  In this paper, we propose a novel multi-task learning method based on the
deep convolutional network. The proposed deep network has four convolutional
layers, three max-pooling layers, and two parallel fully connected layers. To
adjust the deep network to multi-task learning problem, we propose to learn a
low-rank deep network so that the relation among different tasks can be
explored. We proposed to minimize the number of independent parameter rows of
one fully connected layer to explore the relations among different tasks, which
is measured by the nuclear norm of the parameter of one fully connected layer,
and seek a low-rank parameter matrix. Meanwhile, we also propose to regularize
another fully connected layer by sparsity penalty, so that the useful features
learned by the lower layers can be selected. The learning problem is solved by
an iterative algorithm based on gradient descent and back-propagation
algorithms. The proposed algorithm is evaluated over benchmark data sets of
multiple face attribute prediction, multi-task natural language processing, and
joint economics index predictions. The evaluation results show the advantage of
the low-rank deep CNN model over multi-task problems.
