Karsten Borgwardt

We propose a framework for analyzing and comparing distributions, which we use to construct 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), and is called the maximum mean discrepancy (MMD).We present two distributionfree tests based on large deviation bounds for the MMD, and a third test based on the asymptotic distribution of this statistic. The MMD can be computed in quadratic time, although efficient linear time approximations are available. Our statistic is an instance of an integral probability metric, and various classical metrics on distributions are obtained when alternative function classes are used in place of an RKHS. 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.

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.

MOTIVATION: Many problems in data integration in bioinformatics can be posed as one common question: Are two sets of observations generated by the same distribution? We propose a kernel-based statistical test for this problem, based on the fact that two distributions are different if and only if there exists at least one function having different expectation on the two distributions. Consequently we use the maximum discrepancy between function means as the basis of a test statistic. The Maximum Mean Discrepancy (MMD) can take advantage of the kernel trick, which allows us to apply it not only to vectors, but strings, sequences, graphs, and other common structured data types arising in molecular biology. RESULTS: We study the practical feasibility of an MMD-based test on three central data integration tasks: Testing cross-platform comparability of microarray data, cancer diagnosis, and data-content based schema matching for two different protein function classification schemas. In all of these experiments, including high-dimensional ones, MMD is very accurate in finding samples that were generated from the same distribution, and outperforms its best competitors. CONCLUSIONS: We have defined a novel statistical test of whether two samples are from the same distribution, compatible with both multivariate and structured data, that is fast, easy to implement, and works well, as confirmed by our experiments. AVAILABILITY: http://www.dbs.ifi.lmu.de/~borgward/MMD.

We consider the scenario where training and test data are drawn from different distributions, commonly referred to as sample selection bias.Most algorithms for this setting try to first recover sampling distributions and then make appropriate corrections based on the distribution estimate.We present a nonparametric method which directly produces resampling weights without distribution estimation.Our method works by matching distributions between training and testing sets in feature space.Experimental results demonstrate that our method works well in practice.

Motivation: Computational approaches to protein function prediction infer protein function by finding proteins with similar sequence, structure, surface clefts, chemical properties, amino acid motifs, interaction partners or phylogenetic profiles. We present a new approach that combines sequential, structural and chemical information into one graph model of proteins. We predict functional class membership of enzymes and non-enzymes using graph kernels and support vector machine classification on these protein graphs. Results: Our graph model, derivable from protein sequence and structure only, is competitive with vector models that require additional protein information, such as the size of surface pockets. If we include this extra information into our graph model, our classifier yields significantly higher accuracy levels than the vector models. Hyperkernels allow us to select and to optimally combine the most relevant node attributes in our protein graphs. We have laid the foundation for a protein function prediction system that integrates protein information from various sources efficiently and effectively.