Representation Learning: A Review and New Perspectives <s> Sparse Autoencoders <s> THE receptive fields of simple cells in mammalian primary visual cortex can be characterized as being spatially localized, oriented1–4 and bandpass (selective to structure at different spatial scales), comparable to the basis functions of wavelet transforms5,6. One approach to understanding such response properties of visual neurons has been to consider their relationship to the statistical structure of natural images in terms of efficient coding7–12. Along these lines, a number of studies have attempted to train unsupervised learning algorithms on natural images in the hope of developing receptive fields with similar properties13–18, but none has succeeded in producing a full set that spans the image space and contains all three of the above properties. Here we investigate the proposal8,12 that a coding strategy that maximizes sparseness is sufficient to account for these properties. We show that a learning algorithm that attempts to find sparse linear codes for natural scenes will develop a complete family of localized, oriented, bandpass receptive fields, similar to those found in the primary visual cortex. The resulting sparse image code provides a more efficient representation for later stages of processing because it possesses a higher degree of statistical independence among its outputs. <s> BIB001 </s> Representation Learning: A Review and New Perspectives <s> Sparse Autoencoders <s> We describe a novel unsupervised method for learning sparse, overcomplete features. The model uses a linear encoder, and a linear decoder preceded by a sparsifying non-linearity that turns a code vector into a quasi-binary sparse code vector. Given an input, the optimal code minimizes the distance between the output of the decoder and the input patch while being as similar as possible to the encoder output. Learning proceeds in a two-phase EM-like fashion: (1) compute the minimum-energy code vector, (2) adjust the parameters of the encoder and decoder so as to decrease the energy. The model produces "stroke detectors" when trained on handwritten numerals, and Gabor-like filters when trained on natural image patches. Inference and learning are very fast, requiring no preprocessing, and no expensive sampling. Using the proposed unsupervised method to initialize the first layer of a convolutional network, we achieved an error rate slightly lower than the best reported result on the MNIST dataset. Finally, an extension of the method is described to learn topographical filter maps. <s> BIB002 </s> Representation Learning: A Review and New Perspectives <s> Sparse Autoencoders <s> Complexity theory of circuits strongly suggests that deep architectures can be much more efficient (sometimes exponentially) than shallow architectures, in terms of computational elements required to represent some functions. Deep multi-layer neural networks have many levels of non-linearities allowing them to compactly represent highly non-linear and highly-varying functions. However, until recently it was not clear how to train such deep networks, since gradient-based optimization starting from random initialization appears to often get stuck in poor solutions. Hinton et al. recently introduced a greedy layer-wise unsupervised learning algorithm for Deep Belief Networks (DBN), a generative model with many layers of hidden causal variables. In the context of the above optimization problem, we study this algorithm empirically and explore variants to better understand its success and extend it to cases where the inputs are continuous or where the structure of the input distribution is not revealing enough about the variable to be predicted in a supervised task. Our experiments also confirm the hypothesis that the greedy layer-wise unsupervised training strategy mostly helps the optimization, by initializing weights in a region near a good local minimum, giving rise to internal distributed representations that are high-level abstractions of the input, bringing better generalization. <s> BIB003 </s> Representation Learning: A Review and New Perspectives <s> Sparse Autoencoders <s> Motivated in part by the hierarchical organization of the cortex, a number of algorithms have recently been proposed that try to learn hierarchical, or "deep," structure from unlabeled data. While several authors have formally or informally compared their algorithms to computations performed in visual area V1 (and the cochlea), little attempt has been made thus far to evaluate these algorithms in terms of their fidelity for mimicking computations at deeper levels in the cortical hierarchy. This paper presents an unsupervised learning model that faithfully mimics certain properties of visual area V2. Specifically, we develop a sparse variant of the deep belief networks of Hinton et al. (2006). We learn two layers of nodes in the network, and demonstrate that the first layer, similar to prior work on sparse coding and ICA, results in localized, oriented, edge filters, similar to the Gabor functions known to model V1 cell receptive fields. Further, the second layer in our model encodes correlations of the first layer responses in the data. Specifically, it picks up both colinear ("contour") features as well as corners and junctions. More interestingly, in a quantitative