi }$$
π
on
$$[-1,1]^{{\mathbb {N}}}$$
[
-
1
,
1
]
N
we investigate the approximation of the triangular Knothe–Rosenblatt transport
$$T:[-1,1]^{{\mathbb {N}}}\rightarrow [-1,1]^{{\mathbb {N}}}$$
T
:
[
-
1
,
1
]
N
→
[
-
1
,
1
]
N
that pushes forward
$${\rho }$$
ρ
to
$${\pi }$$
π
. Under suitable assumptions, we show that T can be approximated by rational functions without suffering from the curse of dimension. Our results are applicable to posterior measures arising in certain inference problems where the unknown belongs to an (infinite dimensional) Banach space. In particular, we show that it is possible to efficiently approximately sample from certain high-dimensional measures by transforming a lower-dimensional latent variable.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Massachusetts Institute of Technology. Center for Computational Science and Engineering
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1007/s00365-022-09570-9
