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Convergence Rates for Learning Linear Operators from Noisy Data

de Hoop, Maarten V. and Kovachki, Nikola B. and Nelsen, Nicholas H. and Stuart, Andrew M. (2021) Convergence Rates for Learning Linear Operators from Noisy Data. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20220524-180322099

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Abstract

We study the Bayesian inverse problem of learning a linear operator on a Hilbert space from its noisy pointwise evaluations on random input data. Our framework assumes that this target operator is self-adjoint and diagonal in a basis shared with the Gaussian prior and noise covariance operators arising from the imposed statistical model and is able to handle target operators that are compact, bounded, or even unbounded. We establish posterior contraction rates with respect to a family of Bochner norms as the number of data tend to infinity and derive related lower bounds on the estimation error. In the large data limit, we also provide asymptotic convergence rates of suitably defined excess risk and generalization gap functionals associated with the posterior mean point estimator. In doing so, we connect the posterior consistency results to nonparametric learning theory. Furthermore, these convergence rates highlight and quantify the difficulty of learning unbounded linear operators in comparison with the learning of bounded or compact ones. Numerical experiments confirm the theory and demonstrate that similar conclusions may be expected in more general problem settings.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
https://arxiv.org/abs/2108.12515arXivDiscussion Paper
ORCID:
AuthorORCID
de Hoop, Maarten V.0000-0002-6333-0379
Kovachki, Nikola B.0000-0002-3650-2972
Nelsen, Nicholas H.0000-0002-8328-1199
Stuart, Andrew M.0000-0001-9091-7266
Additional Information:The authors thank Kamyar Azizzadenesheli and Joel A. Tropp for helpful discussions about statistical learning. The computations presented in this paper were conducted on the Resnick High Performance Computing Center, a facility supported by the Resnick Sustainability Institute at the California Institute of Technology. MVdH is supported by the Simons Foundation under the MATH + X program, U.S. Department of Energy, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division under grant number DE-SC0020345, the National Science Foundation (NSF) under grant DMS-1815143, and the corporate members of the Geo-Mathematical Imaging Group at Rice University. NHN is supported by the NSF Graduate Research Fellowship Program under grant DGE-1745301. AMS is supported by NSF (grant DMS-1818977). NBK, NHN, and AMS are supported by NSF (grant AGS-1835860) and ONR (grant N00014-19-1-2408).
Group:Resnick Sustainability Institute
Funders:
Funding AgencyGrant Number
Simons FoundationUNSPECIFIED
Department of Energy (DOE)DE-SC0020345
NSFDMS-1815143
Rice UniversityUNSPECIFIED
NSF Graduate Research FellowshipDGE-1745301
NSFDMS-1818977
NSFAGS-1835860
Office of Naval Research (ONR)N00014-19-1-2408
Subject Keywords:operator regression, linear inverse problems, Bayesian inference, posterior consistency, learning theory
Classification Code:AMS subject classifications. 62G20, 62C10, 68T05, 47A62
Record Number:CaltechAUTHORS:20220524-180322099
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20220524-180322099
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:114901
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:24 May 2022 19:34
Last Modified:24 May 2022 19:34

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