Results 11  20
of
464
Gaussian processes for machine learning
 International Journal of Neural Systems
, 2004
"... Gaussian processes (GPs) are natural generalisations of multivariate Gaussian random variables to infinite (countably or continuous) index sets. GPs have been applied in a large number of fields to a diverse range of ends, and very many deep theoretical analyses of various properties are available. ..."
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Cited by 66 (15 self)
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Gaussian processes (GPs) are natural generalisations of multivariate Gaussian random variables to infinite (countably or continuous) index sets. GPs have been applied in a large number of fields to a diverse range of ends, and very many deep theoretical analyses of various properties are available. This paper gives an introduction to Gaussian processes on a fairly elementary level with special emphasis on characteristics relevant in machine learning. It draws explicit connections to branches such as spline smoothing models and support vector machines in which similar ideas have been investigated. Gaussian process models are routinely used to solve hard machine learning problems. They are attractive because of their flexible nonparametric nature and computational simplicity. Treated within a Bayesian framework, very powerful statistical methods can be implemented which offer valid estimates of uncertainties in our predictions and generic model selection procedures cast as nonlinear optimization problems. Their main drawback of heavy computational scaling has recently been alleviated by the introduction of generic sparse approximations [13, 78, 31]. The mathematical literature on GPs is large and often uses deep
Fast Marginal Likelihood Maximisation for Sparse Bayesian Models
 Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics
, 2003
"... The 'sparse Bayesian' modelling approach, as exemplified by the 'relevance vector machine ', enables sparse classification and regression functions to be obtained by linearlyweighting a small nmnber of fixed basis functions from a large dictionary of potential candidates. Such a model conveys ..."
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Cited by 65 (0 self)
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The 'sparse Bayesian' modelling approach, as exemplified by the 'relevance vector machine ', enables sparse classification and regression functions to be obtained by linearlyweighting a small nmnber of fixed basis functions from a large dictionary of potential candidates. Such a model conveys a nmnber of advantages over the related and very popular 'support vector machine', but the necessary 'training' procedure optimisation of the marginal likelihood function is typically much slower. We describe a new and highly accelerated algorithm which exploits recentlyelucidated properties of the marginal likelihood function to enable maximisation via a principled and efficient sequential addition and deletion of candidate basis functions.
The Kernel Recursive Least Squares Algorithm
 IEEE Transactions on Signal Processing
, 2003
"... We present a nonlinear kernelbased version of the Recursive Least Squares (RLS) algorithm. Our KernelRLS (KRLS) algorithm performs linear regression in the feature space induced by a Mercer kernel, and can therefore be used to recursively construct the minimum mean squared error regressor. Spars ..."
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Cited by 62 (2 self)
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We present a nonlinear kernelbased version of the Recursive Least Squares (RLS) algorithm. Our KernelRLS (KRLS) algorithm performs linear regression in the feature space induced by a Mercer kernel, and can therefore be used to recursively construct the minimum mean squared error regressor. Sparsity of the solution is achieved by a sequential sparsification process that admits into the kernel representation a new input sample only if its feature space image cannot be suffciently well approximated by combining the images of previously admitted samples. This sparsification procedure is crucial to the operation of KRLS, as it allows it to operate online, and by effectively regularizing its solutions. A theoretical analysis of the sparsification method reveals its close affinity to kernel PCA, and a datadependent loss bound is presented, quantifying the generalization performance of the KRLS algorithm. We demonstrate the performance and scaling properties of KRLS and compare it to a stateof theart Support Vector Regression algorithm, using both synthetic and real data. We additionally test KRLS on two signal processing problems in which the use of traditional leastsquares methods is commonplace: Time series prediction and channel equalization.
Bayesian inference and optimal design in the sparse linear model
 Workshop on Artificial Intelligence and Statistics
"... The linear model with sparsityfavouring prior on the coefficients has important applications in many different domains. In machine learning, most methods to date search for maximum a posteriori sparse solutions and neglect to represent posterior uncertainties. In this paper, we address problems of ..."
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Cited by 62 (12 self)
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The linear model with sparsityfavouring prior on the coefficients has important applications in many different domains. In machine learning, most methods to date search for maximum a posteriori sparse solutions and neglect to represent posterior uncertainties. In this paper, we address problems of Bayesian optimal design (or experiment planning), for which accurate estimates of uncertainty are essential. To this end, we employ expectation propagation approximate inference for the linear model with Laplace prior, giving new insight into numerical stability properties and proposing a robust algorithm. We also show how to estimate model hyperparameters by empirical Bayesian maximisation of the marginal likelihood, and propose ideas in order to scale up the method to very large underdetermined problems. We demonstrate the versatility of our framework on the application of gene regulatory network identification from microarray expression data, where both the Laplace prior and the active experimental design approach are shown to result in significant improvements. We also address the problem of sparse coding of natural images, and show how our framework can be used for compressive sensing tasks. Part of this work appeared in Seeger et al. (2007b). The gene network identification application appears in Steinke et al. (2007).
Maximum likelihood estimation of a stochastic integrateandfire neural model
 NIPS
, 2003
"... We examine a cascade encoding model for neural response in which a linear filtering stage is followed by a noisy, leaky, integrateandfire spike generation mechanism. This model provides a biophysically more realistic alternative to models based on Poisson (memoryless) spike generation, and can eff ..."
