Operator learning without the adjoint
File(s)
Published version
Author(s)
Boulle, Nicolas
Halikias, Diana
Otto, Samuel E
Townsend, Alex
Type
Journal Article
Abstract
There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint? Current practical approaches suggest that one can accurately recover an operator while only using data generated by the forward action of the operator without access to the adjoint. However, naively, it seems essential to sample the action of the adjoint. In this paper, we partially explain this mystery by proving that without querying the adjoint, one can approximate a family of non-self-adjoint infinite-dimensional compact operators via projection onto a Fourier basis. We then apply the result to recovering Green's functions of elliptic partial differential operators and derive an adjoint-free sample complexity bound. While existing theory justiffes low sample complexity in operator learning, ours is the first adjoint-free analysis that attempts to close the gap between theory and practice.
Date Issued
2024-01-01
Date Acceptance
2024-09-01
Citation
Journal of Machine Learning Research, 2024, 25 (1)
ISSN
1532-4435
Publisher
Microtome Publishing
Journal / Book Title
Journal of Machine Learning Research
Volume
25
Issue
1
Copyright Statement
©2024 Boull´e, Halikias, Otto, and Townsend. License: CC-BY 4.0, see https://creativecommons.org/licenses/by/4.0/. Attribution requirements are provided at http://jmlr.org/papers/v25/24-0162.html.
License URL
Publication Status
Published
Article Number
ARTN 364