Results 1  10
of
27
Hilbert Space Embeddings of Hidden Markov Models
"... Hidden Markov Models (HMMs) are important tools for modeling sequence data. However, they are restricted to discrete latent states, and are largely restricted to Gaussian and discrete observations. And, learning algorithms for HMMs have predominantly relied on local search heuristics, with the excep ..."
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Cited by 33 (10 self)
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Hidden Markov Models (HMMs) are important tools for modeling sequence data. However, they are restricted to discrete latent states, and are largely restricted to Gaussian and discrete observations. And, learning algorithms for HMMs have predominantly relied on local search heuristics, with the exception of spectral methods such as those described below. We propose a nonparametric HMM that extends traditional HMMs to structured and nonGaussian continuous distributions. Furthermore, we derive a localminimumfree kernel spectral algorithm for learning these HMMs. We apply our method to robot vision data, slot car inertial sensor data and audio event classification data, and show that in these applications, embedded HMMs exceed the previous stateoftheart performance. 1.
Kernel Belief Propagation
"... We propose a nonparametric generalization of belief propagation, Kernel Belief Propagation (KBP), for pairwise Markov random fields. Messages are represented as functions in a reproducing kernel Hilbert space (RKHS), and message updates are simple linear operations in the RKHS. KBP makes none of the ..."
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Cited by 18 (8 self)
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We propose a nonparametric generalization of belief propagation, Kernel Belief Propagation (KBP), for pairwise Markov random fields. Messages are represented as functions in a reproducing kernel Hilbert space (RKHS), and message updates are simple linear operations in the RKHS. KBP makes none of the assumptions commonly required in classical BP algorithms: the variables need not arise from a finite domain or a Gaussian distribution, nor must their relations take any particular parametric form. Rather, the relations between variables are represented implicitly, and are learned nonparametrically from training data. KBP has the advantage that it may be used on any domain where kernels are defined (Rd, strings, groups), even where explicit parametric models are not known, or closed form expressions for the BP updates do not exist. The computational cost of message updates in KBP is polynomial in the training data size. We also propose a constant time approximate message update procedure by representing messages using a small number of basis functions. In experiments, we apply KBP to image denoising, depth prediction from still images, and protein configuration prediction: KBP is faster than competing classical and nonparametric approaches (by orders of magnitude, in some cases), while providing significantly more accurate results. 1
Kernel Bayes ’ Rule
"... A nonparametric kernelbased method for realizing Bayes ’ rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mea ..."
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Cited by 9 (4 self)
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A nonparametric kernelbased method for realizing Bayes ’ rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mean of the posterior distribution is computed in the form of a weighted sample. The kernel Bayes ’ rule can be applied to a wide variety of Bayesian inference problems: we demonstrate Bayesian computation without likelihood, and filtering with a nonparametric statespace model. A consistency rate for the posterior estimate is established. 1
Modelling transition dynamics in mdps with rkhs embeddings
 In arXiv
, 2012
"... We propose a new, nonparametric approach to learning and representing transition dynamics in Markov decision processes (MDPs), which can be combined easily with dynamic programming methods for policy optimisation and value estimation. This approach makes use of a recently developed representation of ..."
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Cited by 8 (4 self)
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We propose a new, nonparametric approach to learning and representing transition dynamics in Markov decision processes (MDPs), which can be combined easily with dynamic programming methods for policy optimisation and value estimation. This approach makes use of a recently developed representation of conditional distributions as embeddings in a reproducing kernel Hilbert space (RKHS). Such representations bypass the need for estimating transition probabilities or densities, and apply to any domain on which kernels can be defined. This avoids the need to calculate intractable integrals, since expectations are represented as RKHS inner products whose computation has linear complexity in the number of points used to represent the embedding. We provide guarantees for the proposed applications in MDPs: in the context of a value iteration algorithm, we prove convergence to either the optimal policy, or to the closest projection of the optimal policy in our model class (an RKHS), under reasonable assumptions. In experiments, we investigate a learning task in a typical classical control setting (the underactuated pendulum), and on a navigation problem where only images from a sensor are observed. For policy optimisation we compare with leastsquares policy iteration where a Gaussian process is used for value function estimation. For value estimation we also compare to the NPDP method. Our approach achieves better performance in all experiments.
M.: Conditional mean embeddings as regressors
 ICML
, 2012
"... We demonstrate an equivalence between reproducing kernel Hilbert space (RKHS) embeddings of conditional distributions and vectorvalued regressors. This connection introduces a natural regularized loss function which the RKHS embeddings minimise, providing an intuitive understanding of the embedding ..."
