Lawrence K. Saul

An Introduction to Variational Methods for Graphical Models

This paper presents a tutorial introduction to the use of variational methods for inference and learning in graphical models (Bayesian networks and Markov random fields). We present a number of examples of graphical models, including the QMR-DT database, the sigmoid belief network, the Boltzmann machine, and several variants of hidden Markov models, in which it is infeasible to run exact inference algorithms. We then introduce variational methods, which exploit laws of large numbers to transform the original graphical model into a simplified graphical model in which inference is efficient. Inference in the simpified model provides bounds on probabilities of interest in the original model. We describe a general framework for generating variational transformations based on convex duality. Finally we return to the examples and demonstrate how variational algorithms can be formulated in each case.

Think globally, fit locally: unsupervised learning of low dimensional manifolds

The problem of dimensionality reduction arises in many fields of information processing, including machine learning, data compression, scientific visualization, pattern recognition, and neural computation. Here we describe locally linear embedding (LLE), an unsupervised learning algorithm that computes low dimensional, neighborhood preserving embeddings of high dimensional data. The data, assumed to be sampled from an underlying manifold, are mapped into a single global coordinate system of lower dimensionality. The mapping is derived from the symmetries of locally linear reconstructions, and the actual computation of the embedding reduces to a sparse eigenvalue problem. Notably, the optimizations in LLE---though capable of generating highly nonlinear embeddings---are simple to implement, and they do not involve local minima. In this paper, we describe the implementation of the algorithm in detail and discuss several extensions that enhance its performance. We present results of the algorithm applied to data sampled from known manifolds, as well as to collections of images of faces, lips, and handwritten digits. These examples are used to provide extensive illustrations of the algorithms performance---both successes and failures---and to relate the algorithm to previous and ongoing work in nonlinear dimensionality reduction.

Unsupervised Learning of Image Manifolds by Semidefinite Programming

Can we detect low dimensional structure in high dimensional data sets of images? In this paper, we propose an algorithm for unsupervised learning of image manifolds by semidefinite programming. Given a data set of images, our algorithm computes a low dimensional representation of each image with the property that distances between nearby images are preserved. More generally, it can be used to analyze high dimensional data that lies on or near a low dimensional manifold. We illustrate the algorithm on easily visualized examples of curves and surfaces, as well as on actual images of faces, handwritten digits, and solid objects.

Beyond blacklists: learning to detect malicious web sites from suspicious URLs

Malicious Web sites are a cornerstone of Internet criminal activities. As a result, there has been broad interest in developing systems to prevent the end user from visiting such sites. In this paper, we describe an approach to this problem based on automated URL classification, using statistical methods to discover the tell-tale lexical and host-based properties of malicious Web site URLs. These methods are able to learn highly predictive models by extracting and automatically analyzing tens of thousands of features potentially indicative of suspicious URLs. The resulting classifiers obtain 95-99% accuracy, detecting large numbers of malicious Web sites from their URLs, with only modest false positives.

Unsupervised learning of image manifolds by semidefinite programming

Can we detect low dimensional structure in high dimensional data sets of images and video? The problem of dimensionality reduction arises often in computer vision and pattern recognition. In this paper, we propose a new solution to this problem based on semidefinite programming. Our algorithm can be used to analyze high dimensional data that lies on or near a low dimensional manifold. It overcomes certain limitations of previous work in manifold learning, such as Isomap and locally linear embedding. We illustrate the algorithm on easily visualized examples of curves and surfaces, as well as on actual images of faces, handwritten digits, and solid objects.

Identifying suspicious URLs: an application of large-scale online learning

This paper explores online learning approaches for detecting malicious Web sites (those involved in criminal scams) using lexical and host-based features of the associated URLs. We show that this application is particularly appropriate for online algorithms as the size of the training data is larger than can be efficiently processed in batch and because the distribution of features that typify malicious URLs is changing continuously. Using a real-time system we developed for gathering URL features, combined with a real-time source of labeled URLs from a large Web mail provider, we demonstrate that recently-developed online algorithms can be as accurate as batch techniques, achieving classification accuracies up to 99% over a balanced data set.

Mean field theory for sigmoid belief networks

We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics. Our mean field theory provides a tractable approximation to the true probability distribution in these networks; it also yields a lower bound on the likelihood of evidence. We demonstrate the utility of this framework on a benchmark problem in statistical pattern recognition-the classification of handwritten digits.

Fast solvers and efficient implementations for distance metric learning

In this paper we study how to improve nearest neighbor classification by learning a Mahalanobis distance metric. We build on a recently proposed framework for distance metric learning known as large margin nearest neighbor (LMNN) classification. Our paper makes three contributions. First, we describe a highly efficient solver for the particular instance of semidefinite programming that arises in LMNN classification; our solver can handle problems with billions of large margin constraints in a few hours. Second, we show how to reduce both training and testing times using metric ball trees; the speedups from ball trees are further magnified by learning low dimensional representations of the input space. Third, we show how to learn different Mahalanobis distance metrics in different parts of the input space. For large data sets, the use of locally adaptive distance metrics leads to even lower error rates.

Multiplicative Updates for Nonnegative Quadratic Programming

Many problems in neural computation and statistical learning involve optimizations with nonnegativity constraints. In this article, we study convex problems in quadratic programming where the optimization is confined to an axis-aligned region in the nonnegative orthant. For these problems, we derive multiplicative updates that improve the value of the objective function at each iteration and converge monotonically to the global minimum. The updates have a simple closed form and do not involve any heuristics or free parameters that must be tuned to ensure convergence. Despite their simplicity, they differ strikingly in form from other multiplicative updates used in machine learning. We provide complete proofs of convergence for these updates and describe their application to problems in signal processing and pattern recognition.

Mixed Memory Markov Models: Decomposing Complex Stochastic Processes as Mixtures of Simpler Ones

We study Markov models whose state spaces arise from the Cartesian product of two or more discrete random variables. We show how to parameterize the transition matrices of these models as a convex combination—or mixture—of simpler dynamical models. The parameters in these models admit a simple probabilistic interpretation and can be fitted iteratively by an Expectation-Maximization (EM) procedure. We derive a set of generalized Baum-Welch updates for factorial hidden Markov models that make use of this parameterization. We also describe a simple iterative procedure for approximately computing the statistics of the hidden states. Throughout, we give examples where mixed memory models provide a useful representation of complex stochastic processes.