  In this article, we derive a Bayesian model to learning the sparse and low
rank PARAFAC decomposition for the observed tensor with missing values via the
elastic net, with property to find the true rank and sparse factor matrix which
is robust to the noise. We formulate efficient block coordinate descent
algorithm and admax stochastic block coordinate descent algorithm to solve it,
which can be used to solve the large scale problem. To choose the appropriate
rank and sparsity in PARAFAC decomposition, we will give a solution path by
gradually increasing the regularization to increase the sparsity and decrease
the rank. When we find the sparse structure of the factor matrix, we can fixed
the sparse structure, using a small to regularization to decreasing the
recovery error, and one can choose the proper decomposition from the solution
path with sufficient sparse factor matrix with low recovery error. We test the
power of our algorithm on the simulation data and real data, which show it is
powerful.
