We present a mixture model whose components are Restricted Boltzmann Machines (RBMs). This possibility has not been considered before because computing the partition function of an RBM is intractable, which appears to make learning a mixture of RBMs intractable as well. Surprisingly, when formulated as a third-order Boltzmann machine, such a mixture model can be learned tractably using contrastive divergence. The energy function of the model captures three-way interactions among visible units, hidden units, and a single hidden multinomial unit that represents the cluster labels. The distinguishing feature of this model is that, unlike other mixture models, the mixing proportions are not explicitly parameterized. Instead, they are defined implicitly via the energy function and depend on all the parameters in the model. We present results for the MNIST and NORB datasets showing that the implicit mixture of RBMs learns clusters that reflect the class structure in the data.

The Temporal Restricted Boltzmann Machine (TRBM) is a probabilistic model for sequences that is able to successfully model (i.e., generate nice-looking 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.

We show how to model documents as bags of words using family of two-layer, undirected graphical models. Each member of the family has the same number of binary hidden units but a different number of ``softmax visible units. All of the softmax units in all of the models in the family share the same weights to the binary hidden units. We describe efficient inference and learning procedures for such a family. Each member of the family models the probability distribution of documents of a specific length as a product of topic-specific distributions rather than as a mixture and this gives much better generalization than Latent Dirichlet Allocation for modeling the log probabilities of held-out documents. The low-dimensional topic vectors learned by the undirected family are also much better than LDA topic vectors for retrieving documents that are similar to a query document. The learned topics are more general than those found by LDA because precision is achieved by intersecting many general topics rather than by selecting a single precise topic to generate each word.

We consider the problem of zero-shot learning, where the goal is to learn a classifier $f: X \rightarrow Y$ that must predict novel values of $Y$ that were omitted from the training set. To achieve this, we define the notion of a semantic output code classifier (SOC) which utilizes a knowledge base of semantic properties of $Y$ to extrapolate to novel classes. We provide a formalism for this type of classifier and study its theoretical properties in a PAC framework, showing conditions under which the classifier can accurately predict novel classes. As a case study, we build a SOC classifier for a neural decoding task and show that it can often predict words that people are thinking about from functional magnetic resonance images (fMRI) of their neural activity, even without training examples for those words.

We show how to use unlabeled data and a deep belief net (DBN) to learn a good covariance kernel for a Gaussian process. We first learn a deep generative model of the unlabeled data using the fast, greedy algorithm introduced by Hinton et.al. If the data is high-dimensional and highly-structured, a Gaussian kernel applied to the top layer of features in the DBN works much better than a similar kernel applied to the raw input. Performance at both regression and classification can then be further improved by using backpropagation through the DBN to discriminatively fine-tune the covariance kernel.

We describe an efficient learning procedure for multilayer generative models that combine the best aspects of Markov random fields and deep, directed belief nets. The generative models can be learned one layer at a time and when learning is complete they have a very fast inference procedure for computing a good approximation to the posterior distribution in all of the hidden layers. Each hidden layer has its own MRF whose energy function is modulated by the top-down directed connections from the layer above. To generate from the model, each layer in turn must settle to equilibrium given its top-down input. We show that this type of model is good at capturing the statistics of patches of natural images.

We propose a nonlinear generative model for human motion data that uses an undirected model with binary latent variables and real-valued "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 time-steps. 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.

The latent and Visible variables at each time step receive directed connections from the Visible variables at the last few time-steps. Such an architecture makes online inference efficient and allows us to use a simple approximate learning procedure.

We describe a generative model for handwritten digits that uses two pairs of opposing springs whose stiffnesses are controlled by a motor program. We show how neural networks can be trained to infer the motor programs required to accurately reconstruct the MNIST digits. The inferred motor programs can be used directly for digit classification, but they can also be used in other ways. By adding noise to the motor program inferred from an MNIST image we can generate a large set of very different images of the same class, thus enlarging the training set available to other methods. We can also use the motor programs as additional, highly informative outputs which reduce overfitting when training a feed-forward classifier.

We describe a way of using multiple different types of similarity relationship to learn a low-dimensional embedding of a dataset. Our method chooses different, possibly overlapping representations of similarity by individually reweighting the dimensions of a common underlying latent space. When applied to a single similarity relation that is based on Euclidean distances between the input data points, the method reduces to simple dimensionality reduction. If additional information is available about the dataset or about subsets of it, we can use this information to clean up or otherwise improve the embedding. We demonstrate the potential usefulness of this form of semi-supervised dimensionality reduction on some simple examples.