comparison, the encoding of these more complex "corner" features matches well with the results from the Ito & Komatsu's study of biological V2 responses. This suggests that our sparse variant of deep belief networks holds promise for modeling more higher-order features. <s> BIB004 </s> Representation Learning: A Review and New Perspectives <s> Sparse Autoencoders <s> Recently, many applications for Restricted Boltzmann Machines (RBMs) have been developed for a large variety of learning problems. However, RBMs are usually used as feature extractors for another learning algorithm or to provide a good initialization for deep feed-forward neural network classifiers, and are not considered as a standalone solution to classification problems. In this paper, we argue that RBMs provide a self-contained framework for deriving competitive non-linear classifiers. We present an evaluation of different learning algorithms for RBMs which aim at introducing a discriminative component to RBM training and improve their performance as classifiers. This approach is simple in that RBMs are used directly to build a classifier, rather than as a stepping stone. Finally, we demonstrate how discriminative RBMs can also be successfully employed in a semi-supervised setting. <s> BIB005 </s> Representation Learning: A Review and New Perspectives <s> Sparse Autoencoders <s> For many pattern recognition tasks, the ideal input feature would be invariant to multiple confounding properties (such as illumination and viewing angle, in computer vision applications). Recently, deep architectures trained in an unsupervised manner have been proposed as an automatic method for extracting useful features. However, it is difficult to evaluate the learned features by any means other than using them in a classifier. In this paper, we propose a number of empirical tests that directly measure the degree to which these learned features are invariant to different input transformations. We find that stacked autoencoders learn modestly increasingly invariant features with depth when trained on natural images. We find that convolutional deep belief networks learn substantially more invariant features in each layer. These results further justify the use of "deep" vs. "shallower" representations, but suggest that mechanisms beyond merely stacking one autoencoder on top of another may be important for achieving invariance. Our evaluation metrics can also be used to evaluate future work in deep learning, and thus help the development of future algorithms. <s> BIB006 </s> Representation Learning: A Review and New Perspectives <s> Sparse Autoencoders <s> The predominant methodology in training deep learning advocates the use of stochastic gradient descent methods (SGDs). Despite its ease of implementation, SGDs are difficult to tune and parallelize. These problems make it challenging to develop, debug and scale up deep learning algorithms with SGDs. In this paper, we show that more sophisticated off-the-shelf optimization methods such as Limited memory BFGS (L-BFGS) and Conjugate gradient (CG) with line search can significantly simplify and speed up the process of pretraining deep algorithms. In our experiments, the difference between L-BFGS/CG and SGDs are more pronounced if we consider algorithmic extensions (e.g., sparsity regularization) and hardware extensions (e.g., GPUs or computer clusters). Our experiments with distributed optimization support the use of L-BFGS with locally connected networks and convolutional neural networks. Using L-BFGS, our convolutional network model achieves 0.69% on the standard MNIST dataset. This is a state-of-the-art result on MNIST among algorithms that do not use distortions or pretraining. <s> BIB007
The earliest uses of single-layer autoencoders for building deep architectures by stacking them BIB003 considered the idea of tying the encoder weights and decoder weights to restrict capacity as well as the idea of introducing a form of sparsity regularization BIB002 . Sparsity in the representation can be achieved by penalizing the hidden unit biases (making these additive offset parameters more negative) BIB002 , BIB004 , BIB006 , BIB005 or by directly penalizing the output of the hidden unit activations (making them closer to their saturating value at 0)  , BIB007 ,  . Penalizing the bias runs the danger that the weights could compensate for the bias, which could hurt numerical optimization. When directly penalizing the hidden unit outputs, several variants can be found in the literature, but a clear comparative analysis is still lacking. Although the L1 penalty (i.e., simply the sum of output elements h j in the case of sigmoid nonlinearity) would seem the most natural (because of its use in sparse coding), it is used in few papers involving sparse autoencoders. A close cousin of the L1 penalty is the Student-t penalty (logð1 þ h 2 j Þ), originally proposed for sparse coding BIB001 . Several papers penalize the average output " h j (e.g., over a minibatch), and instead of pushing it to 0 encourage it to approach a fixed target, either through a mean-square error penalty or maybe, more sensibly (because h j behaves like a probability), a Kullback-Liebler divergence with respect to the binomial distribution with probability : À log " h j À ð1 À Þ logð1 À " h j Þ þ constant, for example, with ¼ 0:05.