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421
The Helmholtz Machine
, 1995
"... Discovering the structure inherent in a set of patterns is a fundamental aim of statistical inference or learning. One fruitful approach is to build a parameterized stochastic generative model, independent draws from which are likely to produce the patterns. For all but the simplest generative model ..."
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Cited by 194 (22 self)
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Discovering the structure inherent in a set of patterns is a fundamental aim of statistical inference or learning. One fruitful approach is to build a parameterized stochastic generative model, independent draws from which are likely to produce the patterns. For all but the simplest generative models, each pattern can be generated in exponentially many ways. It is thus intractable to adjust the parameters to maximize the probability of the observed patterns. We describe a way of finessing this combinatorial explosion by maximizing an easily computed lower bound on the probability of the observations. Our method can be viewed as a form of hierarchical selfsupervised learning that may relate to the function of bottomup and topdown cortical processing pathways.
Greedy layerwise training of deep networks
 In NIPS
, 2007
"... 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 multilayer neural networks have many levels of nonlinearities allow ..."
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Cited by 184 (32 self)
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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 multilayer neural networks have many levels of nonlinearities allowing them to compactly represent highly nonlinear and highlyvarying functions. However, until recently it was not clear how to train such deep networks, since gradientbased optimization starting from random initialization appears to often get stuck in poor solutions. Hinton et al. recently introduced a greedy layerwise 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 layerwise 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 highlevel abstractions of the input, bringing better generalization.
Convolutional Deep Belief Networks for Scalable Unsupervised Learning of Hierarchical Representations
"... There has been much interest in unsupervised learning of hierarchical generative models such as deep belief networks. Scaling such models to fullsized, highdimensional images remains a difficult problem. To address this problem, we present the convolutional deep belief network, a hierarchical gene ..."
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Cited by 160 (15 self)
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There has been much interest in unsupervised learning of hierarchical generative models such as deep belief networks. Scaling such models to fullsized, highdimensional images remains a difficult problem. To address this problem, we present the convolutional deep belief network, a hierarchical generative model which scales to realistic image sizes. This model is translationinvariant and supports efficient bottomup and topdown probabilistic inference. Key to our approach is probabilistic maxpooling, a novel technique which shrinks the representations of higher layers in a probabilistically sound way. Our experiments show that the algorithm learns useful highlevel visual features, such as object parts, from unlabeled images of objects and natural scenes. We demonstrate excellent performance on several visual recognition tasks and show that our model can perform hierarchical (bottomup and topdown) inference over fullsized images. 1.
Restricted Boltzmann machines for collaborative filtering
 In Machine Learning, Proceedings of the Twentyfourth International Conference (ICML 2004). ACM
, 2007
"... Most of the existing approaches to collaborative filtering cannot handle very large data sets. In this paper we show how a class of twolayer undirected graphical models, called Restricted Boltzmann Machines (RBM’s), can be used to model tabular data, such as user’s ratings of movies. We present eff ..."
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Cited by 118 (12 self)
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Most of the existing approaches to collaborative filtering cannot handle very large data sets. In this paper we show how a class of twolayer undirected graphical models, called Restricted Boltzmann Machines (RBM’s), can be used to model tabular data, such as user’s ratings of movies. We present efficient learning and inference procedures for this class of models and demonstrate that RBM’s can be successfully applied to the Netflix data set, containing over 100 million user/movie ratings. We also show that RBM’s slightly outperform carefullytuned SVD models. When the predictions of multiple RBM models and multiple SVD models are linearly combined, we achieve an error rate that is well over 6 % better than the score of Netflix’s own system. 1.
Efficient learning of sparse representations with an energybased model
 ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS (NIPS 2006
, 2006
"... 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 nonlinearity that turns a code vector into a quasibinary sparse code vector. Given an input, the optimal code minimizes the distance b ..."
