
Variational Bayes' method for functions with applications to some inverse problems
Bayesian approach as a useful tool for quantifying uncertainties has bee...
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A unified approach to calculation of information operators in semiparametric models
The infinitedimensional information operator for the nuisance parameter...
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Bayesian inverse problems with unknown operators
We consider the Bayesian approach to linear inverse problems when the un...
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Diving into the shallows: a computational perspective on largescale shallow learning
In this paper we first identify a basic limitation in gradient descentb...
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Wellposedness of Bayesian inverse problems in quasiBanach spaces with stable priors
The Bayesian perspective on inverse problems has attracted much mathemat...
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Learning SchattenVon Neumann Operators
We study the learnability of a class of compact operators known as Schat...
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Convergence of the conjugate gradient method with unbounded operators
In the framework of inverse linear problems on infinitedimensional Hilb...
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Stein variational gradient descent on infinitedimensional space and applications to statistical inverse problems
For solving Bayesian inverse problems governed by largescale forward problems, we present an infinitedimensional version of the Stein variational gradient descent (iSVGD) method, which has the ability to generate approximate samples from the posteriors efficiently. Specifically, we introduce the concept of the operatorvalued kernel and the corresponding functionvalued reproducing kernel Hilbert space (RKHS). Through the properties of RKHS, we give an explicit meaning of the infinitedimensional objects (e.g., the Stein operator) and prove that the infinitedimensional objects are indeed the limit of finitedimensional items. Furthermore, by generalizing the change of variables formula, we construct iSVGD with preconditioning operators, yielding more efficient iSVGD. During these generalizations, we introduce a regularity parameter s∈[0,1]. Our analysis shows that the intuitive trivial version (i.e., by directly taking finitedimensional objects as infinitedimensional items) of iSVGD with preconditioning operators (s=0) will yield inaccurate estimates, and the parameter s should be chosen larger than 0 and smaller than 0.5. Finally, the proposed algorithms are applied to an inverse problem governed by the Helmholtz equation. Numerical results confirm the correctness of our theoretical findings and demonstrate the potential usefulness of the proposed approach in the posterior sampling of largescale nonlinear statistical inverse problems.
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