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Cited by 5 (3 self)
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We demonstrate an equivalence between reproducing kernel Hilbert space (RKHS) embeddings of conditional distributions and vectorvalued regressors. This connection introduces a natural regularized loss function which the RKHS embeddings minimise, providing an intuitive understanding of the embeddings and a justification for their use. Furthermore, the equivalence allows the application of vectorvalued regression methods and results to the problem of learning conditional distributions. Using this link we derive a sparse version of the embedding by considering alternative formulations. Further, by applying convergence results for vectorvalued regression to the embedding problem we derive minimax convergence rates which are O(log(n)/n) – compared to current state of the art rates of O(n−1/4) – and are valid under milder and more intuitive assumptions. These minimax upper rates coincide with lower rates up to a logarithmic factor, showing that the embedding method achieves nearly optimal rates. We study our sparse embedding algorithm in a reinforcement learning task where the algorithm shows significant improvement in sparsity over an incomplete Cholesky decomposition. 1. Introduction/Motivation In recent years a framework for embedding probability distributions into reproducing kernel Hilbert spaces (RKHS)
Nonparametric Tree Graphical Models via Kernel Embeddings
"... We introduce a nonparametric representation for graphical model on trees which expresses marginals as Hilbert space embeddings and conditionals as embedding operators. This formulation allows us to define a graphical model solely on the basis of the feature space representation of its variables. Thu ..."
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Cited by 2 (1 self)
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We introduce a nonparametric representation for graphical model on trees which expresses marginals as Hilbert space embeddings and conditionals as embedding operators. This formulation allows us to define a graphical model solely on the basis of the feature space representation of its variables. Thus, this nonparametric model can be applied to general domains where kernels are defined, handling challenging cases such as discrete variables with huge domains, or very complex, nonGaussian continuous distributions. We also derive kernel belief propagation, a Hilbertspace algorithm for performing inference in our model. We show that our method outperforms stateoftheart techniques in a crosslingual document retrieval task and a camera rotation estimation problem. 1
Spectral Approaches to Learning Predictive Representations
, 2011
"... A central problem in artificial intelligence is to choose actions to maximize reward in a partially observable, uncertain environment. To do so, we must obtain an accurate environment model, and then plan to maximize reward. However, for complex domains, specifying a model by hand can be a time cons ..."
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Cited by 1 (1 self)
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A central problem in artificial intelligence is to choose actions to maximize reward in a partially observable, uncertain environment. To do so, we must obtain an accurate environment model, and then plan to maximize reward. However, for complex domains, specifying a model by hand can be a time consuming process. This motivates an alternative approach: learning a model directly from observations. Unfortunately, learning algorithms often recover a model that is too inaccurate to support planning or too large and complex for planning to succeed; or, they require excessive prior domain knowledge or fail to provide guarantees such as statistical consistency. To address this gap, we propose spectral subspace identification algorithms which provably learn compact, accurate, predictive models of partially observable dynamical systems directly from sequences of actionobservation pairs. Our research agenda includes several variations of this general approach: batch algorithms and online algorithms, kernelbased algorithms for learning models in high and infinitedimensional feature spaces, and manifoldbased identification algorithms. All of these approaches share a common framework: they are statistically consistent, computationally efficient, and easy to implement using established matrixalgebra techniques. Additionally, we show that our framework generalizes a variety of successful spectral
LinearTime Estimators for Propensity Scores
"... We present lineartime estimators for three popular covariate shift correction and propensity scoring algorithms: logistic regression(LR), kernel mean matching(KMM) [19], and maximum entropy mean matching(MEMM)[20]. This allows applications in situations where both treatment and control groups are l ..."
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Cited by 1 (0 self)
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We present lineartime estimators for three popular covariate shift correction and propensity scoring algorithms: logistic regression(LR), kernel mean matching(KMM) [19], and maximum entropy mean matching(MEMM)[20]. This allows applications in situations where both treatment and control groups are large. We also show that the last two algorithms differ only in their choice of regularizer (ℓ2 of the Radon Nikodym derivative vs. maximum entropy). Experiments show that all methods scale well. 1
Path Integral Control by Reproducing Kernel Hilbert Space Embedding
"... We present an embedding of stochastic optimal control problems, of the so called path integral form, into reproducing kernel Hilbert spaces. Using consistent, sample based estimates of the embedding leads to a modelfree, nonparametric approach for calculation of an approximate solution to the cont ..."
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Cited by 1 (1 self)
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We present an embedding of stochastic optimal control problems, of the so called path integral form, into reproducing kernel Hilbert spaces. Using consistent, sample based estimates of the embedding leads to a modelfree, nonparametric approach for calculation of an approximate solution to the control problem. This formulation admits a decomposition of the problem into an invariant and task dependent component. Consequently, we make much more efficient use of the sample data compared to previous sample based approaches in this domain, e.g., by allowing sample reuse across tasks. Numerical examples on test problems, which illustrate the sample efficiency, are provided. 1
‖A ‖ 2
, 2010
"... The supplementary material contains proofs of the main theorems (Section 1), and two additional experiments (Section 2): a reconstruction of camera orientation from images; and an additional set of document retrieval experiments, using a language graph constructed via the ChowLiu algorithm. ..."
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The supplementary material contains proofs of the main theorems (Section 1), and two additional experiments (Section 2): a reconstruction of camera orientation from images; and an additional set of document retrieval experiments, using a language graph constructed via the ChowLiu algorithm.