We show how to use “complementary priors ” to eliminate the explaining away effects that make inference difficult in densely-connected belief nets that have many hidden layers. Using complementary priors, we derive a fast, greedy algorithm that can learn deep, directed belief networks one layer at a time, provided the top two layers form an undirected associative memory. The fast, greedy algorithm is used to initialize a slower learning procedure that fine-tunes the weights using a contrastive version of the wake-sleep algorithm. After fine-tuning, a network with three hidden layers forms a very good generative model of the joint distribution of handwritten digit images and their labels. This generative model gives better digit classification than the best discriminative learning algorithms. The low-dimensional manifolds on which the digits lie are modelled by long ravines in the free-energy landscape of the top-level associative memory and it is easy to explore these ravines by using the directed connections to display what the associative memory has in mind. 1

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 self-supervised learning that may relate to the function of bottom-up and top-down cortical processing pathways.

Most of the existing approaches to collaborative filtering cannot handle very large data sets. In this paper we show how a class of two-layer 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 carefully-tuned 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.

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 Pseudo-Likelihood 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.

We present an energy-based model that uses a product of generalised Student-t distributions to capture the statistical structure in datasets. This model is inspired by and particularly applicable to “natural ” datasets such as images. We begin by providing the mathematical framework, where we discuss complete and overcomplete models, and provide algorithms for training these models from data. Using patches of natural scenes we demonstrate that our approach represents a viable alternative to “independent components analysis ” as an interpretive model of biological visual systems. Although the two approaches are similar in flavor there are also important differences, particularly when the representations are overcomplete. By constraining the interactions within our model we are also able to study the topographic organization of Gabor-like receptive fields that are learned by our model. Finally, we discuss the relation of our new approach to previous work — in particular Gaussian Scale Mixture models, and variants of independent components analysis. 1

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.

Deep Belief Networks (DBN’s) are generative models that contain many layers of hidden variables. Efficient greedy algorithms for learning and approximate inference have allowed these models to be applied successfully in many application domains. The main building block of a DBN is a bipartite undirected graphical model called a restricted Boltzmann machine (RBM). Due to the presence of the partition function, model selection, complexity control, and exact maximum likelihood learning in RBM’s are intractable. We show that Annealed Importance Sampling (AIS) can be used to efficiently estimate the partition function of an RBM, and we present a novel AIS scheme for comparing RBM’s with different architectures. We further show how an AIS estimator, along with approximate inference, can be used to estimate a lower bound on the log-probability that a DBN model with multiple hidden layers assigns to the test data. This is, to our knowledge, the first step towards obtaining quantitative results that would allow us to directly assess the performance of Deep Belief Networks as generative models of data. 1.

Abstract: We propose a family of statistical models for social network evolution over time, which represents an extension of Exponential Random Graph Models (ERGMs). Many of the methods for ERGMs are readily adapted for these models, including maximum likelihood estimation algorithms. We discuss models of this type and their properties, and give examples, as well as a demonstration of their use for hypothesis testing and classification. We believe our temporal ERG models represent a useful new framework for modeling time-evolving social networks, and rewiring networks from other domains such as gene regulation circuitry, and communication networks. Received November 2009. 1.

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 three-way 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 three-way interactions greatly improve the model’s ability to blend motion styles or to transition smoothly between them. 1.

We describe a family of non-linear sequence models that is substantially more powerful than hidden Markov models or linear dynamical systems. Our models have simple approximate inference and learning procedures that work well in practice. Multilevel representations of sequential data can be learned one hidden layer at a time, and adding extra hidden layers improves the resulting generative models. The models can be trained with very high-dimensional, very non-linear data such as raw pixel sequences. Their performance is demonstrated using synthetic video sequences of two balls bouncing in a box.