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Solving Linear and Bilinear Inverse Problems using Approximate Message Passing Methods

Sarkar, Subrata
http://rave.ohiolink.edu/etdc/view?acc_num=osu1595529156778986

Abstract Details

2020, Doctor of Philosophy, Ohio State University, Electrical and Computer Engineering.
Recovering an unknown vector from its noisy linear measurements is an important problem that arises in many fields. In linear inverse problems, the forward operator is perfectly known, whereas in bilinear inverse problems, the forward operator has some unknown parameters. Most existing recovery algorithms are either slow to converge or give a final solution that is not accurate. In this dissertation, we develop algorithms using Approximate Message Passing (AMP) methods to solve linear and bilinear inverse problems. First, we consider the computationally efficient Vector Approximate Message Passing (VAMP) algorithm, which is based on the Expectation Propagation framework. VAMP minimizes a cost function, also known as the Gibbs free energy in statistical physics, under moment-matching constraints. It iteratively calls a denoiser that is chosen based on prior knowledge about the unknown vector that we want to recover. VAMP has a remarkable property, that when the sensing matrix is a typical instance of a large right-rotationally invariant random matrix, the per-iteration macroscopic behaviour of VAMP can be exactly predicted by a set of scalar equations that yield the so called state-evolution (SE). The state-evolution can be used to predict the MSE performance of VAMP at each iteration. The SE was first proven for separable Lipschitz denoisers. In this work, we extend the state-evolution to a larger class of non-separable Lipschitz denoisers. Empirical results show that the SE also accurately predicts VAMP's performance in bilinear problems solved that have been converted to linear problems using the ''lifting'' technique. In bilinear inverse problems, the forward operator is written as a linear combination of known matrices with unknown weights. Problems such as dictionary learning, CS with matrix uncertainty, self-calibration, are all instances of bilinear inverse problems. We propose a new algorithm called Bilinear Adaptive Vector Approximate Message Passing (BAd-VAMP). Our method is based on the EM-VAMP algorithm, where Expectation Maximization (EM) is combined with VAMP to estimate the unknown parameters in the likelihood and prior. We show that our BAd-VAMP method is efficient and robust to the conditioning of the forward operator, which is a common problem with algorithms based on the earlier AMP algorithm. Finally, we consider the problem of magnetic resonance imaging (MRI). MRI is a safe, non-invasive method to capture internal images of our body. MRI's long data-acquisition time is one of its biggest drawbacks. In accelerated MRI, a small number of samples in k-space are acquired and the goal is to accurately reconstruct the image from these few measurements. We develop two algorithms for MRI reconstruction using Approximate Message Passing methods with non-separable denoisers. Our first algorithm is a modification of the existing AMP algorithm, where we propose a scaling in k-space to debias the denoiser input error. Our second algorithm is a modification of VAMP where we propose a robust damping scheme to stabilize VAMP in MRI. We compare our algorithms against conventional plug-and-play algorithms on the fastMRI knee dataset.
Philip Schniter (Advisor)
Lee Potter (Committee Member)
Rizwan Ahmad (Committee Member)
128 p.
Electrical Engineering
Approximate message passing; expectation propagation; expectation maximization; self-calibration; computed tomography; dictionary learning; MRI;

Recommended Citations

Citations

  • Sarkar, S. (2020). Solving Linear and Bilinear Inverse Problems using Approximate Message Passing Methods [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1595529156778986

    APA Style (7th edition)

  • Sarkar, Subrata. Solving Linear and Bilinear Inverse Problems using Approximate Message Passing Methods. 2020. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1595529156778986.

    MLA Style (8th edition)

  • Sarkar, Subrata. "Solving Linear and Bilinear Inverse Problems using Approximate Message Passing Methods." Doctoral dissertation, Ohio State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1595529156778986

    Chicago Manual of Style (17th edition)

osu1595529156778986
289
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This open access ETD is published by The Ohio State University and OhioLINK.