Saul, Lawrence K

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2001 - 200517
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  • Publication
    Multiplicative Updates for Large Margin Classifiers
    (2003-08-24) Saul, Lawrence K; Sha, Fei; Lee, Daniel D
    Various problems in nonnegative quadratic programming arise in the training of large margin classifiers. We derive multiplicative updates for these problems that converge monotonically to the desired solutions for hard and soft margin classifiers. The updates differ strikingly in form from other multiplicative updates used in machine learning. In this paper, we provide complete proofs of convergence for these updates and extend previous work to incorporate sum and box constraints in addition to nonnegativity.
  • Publication
    Real-Time Pitch Determination of One or More Voices by Nonnegative Matrix Factorization
    (2004-12-13) Sha, Fei; Saul, Lawrence K
    An auditory "scene", composed of overlapping acoustic sources, can be viewed as a complex object whose constituent parts are the individual sources. Pitch is known to be an important cue for auditory scene analysis. In this paper, with the goal of building agents that operate in human environments, we describe a real-time system to identify the presence of one or more voices and compute their pitch. The signal processing in the front end is based on instantaneous frequency estimation, a method for tracking the partials of voiced speech, while the pattern-matching in the back end is based on nonnegative matrix factorization, an unsupervised algorithm for learning the parts of complex objects. While supporting a framework to analyze complicated auditory scenes, our system maintains real-time operability and state-of-the-art performance in clean speech.
  • Publication
    Real time voice processing with audiovisual feedback : toward autonomous agents with perfect pitch
    (2002-12-09) Saul, Lawrence K; Lee, Daniel D; Isbell, Charles L; LeCun, Yann
    We have implemented a real time front end for detecting voiced speech and estimating its fundamental frequency. The front end performs the signal processing for voice-driven agents that attend to the pitch contours of human speech and provide continuous audiovisual feedback. The algorithm we use for pitch tracking has several distinguishing features: it makes no use of FFTs or autocorrelation at the pitch period; it updates the pitch incrementally on a sample-by-sample basis; it avoids peak picking and does not require interpolation in time or frequency to obtain high resolution estimates; and it works reliably over a four octave range, in real time, without the need for postprocessing to produce smooth contours. The algorithm is based on two simple ideas in neural computation: the introduction of a purposeful nonlinearity, and the error signal of a least squares fit. The pitch tracker is used in two real time multimedia applications: a voice-to-MIDI player that synthesizes electronic music from vocalized melodies, and an audiovisual Karaoke machine with multimodal feedback. Both applications run on a laptop and display the user’s pitch scrolling across the screen as he or she sings into the computer.
  • Publication
    Statistical signal processing with nonnegativity constraints
    (2003-09-01) Saul, Lawrence K; Sha, Fei; Lee, Daniel D
    Nonnegativity constraints arise frequently in statistical learning and pattern recognition. Multiplicative updates provide natural solutions to optimizations involving these constraints. One well known set of multiplicative updates is given by the Expectation-Maximization algorithm for hidden Markov models, as used in automatic speech recognition. Recently, we have derived similar algorithms for nonnegative deconvolution and nonnegative quadratic programming. These algorithms have applications to low-level problems in voice processing, such as fundamental frequency estimation, as well as high-level problems, such as the training of large margin classifiers. In this paper, we describe these algorithms and the ideas that connect them.
  • Publication
    Multiplicative updates for nonnegative quadratic programming in support vector machines
    (2002-12-10) Sha, Fei; Saul, Lawrence K; Lee, Daniel D
    We derive multiplicative updates for solving the nonnegative quadratic programming problem in support vector machines (SVMs). The updates have a simple closed form, and we prove that they converge monotonically to the solution of the maximum margin hyperplane. The updates optimize the traditionally proposed objective function for SVMs. They do not involve any heuristics such as choosing a learning rate or deciding which variables to update at each iteration. They can be used to adjust all the quadratic programming variables in parallel with a guarantee of improvement at each iteration. We analyze the asymptotic convergence of the updates and show that the coefficients of non-support vectors decay geometrically to zero at a rate that depends on their margins. In practice, the updates converge very rapidly to good classifiers.
