التفاصيل البيبلوغرافية
| العنوان: |
Sparse Approximation of Triangular Transports, Part I: The Finite-Dimensional Case |
| المؤلفون: |
Zech, Jakob, Marzouk, Youssef |
| المساهمون: |
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| المصدر: |
Springer US |
| بيانات النشر: |
Springer US |
| سنة النشر: |
2022 |
| المجموعة: |
DSpace@MIT (Massachusetts Institute of Technology) |
| الوصف: |
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. |
| نوع الوثيقة: |
article in journal/newspaper |
| وصف الملف: |
application/pdf |
| اللغة: |
English |
| Relation: |
https://doi.org/10.1007/s00365-022-09569-2; https://hdl.handle.net/1721.1/141313; Zech, Jakob and Marzouk, Youssef. 2022. "Sparse Approximation of Triangular Transports, Part I: The Finite-Dimensional Case."; PUBLISHER_CC |
| الاتاحة: |
https://hdl.handle.net/1721.1/141313 |
| Rights: |
Creative Commons Attribution ; https://creativecommons.org/licenses/by/4.0/ ; The Author(s) |
| رقم الانضمام: |
edsbas.BD828288 |
| قاعدة البيانات: |
BASE |