Sparse Approximation of Triangular Transports, Part II: The Infinite-Dimensional Case
Abstract For two probability measures $${\rho }$$ ρ and $${\pi }$$...
| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer US
2022
|
| Online Access: | https://hdl.handle.net/1721.1/141312 |
| _version_ | 1811084460181946368 |
|---|---|
| author | Zech, Jakob Marzouk, Youssef |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Zech, Jakob Marzouk, Youssef |
| author_sort | Zech, Jakob |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T12:51:06Z |
| format | Article |
| id | mit-1721.1/141312 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:51:06Z |
| publishDate | 2022 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1413122023-07-19T16:18:26Z Sparse Approximation of Triangular Transports, Part II: The Infinite-Dimensional Case Zech, Jakob Marzouk, Youssef Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Massachusetts Institute of Technology. Center for Computational Science and Engineering 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. 2022-03-21T12:41:36Z 2022-03-21T12:41:36Z 2022-03-17 2022-03-20T04:14:59Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/141312 Zech, Jakob and Marzouk, Youssef. 2022. "Sparse Approximation of Triangular Transports, Part II: The Infinite-Dimensional Case." PUBLISHER_CC en https://doi.org/10.1007/s00365-022-09570-9 Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer US Springer US |
| spellingShingle | Zech, Jakob Marzouk, Youssef Sparse Approximation of Triangular Transports, Part II: The Infinite-Dimensional Case |
| title | Sparse Approximation of Triangular Transports, Part II: The Infinite-Dimensional Case |
| title_full | Sparse Approximation of Triangular Transports, Part II: The Infinite-Dimensional Case |
| title_fullStr | Sparse Approximation of Triangular Transports, Part II: The Infinite-Dimensional Case |
| title_full_unstemmed | Sparse Approximation of Triangular Transports, Part II: The Infinite-Dimensional Case |
| title_short | Sparse Approximation of Triangular Transports, Part II: The Infinite-Dimensional Case |
| title_sort | sparse approximation of triangular transports part ii the infinite dimensional case |
| url | https://hdl.handle.net/1721.1/141312 |
| work_keys_str_mv | AT zechjakob sparseapproximationoftriangulartransportspartiitheinfinitedimensionalcase AT marzoukyoussef sparseapproximationoftriangulartransportspartiitheinfinitedimensionalcase |