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89
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 195 (21 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.
Learning multiple layers of features from tiny images
, 2009
"... April 8, 2009Groups at MIT and NYU have collected a dataset of millions of tiny colour images from the web. It is, in principle, an excellent dataset for unsupervised training of deep generative models, but previous researchers who have tried this have found it di cult to learn a good set of lters f ..."
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Cited by 110 (5 self)
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April 8, 2009Groups at MIT and NYU have collected a dataset of millions of tiny colour images from the web. It is, in principle, an excellent dataset for unsupervised training of deep generative models, but previous researchers who have tried this have found it di cult to learn a good set of lters from the images. We show how to train a multilayer generative model that learns to extract meaningful features which resemble those found in the human visual cortex. Using a novel parallelization algorithm to distribute the work among multiple machines connected on a network, we show how training such a model can be done in reasonable time. A second problematic aspect of the tiny images dataset is that there are no reliable class labels which makes it hard to use for object recognition experiments. We created two sets of reliable labels. The CIFAR10 set has 6000 examples of each of 10 classes and the CIFAR100 set has 600 examples of each of 100 nonoverlapping classes. Using these labels, we show that object recognition is signi cantly
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 94 (15 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 + �
Using Fast Weights to Improve Persistent Contrastive Divergence
"... The most commonly used learning algorithm for restricted Boltzmann machines is contrastive divergence which starts a Markov chain at a data point and runs the chain for only a few iterations to get a cheap, low variance estimate of the sufficient statistics under the model. Tieleman (2008) showed th ..."
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Cited by 35 (13 self)
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The most commonly used learning algorithm for restricted Boltzmann machines is contrastive divergence which starts a Markov chain at a data point and runs the chain for only a few iterations to get a cheap, low variance estimate of the sufficient statistics under the model. Tieleman (2008) showed that better learning can be achieved by estimating the model’s statistics using a small set of persistent ”fantasy particles ” that are not reinitialized to data points after each weight update. With sufficiently small weight updates, the fantasy particles represent the equilibrium distribution accurately but to explain why the method works with much larger weight updates it is necessary to consider the interaction between the weight updates and the Markov chain. We show that the weight updates force the Markov chain to mix fast, and using this insight we develop an even faster mixing chain that uses an auxiliary set of ”fast weights ” to implement a temporary overlay on the energy landscape. The fast weights learn rapidly but also decay rapidly and do not contribute to the normal energy landscape that defines the model. 1.
Factored Conditional Restricted Boltzmann Machines for Modeling Motion Style
"... The Conditional Restricted Boltzmann Machine (CRBM) is a recently proposed model for time series that has a rich, distributed hidden state and permits simple, exact inference. We present a new model, based on the CRBM that preserves its most important computational properties and includes multiplica ..."
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Cited by 30 (8 self)
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The Conditional Restricted Boltzmann Machine (CRBM) is a recently proposed model for time series that has a rich, distributed hidden state and permits simple, exact inference. We present a new model, based on the CRBM that preserves its most important computational properties and includes multiplicative threeway interactions that allow the effective interaction weight between two units to be modulated by the dynamic state of a third unit. We factorize the threeway weight tensor implied by the multiplicative model, reducing the number of parameters from O(N 3) to O(N 2). The result is an efficient, compact model whose effectiveness we demonstrate by modeling human motion. Like the CRBM, our model can capture diverse styles of motion with a single set of parameters, and the threeway interactions greatly improve the model’s ability to blend motion styles or to transition smoothly between them. 1.
The Recurrent Temporal Restricted Boltzmann Machine
"... The Temporal Restricted Boltzmann Machine (TRBM) is a probabilistic model for sequences that is able to successfully model (i.e., generate nicelooking samples of) several very high dimensional sequences, such as motion capture data and the pixels of low resolution videos of balls bouncing in a box. ..."
