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Scalable Bayesian inversion with Poisson data

Zhang, Chen; (2020) Scalable Bayesian inversion with Poisson data. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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Abstract

Poisson data arise in many important inverse problems, e.g., medical imaging. The stochastic nature of noisy observation processes and imprecise prior information implies that there exists an ensemble of solutions consistent with the given Poisson data to various extents. Existing approaches, e.g., maximum likelihood and penalised maximum likelihood, incorporate the statistical information for point estimates, but fail to provide the important uncertainty information of various possible solu- tions. While full Bayesian approaches can solve this problem, the posterior distributions are often intractable due to their complicated form and the curse of dimensionality. In this thesis, we investigate approximate Bayesian inference techniques, i.e., variational inference (VI), expectation propagation (EP) and Bayesian deep learning (BDL), for scalable posterior exploration. The scalability relies on leveraging 1) mathematical structures emerging in the problems, i.e., the low rank structure of forward operators and the rank 1 projection form of factors in the posterior distribution, and 2) efficient feed forward processes of neural networks and further reduced training time by flexibility of dimensions with incorporating forward and adjoint operators. Apart from the scalability, we also address theoretical analysis, algorithmic design and practical implementation. For VI, we derive explicit functional form and analyse the convergence of algorithms, which are long-standing problems in the literature. For EP, we discuss how to incorporate nonnegative constraints and how to design stable moment evaluation schemes, which are vital and nontrivial practical concerns. For BDL, specifically conditional variational auto-encoders (CVAEs), we investigate how to apply them for uncertainty quantification of inverse problems and develop flexible and novel frameworks for general Bayesian Inversion. Finally, we justify these contributions with numerical experiments and show the competitiveness of our proposed methods by comparing with state-of-the-art benchmarks.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Scalable Bayesian inversion with Poisson data
Event: UCL (University College London)
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Copyright © The Author 2020. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/10095835
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