Bayesian methods for sparse and low-rank matrix problems
Abstract: Many scientific and engineering problems require us to process measurements and data in order to extract information. Since we base decisions on information,it is important to design accurate and efficient processing algorithms. This is often done by modeling the signal of interest and the noise in the problem. One type ofmodeling is Compressed Sensing, where the signal has a sparse or low-rank representation. In this thesis we study different approaches to designing algorithms for sparse and low-rank problems.Greedy methods are fast methods for sparse problems which iteratively detects and estimates the non-zero components. By modeling the detection problem as an array processing problem and a Bayesian filtering problem, we improve the detection accuracy. Bayesian methods approximate the sparsity by probability distributions which are iteratively modified. We show one approach to making the Bayesian method the Relevance Vector Machine robust against sparse noise.Bayesian methods for low-rank matrix estimation typically use probability distributions which only depends on the singular values or a factorization approach. Here we introduce a new method, the Relevance Singular Vector Machine, which uses precision matrices with prior distributions to promote low-rank. The method is also applied to the robust Principal Component Analysis (PCA) problem, where a low-rank matrix is contaminated by sparse noise.In many estimation problems, there exists theoretical lower bounds on how well an algorithm can perform. When the performance of an algorithm matches a lowerbound, we know that the algorithm has optimal performance and that the lower bound is tight. When no algorithm matches a lower bound, there exists room for better algorithms and/or tighter bounds. In this thesis we derive lower bounds for three different Bayesian low-rank matrix models.In some problems, only the amplitudes of the measurements are recorded. Despitebeing non-linear, some problems can be transformed to linear problems. Earlier works have shown how sparsity can be utilized in the problem, here we show how the low-rank can be used.In some situations, the number of measurements and/or the number of parametersis very large. Such Big Data problems require us to design new algorithms. We show how the Basis Pursuit algorithm can be modified for problems with a very large number of parameters.
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