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Arthur Gretton

Publications and source records attributed to Arthur Gretton.

152 records · Page 9Linked to original sources
arXiv2012

Hypothesis testing using pairwise distances and associated kernels (with Appendix)

We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, distances between embeddings of distributions to reproducing kernel Hilbert spaces (RKHS), as established in machine learning. The equivalence holds when energy distances are computed with semimetrics of negative type, in which case a kernel may be defined such that the RKHS distance between distributions corresponds exactly to the energy distance. We determine the class of probability distributions for which kernels induced by semimetrics are characteristic (that is, for which embeddings of the distributions to an RKHS are injective). Finally, we investigate the performance of this family of kernels in two-sample and independence tests: we show in particular that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels, and that other choices from this family can yield more powerful tests.

cs.LG↗
arXiv2011

Kernel Bayes' rule

A nonparametric kernel-based method for realizing Bayes' rule is proposed, based on representations of probabilities in reproducing kernel Hilbert spaces. Probabilities are uniquely characterized by the mean of the canonical map to the RKHS. The prior and conditional probabilities are expressed in terms of RKHS functions of an empirical sample: no explicit parametric model is needed for these quantities. The posterior is likewise an RKHS mean of a weighted sample. The estimator for the expectation of a function of the posterior is derived, and rates of consistency are shown. Some representative applications of the kernel Bayes' rule are presented, including Baysian computation without likelihood and filtering with a nonparametric state-space model.

stat.ML↗
arXiv2011

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 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.

cs.LG↗
arXiv2010

Hilbert space embeddings and metrics on probability measures

A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing, and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). A pseudometric on the space of probability measures can be defined as the distance between distribution embeddings: we denote this as $γ_k$, indexed by the kernel function $k$ that defines the inner product in the RKHS. We present three theoretical properties of $γ_k$. First, we consider the question of determining the conditions on the kernel $k$ for which $γ_k$ is a metric: such $k$ are denoted {\em characteristic kernels}. Unlike pseudometrics, a metric is zero only when two distributions coincide, thus ensuring the RKHS embedding maps all distributions uniquely (i.e., the embedding is injective). While previously published conditions may apply only in restricted circumstances (e.g. on compact domains), and are difficult to check, our conditions are straightforward and intuitive: bounded continuous strictly positive definite kernels are characteristic. Alternatively, if a bounded continuous kernel is translation-invariant on $\bb{R}^d$, then it is characteristic if and only if the support of its Fourier transform is the entire $\bb{R}^d$. Second, we show that there exist distinct distributions that are arbitrarily close in $γ_k$. Third, to understand the nature of the topology induced by $γ_k$, we relate $γ_k$ to other popular metrics on probability measures, and present conditions on the kernel $k$ under which $γ_k$ metrizes the weak topology.

stat.ML↗
arXiv2009

On integral probability metrics, ϕ-divergences and binary classification

A class of distance measures on probabilities -- the integral probability metrics (IPMs) -- is addressed: these include the Wasserstein distance, Dudley metric, and Maximum Mean Discrepancy. IPMs have thus far mostly been used in more abstract settings, for instance as theoretical tools in mass transportation problems, and in metrizing the weak topology on the set of all Borel probability measures defined on a metric space. Practical applications of IPMs are less common, with some exceptions in the kernel machines literature. The present work contributes a number of novel properties of IPMs, which should contribute to making IPMs more widely used in practice, for instance in areas where $ϕ$-divergences are currently popular. First, to understand the relation between IPMs and $ϕ$-divergences, the necessary and sufficient conditions under which these classes intersect are derived: the total variation distance is shown to be the only non-trivial $ϕ$-divergence that is also an IPM. This shows that IPMs are essentially different from $ϕ$-divergences. Second, empirical estimates of several IPMs from finite i.i.d. samples are obtained, and their consistency and convergence rates are analyzed. These estimators are shown to be easily computable, with better rates of convergence than estimators of $ϕ$-divergences. Third, a novel interpretation is provided for IPMs by relating them to binary classification, where it is shown that the IPM between class-conditional distributions is the negative of the optimal risk associated with a binary classifier. In addition, the smoothness of an appropriate binary classifier is proved to be inversely related to the distance between the class-conditional distributions, measured in terms of an IPM.

cs.IT↗
arXiv2008

A Kernel Method for the Two-Sample Problem

We propose a framework for analyzing and comparing distributions, allowing us to design statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS). We present two tests based on large deviation bounds for the test statistic, while a third is based on the asymptotic distribution of this statistic. The test statistic can be computed in quadratic time, although efficient linear time approximations are available. Several classical metrics on distributions are recovered when the function space used to compute the difference in expectations is allowed to be more general (eg. a Banach space). We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests.

cs.LG↗
arXiv2007

Supervised Feature Selection via Dependence Estimation

We introduce a framework for filtering features that employs the Hilbert-Schmidt Independence Criterion (HSIC) as a measure of dependence between the features and the labels. The key idea is that good features should maximise such dependence. Feature selection for various supervised learning problems (including classification and regression) is unified under this framework, and the solutions can be approximated using a backward-elimination algorithm. We demonstrate the usefulness of our method on both artificial and real world datasets.

cs.LG↗