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Cited by 18 (2 self)
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The Temporal Restricted Boltzmann Machine (TRBM) is a probabilistic model for sequences that is able to successfully model (i.e., generate nicelooking samples of) several very high dimensional sequences, such as motion capture data and the pixels of low resolution videos of balls bouncing in a box. The major disadvantage of the TRBM is that exact inference is extremely hard, since even computing a Gibbs update for a single variable of the posterior is exponentially expensive. This difficulty has necessitated the use of a heuristic inference procedure, that nonetheless was accurate enough for successful learning. In this paper we introduce the Recurrent TRBM, which is a very slight modification of the TRBM for which exact inference is very easy and exact gradient learning is almost tractable. We demonstrate that the RTRBM is better than an analogous TRBM at generating motion capture and videos of bouncing balls. 1
Learning in markov random fields using tempered transitions
 In Advances in Neural Information Processing Systems
"... Markov random fields (MRF’s), or undirected graphical models, provide a powerful framework for modeling complex dependencies among random variables. Maximum likelihood learning in MRF’s is hard due to the presence of the global normalizing constant. In this paper we consider a class of stochastic ap ..."
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Cited by 18 (2 self)
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Markov random fields (MRF’s), or undirected graphical models, provide a powerful framework for modeling complex dependencies among random variables. Maximum likelihood learning in MRF’s is hard due to the presence of the global normalizing constant. In this paper we consider a class of stochastic approximation algorithms of the RobbinsMonro type that use Markov chain Monte Carlo to do approximate maximum likelihood learning. We show that using MCMC operators based on tempered transitions enables the stochastic approximation algorithm to better explore highly multimodal distributions, which considerably improves parameter estimates in large, denselyconnected MRF’s. Our results on MNIST and NORB datasets demonstrate that we can successfully learn good generative models of highdimensional, richly structured data that perform well on digit and object recognition tasks. 1
Inductive Principles for Restricted Boltzmann Machine Learning
"... Recent research has seen the proposal of several new inductive principles designed specifically to avoid the problems associated with maximum likelihood learning in models with intractable partition functions. In this paper, we study learning methods for binary restricted Boltzmann machines (RBMs) b ..."
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Cited by 18 (6 self)
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Recent research has seen the proposal of several new inductive principles designed specifically to avoid the problems associated with maximum likelihood learning in models with intractable partition functions. In this paper, we study learning methods for binary restricted Boltzmann machines (RBMs) based on ratio matching and generalized score matching. We compare these new RBM learning methods to a range of existing learning methods including stochastic maximum likelihood, contrastive divergence, and pseudolikelihood. We perform an extensive empirical evaluation across multiple tasks and data sets. 1
Herding Dynamical Weights to Learn
"... A new “herding ” algorithm is proposed which directly converts observed moments into a sequence of pseudosamples. The pseudosamples respect the moment constraints and may be used to estimate (unobserved) quantities of interest. The procedure allows us to sidestep the usual approach of first learnin ..."
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Cited by 16 (6 self)
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A new “herding ” algorithm is proposed which directly converts observed moments into a sequence of pseudosamples. The pseudosamples respect the moment constraints and may be used to estimate (unobserved) quantities of interest. The procedure allows us to sidestep the usual approach of first learning a joint model (which is intractable) and then sampling from that model (which can easily get stuck in a local mode). Moreover, the algorithm is fully deterministic, avoiding random number generation) and does not need expensive operations such as exponentiation. 1.
Learning Deep Boltzmann Machines using Adaptive MCMC
"... When modeling highdimensional richly structured data, it is often the case that the distribution defined by the Deep Boltzmann Machine (DBM) has a rough energy landscape with many local minima separated by high energy barriers. The commonly used Gibbs sampler tends to get trapped in one local mode, ..."
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Cited by 12 (1 self)
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When modeling highdimensional richly structured data, it is often the case that the distribution defined by the Deep Boltzmann Machine (DBM) has a rough energy landscape with many local minima separated by high energy barriers. The commonly used Gibbs sampler tends to get trapped in one local mode, which often results in unstable learning dynamics and leads to poor parameter estimates. In this paper, we concentrate on learning DBM’s using adaptive MCMC algorithms. We first show a close connection between Fast PCD and adaptive MCMC. We then develop a Coupled Adaptive Simulated Tempering algorithm that can be used to better explore a highly multimodal energy landscape. Finally, we demonstrate that the proposed algorithm considerably improves parameter estimates, particularly when learning largescale DBM’s. 1.