  Matrix completion has received vast amount of attention and research due to
its wide applications in various study fields. Existing methods of matrix
completion consider only nonlinear (or linear) relations among entries in a
data matrix and ignore linear (or nonlinear) relationships latent. This paper
introduces a new latent variables model for data matrix which is a combination
of linear and nonlinear models and designs a novel deep-neural-network-based
matrix completion algorithm to address both linear and nonlinear relations
among entries of data matrix. The proposed method consists of two branches. The
first branch learns the latent representations of columns and reconstructs the
columns of the partially observed matrix through a series of hidden neural
network layers. The second branch does the same for the rows. In addition,
based on multi-task learning principles, we enforce these two branches work
together and introduce a new regularization technique to reduce over-fitting.
More specifically, the missing entries of data are recovered as a main task and
manifold learning is performed as an auxiliary task. The auxiliary task
constrains the weights of the network so it can be considered as a regularizer,
improving the main task and reducing over-fitting. Experimental results
obtained on the synthetic data and several real-world data verify the
effectiveness of the proposed method compared with state-of-the-art matrix
completion methods.