  • Publication
    Multiplicative Updates for Classification by Mixture Models
    (2001-12-03) Saul, Lawrence K; Lee, Daniel D
    We investigate a learning algorithm for the classification of nonnegative data by mixture models. Multiplicative update rules are derived that directly optimize the performance of these models as classifiers. The update rules have a simple closed form and an intuitive appeal. Our algorithm retains the main virtues of the Expectation-Maximization (EM) algorithm—its guarantee of monotonic improvement, and its absence of tuning parameters—with the added advantage of optimizing a discriminative objective function. The algorithm reduces as a special case to the method of generalized iterative scaling for log-linear models. The learning rate of the algorithm is controlled by the sparseness of the training data. We use the method of nonnegative matrix factorization (NMF) to discover sparse distributed representations of the data. This form of feature selection greatly accelerates learning and makes the algorithm practical on large problems. Experiments show that discriminatively trained mixture models lead to much better classification than comparably sized models trained by EM.
  • Publication
    Multiband statistical learning for f0 estimation in speech
    (2004-05-17) Sha, Fei; Burgoyne, J. Ashley; Saul, Lawrence K
    We investigate a simple algorithm that combines multiband processing and least squares fits to estimate f0 contours in speech. The algorithm is untraditional in several respects: it makes no use of FFTs or autocorrelation at the pitch period; it updates the pitch incrementally on a sample-by-sample basis; it avoids peak picking and does not require interpolation in time or frequency to obtain high resolution estimates; and it works reliably, in real time, without the need for postprocessing to produce smooth contours. We show that a baseline implementation of the algorithm, though already quite accurate, is significantly improved by incorporating a model of statistical learning into its final stages. Model parameters are estimated from training data to minimize the likelihood of gross errors in f0 as well as errors in classifying voiced versus unvoiced speech. Experimental results on several databases confirm the benefits of statistical learning.
  • Publication
    Learning High Dimensional Correspondences from Low Dimensional Manifolds
    (2003-08-21) Lee, Daniel D; Ham, Ji Hun; Saul, Lawrence K
    Many different high dimensional data sets are characterized by the same underlying modes of variability. When these modes of variability are continuous and few in number, they can be viewed as parameterizing a low dimensional manifold. The manifold provides a compact shared representation of the data, suggesting correspondences between the high dimensional examples from different data sets. These correspondences, though naturally induced by the underlying manifold, are difficult to learn using traditional methods in supervised learning. In this paper, we generalize three methods in unsupervised learning—principal components analysis, factor analysis, and locally linear embedding—to discover subspaces and manifolds that provide common low dimensional representations of different high dimensional data sets. We use the shared representations discovered by these algorithms to put high dimensional examples from different data sets into correspondence. Finally, we show that a notion of "self-correspondence" between examples in the same data set can be used to improve the performance of these algorithms on small data sets. The algorithms are demonstrated on images and text.
  • Publication
    Think Globally, Fit Locally : Unsupervised Learning of Low Dimensional Manifolds
    (2003-06-01) Saul, Lawrence K; Roweis, Sam T
    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 algorithm’s performance--both successes and failures--and to relate the algorithm to previous and ongoing work in nonlinear dimensionality reduction.
  • Publication
    Visualization of Low Dimensional Structure in Tonal Pitch Space
    (2005-09-06) Burgoyne, J. Ashley; Saul, Lawrence K
    In his 2001 monograph Tonal Pitch Space, Fred Lerdahl defined a distance function over tonal and post-tonal harmonies distilled from years of research on music cognition. Although this work references the toroidal structure commonly associated with harmonic space, it stops short of presenting an explicit embedding of this torus. It is possible to use statistical techniques to recreate such an embedding from the distance function, yielding a more complex structure than the standard toroidal model has heretofore assumed. Nonlinear techniques can reduce the dimensionality of this structure and be tuned to emphasize global or local anatomy. The resulting manifolds highlight the relationships inherent in the tonal system and offer a basis for future work in machine-assisted analysis and music theory.