Computer vision has grown tremendously in the past two decades. Despite all efforts, existing attempts at matching parts of the human visual system's extraordinary ability to understand visual scenes lack either scope or power. By combining the advantages of general low-level generative models and powerful layer-based and hierarchical models, this work aims at being a first step toward richer, more flexible models of images. After comparing various types of restricted Boltzmann machines (RBMs) able to model continuous-valued data, we introduce our basic model, the masked RBM, which explicitly models occlusion boundaries in image patches by factoring the appearance of any patch region from its shape. We then propose a generative model of larger images using a field of such RBMs. Finally, we discuss how masked RBMs could be stacked to form a deep model able to generate more complicated structures and suitable for various tasks such as segmentation or object recognition.

Deep belief networks (DBN) are generative models with many layers of hidden causal variables, recently introduced by Hinton, Osindero, and Teh ( 2006 ), along with a greedy layer-wise unsupervised learning algorithm. Building on Le Roux and Bengio ( 2008 ) and Sutskever and Hinton ( 2008 ), we show that deep but narrow generative networks do not require more parameters than shallow ones to achieve universal approximation. Exploiting the proof technique, we prove that deep but narrow feedforward neural networks with sigmoidal units can represent any Boolean expression.

Deep belief networks (DBN) are generative neural network models with many layers of hidden explanatory factors, recently introduced by Hinton, Osindero, and Teh (2006) along with a greedy layer-wise unsupervised learning algorithm. The building block of a DBN is a probabilistic model called a restricted Boltzmann machine (RBM), used to represent one layer of the model. Restricted Boltzmann machines are interesting because inference is easy in them and because they have been successfully used as building blocks for training deeper models. We first prove that adding hidden units yields strictly improved modeling power, while a second theorem shows that RBMs are universal approximators of discrete distributions. We then study the question of whether DBNs with more layers are strictly more powerful in terms of representational power. This suggests a new and less greedy criterion for training RBMs within DBNs.

In this letter, we show a direct relation between spectral embedding methods and kernel principal components analysis and how both are special cases of a more general learning problem: learning the principal eigenfunctions of an operator defined from a kernel and the unknown data-generating density. Whereas spectral embedding methods provided only coordinates for the training points, the analysis justifies a simple extension to out-of-sample examples (the Nyström formula) for multidimensional scaling (MDS), spectral clustering, Laplacian eigenmaps, locally linear embedding (LLE), and Isomap. The analysis provides, for all such spectral embedding methods, the definition of a loss function, whose empirical average is minimized by the traditional algorithms. The asymptotic expected value of that loss defines a generalization performance and clarifies what these algorithms are trying to learn. Experiments with LLE, Isomap, spectral clustering, and MDS show that this out-of-sample embedding formula generalizes well, with a level of error comparable to the effect of small perturbations of the training set on the embedding.