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Cited by 59 (20 self)
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We examine a cascade encoding model for neural response in which a linear filtering stage is followed by a noisy, leaky, integrateandfire spike generation mechanism. This model provides a biophysically more realistic alternative to models based on Poisson (memoryless) spike generation, and can effectively reproduce a variety of spiking behaviors seen in vivo. We describe the maximum likelihood estimator for the model parameters, given only extracellular spike train responses (not intracellular voltage data). Specifically, we prove that the log likelihood function is concave and thus has an essentially unique global maximum that can be found using gradient ascent techniques. We develop an efficient algorithm for computing the maximum likelihood solution, demonstrate the effectiveness of the resulting estimator with numerical simulations, and discuss a method of testing the model’s validity using timerescaling and density evolution techniques. Paninski et al., November 30, 2004 2 1
Building Support Vector Machines with Reduced Classifier Complexity
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2006
"... Support vector machines (SVMs), though accurate, are not preferred in applications requiring great classification speed, due to the number of support vectors being large. To overcome this problem we devise a primal method with the following properties: (1) it decouples the idea of basis functions ..."
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Cited by 58 (1 self)
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Support vector machines (SVMs), though accurate, are not preferred in applications requiring great classification speed, due to the number of support vectors being large. To overcome this problem we devise a primal method with the following properties: (1) it decouples the idea of basis functions from the concept of support vectors; (2) it greedily finds a set of kernel basis functions of a specified maximum size (d max ) to approximate the SVM primal cost function well; (3) it is efficient and roughly scales as O(nd max ) where n is the number of training examples; and, (4) the number of basis functions it requires to achieve an accuracy close to the SVM accuracy is usually far less than the number of SVM support vectors.
The generalized LASSO
 IEEE Transactions on Neural Networks
"... Abstract—In the last few years, the support vector machine (SVM) method has motivated new interest in kernel regression techniques. Although the SVM has been shown to exhibit excellent generalization properties in many experiments, it suffers from several drawbacks, both of a theoretical and a techn ..."
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Cited by 47 (0 self)
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Abstract—In the last few years, the support vector machine (SVM) method has motivated new interest in kernel regression techniques. Although the SVM has been shown to exhibit excellent generalization properties in many experiments, it suffers from several drawbacks, both of a theoretical and a technical nature: the absence of probabilistic outputs, the restriction to Mercer kernels, and the steep growth of the number of support vectors with increasing size of the training set. In this paper, we present a different class of kernel regressors that effectively overcome the above problems. We call this approach generalized LASSO regression. It has a clear probabilistic interpretation, can handle learning sets that are corrupted by outliers, produces extremely sparse solutions, and is capable of dealing with largescale problems. For regression functionals which can be modeled as iteratively reweighted leastsquares (IRLS) problems, we present a highly efficient algorithm with guaranteed global convergence. This defies a unique framework for sparse regression models in the very rich class of IRLS models, including various types of robust regression models and logistic regression. Performance studies for many standard benchmark datasets effectively demonstrate the advantages of this model over related approaches. Index Terms—Kernel regression, probabilistic interpretation, robust loss functions, sparisity, support vector machines (SVMs). tion by the following expansion, employing a Mercer kernel In the SVM regression setting, the approximation quality is measured in terms of aninsensitive loss function. Deviations smaller than a predefined value produce no costs at all, while larger deviations are penalized linearly for for. Implicit with this loss function is a model of noise in which the data are corrupted by uniform noise on the interval, cf. the discussion in Section IIIC. The model complexity is controlled by: 1) choosing an adequate kernel function, i.e., an adequate mapping from the input space into some feature space
Learning to track 3D human motion from silhouettes
 In International Conference on Machine Learning
, 2004
"... We describe a sparse Bayesian regression method for recovering 3D human body motion directly from silhouettes extracted from monocular video sequences. No detailed body shape model is needed, and realism is ensured by training on real human motion capture data. The tracker estimates 3D body pose by ..."
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Cited by 44 (3 self)
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We describe a sparse Bayesian regression method for recovering 3D human body motion directly from silhouettes extracted from monocular video sequences. No detailed body shape model is needed, and realism is ensured by training on real human motion capture data. The tracker estimates 3D body pose by using Relevance Vector Machine regression to combine a learned autoregressive dynamical model with robust shape descriptors extracted automatically from image silhouettes. We studied several different combination methods, the most effective being to learn a nonlinear observationupdate correction based on joint regression with respect to the predicted state and the observations. We demonstrate the method on a 54parameter full body pose model, both quantitatively using motion capture based test sequences, and qualitatively on a test video sequence. 1.
Sparse Representation For Computer Vision and Pattern Recognition
, 2009
"... Techniques from sparse signal representation are beginning to see significant impact in computer vision, often on nontraditional applications where the goal is not just to obtain a compact highfidelity representation of the observed signal, but also to extract semantic information. The choice of ..."
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Cited by 44 (1 self)
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Techniques from sparse signal representation are beginning to see significant impact in computer vision, often on nontraditional applications where the goal is not just to obtain a compact highfidelity representation of the observed signal, but also to extract semantic information. The choice of dictionary plays a key role in bridging this gap: unconventional dictionaries consisting of, or learned from, the training samples themselves provide the key to obtaining stateoftheart results and to attaching semantic meaning to sparse signal representations. Understanding the good performance of such unconventional dictionaries in turn demands new algorithmic and analytical techniques. This review paper highlights a few representative examples of how the interaction between sparse signal representation and computer vision can enrich both fields, and raises a number of open questions for further study.