A nonparametric Bayesian model for kernel matrix completion

We present a nonparametric Bayesian model for completing low-rank, positive semidefinite matrices. Given an N × N matrix with underlying rank r, and noisy measured values and missing values with a symmetric pattern, the proposed Bayesian hierarchical model nonparametrically uncovers the underlying rank from all positive semidefinite matrices, and completes the matrix by approximating the missing values. We analytically derive all posterior distributions for the fully conjugate model hierarchy and discuss variational Bayes and MCMC Gibbs sampling for inference, as well as an efficient measurement selection procedure. We present results on a toy problem, and a music recommendation problem, where we complete the kernel matrix of 2,250 pieces of music.