Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Efficient Nonparametric Bayesian Inference For X-Ray Transforms   We consider the statistical inverse problem of recovering a function $f: M
\to \mathbb R$, where $M$ is a smooth compact Riemannian manifold with
boundary, from measurements of general $X$-ray transforms $I_a(f)$ of $f$,
corrupted by additive Gaussian noise. For $M$ equal to the unit disk with
`flat' geometry and $a=0$ this reduces to the standard Radon transform, but our
general setting allows for anisotropic media $M$ and can further model local
`attenuation' effects -- both highly relevant in practical imaging problems
such as SPECT tomography. We propose a nonparametric Bayesian inference
approach based on standard Gaussian process priors for $f$. The posterior
reconstruction of $f$ corresponds to a Tikhonov regulariser with a reproducing
kernel Hilbert space norm penalty that does not require the calculation of the
singular value decomposition of the forward operator $I_a$. We prove
Bernstein-von Mises theorems that entail that posterior-based inferences such
as credible sets are valid and optimal from a frequentist point of view for a
large family of semi-parametric aspects of $f$. In particular we derive the
asymptotic distribution of smooth linear functionals of the Tikhonov
regulariser, which is shown to attain the semi-parametric Cram\'er-Rao
information bound. The proofs rely on an invertibility result for the `Fisher
information' operator $I_a^*I_a$ between suitable function spaces, a result of
independent interest that relies on techniques from microlocal analysis. We
illustrate the performance of the proposed method via simulations in various
settings.

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The categories of this paper are: cs
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
27
The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat