Extracting low-rank and/or sparse structures using matrix factorization techniques has been extensively studied in the machine learning community. Kernelized matrix factorization (KMF) is a powerful tool to incorporate side information into the low-rank approximation model, which has been applied to solve the problems of data mining, recommender systems, image restoration, and machine vision. However, most existing KMF models rely on specifying the rows and columns of the data matrix through a Gaussian process prior and have to tune manually the rank. There are also computational issues of existing models based on regularization or the Markov chain Monte Carlo. In this article, we develop a hierarchical kernelized sparse Bayesian matrix factorization (KSBMF) model to integrate side information. The KSBMF automatically infers the parameters and latent variables including the reduced rank using the variational Bayesian inference. In addition, the model simultaneously achieves low-rankness through sparse Bayesian learning and columnwise sparsity through an enforced constraint on latent factor matrices. We further connect the KSBMF with the nonlocal image processing framework to develop two algorithms for image denoising and inpainting. Experimental results demonstrate that KSBMF outperforms the state-of-the-art approaches for these image-restoration tasks under various levels of corruption.

In this work, we propose a probabilistic framework for relational data modelling and latent structure exploring. Given the possible feature information for the nodes in a network, our model builds up a deep architecture that can approximate to the possible nonlinear mappings between the nodes' feature information and latent representations. For each node, we incorporate all its neighborhoods' high-order structure information to generate latent representation, such that these latent representations are ``smooth'' in terms of the network. Since the latent representations are generated from Dirichlet distributions, we further develop a data augmentation trick to enable efficient Gibbs sampling for Ber-Poisson likelihood with Dirichlet random variables. Our model can be ready to apply to large sparse network as its computations cost scales to the number of positive links in the networks. The superior performance of our model is demonstrated through improved link prediction performance on a range of real-world datasets.

Singular value thresholding (SVT)- or nuclear norm minimization (NNM)-based nonlocal image denoising methods often rely on the precise estimation of the noise variance. However, most existing methods either assume that the noise variance is known or require an extra step to estimate it. Under the iterative regularization framework, the error in the noise variance estimate propagates and accumulates with each iteration, ultimately degrading the overall denoising performance. In addition, the essence of these methods is still least squares estimation, which can cause a very high mean-squared error (MSE) and is inadequate for handling missing data or outliers. In order to address these deficiencies, we present a hybrid denoising model based on variational Bayesian inference and Stein's unbiased risk estimator (SURE), which consists of two complementary steps. In the first step, the variational Bayesian SVT performs a low-rank approximation of the nonlocal image patch matrix to simultaneously remove the noise and estimate the noise variance. In the second step, we modify the conventional SURE full-rank SVT and its divergence formulas for rank-reduced eigen-triplets to remove the residual artifacts. The proposed hybrid BSSVT method achieves better performance in recovering the true image compared with state-of-the-art methods.