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299
Fast approximate energy minimization via graph cuts
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2001
"... In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function’s smoothness term must only involve pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when v ..."
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Cited by 2120 (61 self)
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In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function’s smoothness term must only involve pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when very large moves are allowed. The first move we consider is an αβswap: for a pair of labels α, β, this move exchanges the labels between an arbitrary set of pixels labeled α and another arbitrary set labeled β. Our first algorithm generates a labeling such that there is no swap move that decreases the energy. The second move we consider is an αexpansion: for a label α, this move assigns an arbitrary set of pixels the label α. Our second
An introduction to variational methods for graphical models
 TO APPEAR: M. I. JORDAN, (ED.), LEARNING IN GRAPHICAL MODELS
"... ..."
The Infinite Hidden Markov Model
 Machine Learning
, 2002
"... We show that it is possible to extend hidden Markov models to have a countably infinite number of hidden states. By using the theory of Dirichlet processes we can implicitly integrate out the infinitely many transition parameters, leaving only three hyperparameters which can be learned from data. Th ..."
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Cited by 637 (41 self)
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We show that it is possible to extend hidden Markov models to have a countably infinite number of hidden states. By using the theory of Dirichlet processes we can implicitly integrate out the infinitely many transition parameters, leaving only three hyperparameters which can be learned from data. These three hyperparameters define a hierarchical Dirichlet process capable of capturing a rich set of transition dynamics. The three hyperparameters control the time scale of the dynamics, the sparsity of the underlying statetransition matrix, and the expected number of distinct hidden states in a finite sequence. In this framework it is also natural to allow the alphabet of emitted symbols to be infiniteconsider, for example, symbols being possible words appearing in English text.
Regularization Theory and Neural Networks Architectures
 Neural Computation
, 1995
"... We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Ba ..."
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Cited by 395 (32 self)
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We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Basis Functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to different classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends Radial Basis Functions (RBF) to Hyper Basis Functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, som...
Regularization networks and support vector machines
 Advances in Computational Mathematics
, 2000
"... Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization a ..."
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Cited by 366 (38 self)
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Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples – in particular the regression problem of approximating a multivariate function from sparse data. Radial Basis Functions, for example, are a special case of both regularization and Support Vector Machines. We review both formulations in the context of Vapnik’s theory of statistical learning which provides a general foundation for the learning problem, combining functional analysis and statistics. The emphasis is on regression: classification is treated as a special case.
A Theory of Networks for Approximation and Learning
 Laboratory, Massachusetts Institute of Technology
, 1989
"... Learning an inputoutput mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multidimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, t ..."
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Cited by 235 (24 self)
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Learning an inputoutput mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multidimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nonlinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. Wedevelop a theoretical framework for approximation based on regularization techniques that leads to a class of threelayer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the wellknown Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods suchasParzen windows and potential functions and to several neural network algorithms, suchas Kanerva's associative memory,backpropagation and Kohonen's topology preserving map. They also haveaninteresting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.
Variational learning for switching statespace models
 Neural Computation
, 1998
"... We introduce a new statistical model for time series which iteratively segments data into regimes with approximately linear dynamics and learns the parameters of each of these linear regimes. This model combines and generalizes two of the most widely used stochastic time series models  hidden Ma ..."
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Cited by 173 (5 self)
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We introduce a new statistical model for time series which iteratively segments data into regimes with approximately linear dynamics and learns the parameters of each of these linear regimes. This model combines and generalizes two of the most widely used stochastic time series models  hidden Markov models and linear dynamical systems  and is closely related to models that are widely used in the control and econometrics literatures. It can also be derived by extending the mixture of experts neural network (Jacobs et al., 1991) to its fully dynamical version, in which both expert and gating networks are recurrent. Inferring the posterior probabilities of the hidden states of this model is computationally intractable, and therefore the exact Expectation Maximization (EM) algorithm cannot be applied. However, we present a variational approximation that maximizes a lower bound on the log likelihood and makes use of both the forwardbackward recursions for hidden Markov models and the Kalman lter recursions for linear dynamical systems. We tested the algorithm both on artificial data sets and on a natural data set of respiration force from a patient with sleep apnea. The results suggest that variational approximations are a viable method for inference and learning in switching statespace models.
Learning Dynamic Bayesian Networks
 In Adaptive Processing of Sequences and Data Structures, Lecture Notes in Artificial Intelligence
, 1998
"... Suppose we wish to build a model of data from a finite sequence of ordered observations, {Y1, Y2,..., Yt}. In most realistic scenarios, from modeling stock prices to physiological data, the observations are not related deterministically. Furthermore, there is added uncertainty resulting from the li ..."
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Cited by 166 (0 self)
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Suppose we wish to build a model of data from a finite sequence of ordered observations, {Y1, Y2,..., Yt}. In most realistic scenarios, from modeling stock prices to physiological data, the observations are not related deterministically. Furthermore, there is added uncertainty resulting from the limited size of our data set and any mismatch between our model and the true process. Probability theory provides a powerful tool for expressing both randomness and uncertainty in our model [23]. We can express the uncertainty in our prediction of the future outcome Yt+l via a probability density P(Yt+llY1,..., Yt). Such a probability density can then be used to make point predictions, define error bars, or make decisions that are expected to minimize some loss function. This chapter presents a probabilistic framework for learning models of temporal data. We express these models using the Bayesian network formalism (a.k.a. probabilistic graphical models or belief networks)a marriage of probability theory and graph theory in which dependencies between variables are expressed graphically. The graph not only allows the user to understand which variables
Mean Field Theory for Sigmoid Belief Networks
 Journal of Artificial Intelligence Research
, 1996
"... We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics. ..."
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Cited by 154 (13 self)
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We develop a mean field theory for sigmoid belief networks based on ideas from statistical mechanics.
Stochastic Variational Inference
 JOURNAL OF MACHINE LEARNING RESEARCH (2013, IN PRESS)
, 2013
"... We develop stochastic variational inference, a scalable algorithm for approximating posterior distributions. We develop this technique for a large class of probabilistic models and we demonstrate it with two probabilistic topic models, latent Dirichlet allocation and the hierarchical Dirichlet proce ..."
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Cited by 131 (27 self)
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We develop stochastic variational inference, a scalable algorithm for approximating posterior distributions. We develop this technique for a large class of probabilistic models and we demonstrate it with two probabilistic topic models, latent Dirichlet allocation and the hierarchical Dirichlet process topic model. Using stochastic variational inference, we analyze several large collections of documents: 300K articles from Nature, 1.8M articles from The New York Times, and 3.8M articles from Wikipedia. Stochastic inference can easily handle data sets of this size and outperforms traditional variational inference, which can only handle a smaller subset. (We also show that the Bayesian nonparametric topic model outperforms its parametric counterpart.) Stochastic variational inference lets us apply complex Bayesian models to massive data sets.