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.
Since in training one may just get the identity operator for @math and @math , which keeps the achieved representation the same as the input, some additional regularization is required. One option is to make the dimension of the representation smaller than the input. Another option is using denoising autoencoders @cite_32 . In these architectures, the input is disrupted by some noise (e.g., additive white Gaussian noise or erasures using Dropout) and the encoder is expected to reconstruct the clean version of the input.
@cite_32
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Denoising auto-encoders (DAEs) have been successfully used to learn new representations for a wide range of machine learning tasks. During training, DAEs make many passes over the training dataset and reconstruct it from partial corruption generated from a pre-specified corrupting distribution. This process learns robust representation, though at the expense of requiring many training epochs, in which the data is explicitly corrupted. In this paper we present the marginalized Denoising Auto-encoder (mDAE), which (approximately) marginalizes out the corruption during training. Effectively, the mDAE takes into account infinitely many corrupted copies of the training data in every epoch, and therefore is able to match or outperform the DAE with much fewer training epochs. We analyze our proposed algorithm and show that it can be understood as a classic auto-encoder with a special form of regularization. In empirical evaluations we show that it attains 1-2 order-of-magnitude speedup in training time over other competing approaches. Deep autoencoders, and other deep neural networks, have demonstrated their effectiveness in discovering non-linear features across many problem domains. However, in many real-world problems, large outliers and pervasive noise are commonplace, and one may not have access to clean training data as required by standard deep denoising autoencoders. Herein, we demonstrate novel extensions to deep autoencoders which not only maintain a deep autoencoders' ability to discover high quality, non-linear features but can also eliminate outliers and noise without access to any clean training data. Our model is inspired by Robust Principal Component Analysis, and we split the input data X into two parts, @math , where @math can be effectively reconstructed by a deep autoencoder and @math contains the outliers and noise in the original data X. Since such splitting increases the robustness of standard deep autoencoders, we name our model a "Robust Deep Autoencoder (RDA)". Further, we present generalizations of our results to grouped sparsity norms which allow one to distinguish random anomalies from other types of structured corruptions, such as a collection of features being corrupted across many instances or a collection of instances having more corruptions than their fellows. Such "Group Robust Deep Autoencoders (GRDA)" give rise to novel anomaly detection approaches whose superior performance we demonstrate on a selection of benchmark problems. 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.