Dropout is a popular regularization technique in neural networks. Yet, the reason for its success is still not fully understood. This paper provides a new interpretation of Dropout from a frame theory perspective. By drawing a connection to recent developments in analog channel coding, we suggest that for a certain family of autoencoders with a linear encoder, the minimizer of an optimization with dropout regularization on the encoder is an equiangular tight frame (ETF). Since this optimization is non-convex, we add another regularization that promotes such structures by minimizing the cross-correlation between filters in the network. We demonstrate its applicability in convolutional and fully connected layers in both feed-forward and recurrent networks. All these results suggest that there is indeed a relationship between dropout and ETF structure of the regularized linear operations.
Autoencoders may be trained end-to-end or gradually layer by layer. In the latter case, they are ''stacked'' together, which leads to a deeper encoder. In @cite_58 , this is done with convolutional autoencoders, and in @cite_56 with denoising ones.
@cite_58 @cite_56
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We show how a deep denoising autoencoder with lateral connections can be used as an auxiliary unsupervised learning task to support supervised learning. The proposed model is trained to minimize simultaneously the sum of supervised and unsupervised cost functions by back-propagation, avoiding the need for layer-wise pretraining. It improves the state of the art significantly in the permutation-invariant MNIST classification task. We explore an original strategy for building deep networks, based on stacking layers of denoising autoencoders which are trained locally to denoise corrupted versions of their inputs. The resulting algorithm is a straightforward variation on the stacking of ordinary autoencoders. It is however shown on a benchmark of classification problems to yield significantly lower classification error, thus bridging the performance gap with deep belief networks (DBN), and in several cases surpassing it. Higher level representations learnt in this purely unsupervised fashion also help boost the performance of subsequent SVM classifiers. Qualitative experiments show that, contrary to ordinary autoencoders, denoising autoencoders are able to learn Gabor-like edge detectors from natural image patches and larger stroke detectors from digit images. This work clearly establishes the value of using a denoising criterion as a tractable unsupervised objective to guide the learning of useful higher level representations. Denoising autoencoders have been previously shown to be competitive alternatives to restricted Boltzmann machines for unsupervised pretraining of each layer of a deep architecture. We show that a simple denoising autoencoder training criterion is equivalent to matching the score (with respect to the data) of a specific energy-based model to that of a nonparametric Parzen density estimator of the data. This yields several useful insights. It defines a proper probabilistic model for the denoising autoencoder technique, which makes it in principle possible to sample from them or rank examples by their energy. It suggests a different way to apply score matching that is related to learning to denoise and does not require computing second derivatives. It justifies the use of tied weights between the encoder and decoder and suggests ways to extend the success of denoising autoencoders to a larger family of energy-based models. We present in this paper a novel approach for training deterministic auto-encoders. We show that by adding a well chosen penalty term to the classical reconstruction cost function, we can achieve results that equal or surpass those attained by other regularized auto-encoders as well as denoising auto-encoders on a range of datasets. This penalty term corresponds to the Frobenius norm of the Jacobian matrix of the encoder activations with respect to the input. We show that this penalty term results in a localized space contraction which in turn yields robust features on the activation layer. Furthermore, we show how this penalty term is related to both regularized auto-encoders and denoising auto-encoders and how it can be seen as a link between deterministic and non-deterministic auto-encoders. We find empirically that this penalty helps to carve a representation that better captures the local directions of variation dictated by the data, corresponding to a lower-dimensional non-linear manifold, while being more invariant to the vast majority of directions orthogonal to the manifold. Finally, we show that by using the learned features to initialize a MLP, we achieve state of the art classification error on a range of datasets, surpassing other methods of pretraining.