Bayesian nonparametric modeling and its applications

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Bayesian nonparametric methods (or nonparametric Bayesian methods) take the benefit of unlimited parameters and unbounded dimensions to reduce the constraints on the parameter assumption and avoid over-fitting implicitly. They have proven to be extremely useful due to their flexibility and applicability to a wide range of problems. In this thesis, we study the Bayesain nonparametric theory with Lévy process and completely random measures (CRM). Several Bayesian nonparametric techniques are presented for computer vision and pattern recognition problems. In particular, our research and contributions focus on the following problems. Firstly, we propose a novel example-based face hallucination method, based on a nonparametric Bayesian model with the assumption that all human faces have similar local pixel structures. We use distance dependent Chinese restaurant process (ddCRP) to cluster the low-resolution (LR) face image patches and give a matrix-normal prior for learning the mapping dictionaries from LR to the corresponding high-resolution (HR) patches. The ddCRP is employed to assist in learning the clusters and mapping dictionaries without setting the number of clusters in advance, such that each dictionary can better reflect the details of the image patches. Experimental results show that our method is efficient and can achieve competitive performance for face hallucination problem. Secondly, we address sparse nonnegative matrix factorization (NMF) problems by using a graph-regularized Beta process (BP) model. BP is a nonparametric method which lets itself naturally model sparse binary matrices with an infinite number of columns. In order to maintain the positivity of the factorized matrices, an exponential prior is proposed. The graph in our model regularizes the similar training samples having similar sparse coefficients. In this way, the structure of the data can be better represented. We demonstrate the effectiveness of our method on different databases. Thirdly, we consider face recognition problem by a nonparametric Bayesian model combined with Sparse Coding Recognition (SCR) framework. In order to get an appropriate dictionary with sparse coefficients, we use a graph regularized Beta process prior for the dictionary learning. The graph in our model regularizes training samples in a same class to have similar sparse coefficients and share similar dictionary atoms. In this way, the proposed method is more robust to noise and occlusion of the testing images. The models in this thesis can also find many other applications like super-resolution, image recognition, text analysis, image compressive sensing and so on.
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