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Cited by 116 (14 self)
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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 nonlinearity that turns a code vector into a quasibinary 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 twophase EMlike fashion: (1) compute the minimumenergy 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 Gaborlike 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. 1
Unsupervised learning of invariant feature hierarchies with application to object recognition.” CVPR, 2007. 1 Data Driven HMC Algorithm. DDHMC (motionbased proposals) 1: Initialize chain with τo 2: for i = 1 to nsamples do 3: // 1. DataDriven: Get Propo
 Initialize the Acceptance, H(qo, po), and the Proposal, H ′ (qo, po ) Hamiltonians , τq) 14: po = DMotion(τ ′ i , τq) 15: qo = DF orm(τ ′ i , τq) 16: draw po ∼ N (0, 1) 17: // 2. Perturbation on H ′ using Leapfrog 18: for j=1 to l do 13: qo = DF orm(τ ′ i
"... We present an unsupervised method for learning a hierarchy of sparse feature detectors that are invariant to small shifts and distortions. The resulting feature extractor consists of multiple convolution filters, followed by a pointwise sigmoid nonlinearity, and a featurepooling layer that compute ..."
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Cited by 109 (13 self)
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We present an unsupervised method for learning a hierarchy of sparse feature detectors that are invariant to small shifts and distortions. The resulting feature extractor consists of multiple convolution filters, followed by a pointwise sigmoid nonlinearity, and a featurepooling layer that computes the max of each filter output within adjacent windows. A second level of larger and more invariant features is obtained by training the same algorithm on patches of features from the first level. Training a supervised classifier on these features yields 0.64 % error on MNIST, and 54 % average recognition rate on Caltech 101
Modeling human motion using binary latent variables
 Advances in Neural Information Processing Systems
, 2006
"... We propose a nonlinear generative model for human motion data that uses an undirected model with binary latent variables and realvalued “visible ” variables that represent joint angles. The latent and visible variables at each time step receive directed connections from the visible variables at th ..."
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Cited by 90 (20 self)
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We propose a nonlinear generative model for human motion data that uses an undirected model with binary latent variables and realvalued “visible ” variables that represent joint angles. The latent and visible variables at each time step receive directed connections from the visible variables at the last few timesteps. Such an architecture makes online inference efficient and allows us to use a simple approximate learning procedure. After training, the model finds a single set of parameters that simultaneously capture several different kinds of motion. We demonstrate the power of our approach by synthesizing various motion sequences and by performing online filling in of data lost during motion capture. Website:
Spiking Boltzmann machines
 In Advances in Neural Information Processing Systems
, 1998
"... A Boltzmann Machine is a network of symmetrically connected, neuronlike units that make stochastic decisions about whether to be on or off. Boltzmann machines have a simple learning algorithm that allows them to discover interesting features in datasets composed of binary vectors. The learning algor ..."
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Cited by 85 (14 self)
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A Boltzmann Machine is a network of symmetrically connected, neuronlike units that make stochastic decisions about whether to be on or off. Boltzmann machines have a simple learning algorithm that allows them to discover interesting features in datasets composed of binary vectors. The learning algorithm is very slow in networks with many layers of feature detectors, but it can be made much faster by learning one layer of feature detectors at a time. Boltzmann machines are used to solve two quite different computational problems. For a search problem, the weights on the connections are fixed and are used to represent the cost function of an optimization problem. The stochastic dynamics of a Boltzmann machine then allow it to sample binary state vectors that represent good solutions to the optimization problem. For a learning problem, the Boltzmann machine is shown a set of binary data vectors and it must find weights on the connections so that the data vectors are good solutions to the optimization problem defined by those weights. To solve a learning problem, Boltzmann machines make many small updates to their weights, and each update requires them to solve many different search problems. The stochastic dynamics of a Boltzmann machine When unit i is given the opportunity to update its binary state, it first computes its total input, zi, which is the sum of its own bias, bi, and the weights on connections coming from other active units: zi = bi + �
Training restricted Boltzmann machines using approximations to the likelihood gradient
 Proceedings of the 25th international conference on Machine learning
, 2008
"... A new algorithm for training Restricted Boltzmann Machines is introduced. The algorithm, named Persistent Contrastive Divergence, is different from the standard Contrastive Divergence algorithms in that it aims to draw samples from almost exactly the model distribution. It is compared to some standa ..."
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Cited by 84 (2 self)
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A new algorithm for training Restricted Boltzmann Machines is introduced. The algorithm, named Persistent Contrastive Divergence, is different from the standard Contrastive Divergence algorithms in that it aims to draw samples from almost exactly the model distribution. It is compared to some standard Contrastive Divergence and PseudoLikelihood algorithms on the tasks of modeling and classifying various types of data. The Persistent Contrastive Divergence algorithm outperforms the other algorithms, and is equally fast and simple.