  Unregularized deep neural networks (DNNs) can be easily overfit with a
limited sample size. We argue that this is mostly due to the disriminative
nature of DNNs which directly model the conditional probability (or score) of
labels given the input. The ignorance of input distribution makes DNNs
difficult to generalize to unseen data. Recent advances in regularization
techniques, such as pretraining and dropout, indicate that modeling input data
distribution (either explicitly or implicitly) greatly improves the
generalization ability of a DNN. In this work, we explore the manifold
hypothesis which assumes that instances within the same class lie in a smooth
manifold. We accordingly propose two simple regularizers to a standard
discriminative DNN. The first one, named Label-Aware Manifold Regularization,
assumes the availability of labels and penalizes large norms of the loss
function w.r.t. data points. The second one, named Label-Independent Manifold
Regularization, does not use label information and instead penalizes the
Frobenius norm of the Jacobian matrix of prediction scores w.r.t. data points,
which makes semi-supervised learning possible. We perform extensive control
experiments on fully supervised and semi-supervised tasks using the MNIST,
CIFAR10 and SVHN datasets and achieve excellent results.
