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@ author = {Gretton, A. and L. Györfi},
title = {Nonparametric Indepedence Tests: Space Partitioning and Kernel Approaches},
year = {2008},
month = {10},
institution like {%Max-Planck Institute for Biological Cybernetics%},
location = {19th International Conference on Algorithmic Learning Theory (ALT08)},
URL = {http://www-alg.ist.hokudai.ac.jp/~thomas/ALT08/alt08.jhtml}
}
@ author = {Fukumizu, K. and A. Gretton and A. Smola},
title = {Painless Embeddings of Distributions: the Function Space View (Part 1)},
year = {2008},
month = {07},
institution like {%Max-Planck-Institute for Biological Cybernetics%},
abstract = {This tutorial will give an introduction to the recent understanding and methodology of the kernel method: dealing with higher order statistics by embedding painlessly random variables/probability distributions.
In the early days of kernel machines research, the "kernel trick" was considered a useful way of constructing nonlinear algorithms from linear ones. More recently, however, it has become clear that a potentially more far reaching use of kernels is as a linear way of dealing with higher order statistics by embedding distributions in a suitable reproducing kernel Hilbert space (RKHS). Notably, unlike the straightforward expansion of higher order moments or conventional characteristic function approach, the use of kernels or RKHS provides a painless, tractable way of embedding distributions.
This line of reasoning leads naturally to the questions: what does it mean to embed a distribution in an RKHS? when is this embedding injective (and thus, when do different distributions have unique mappings)? what implications are there for learning algorithms that make use of these embeddings? This tutorial aims at answering these questions.
There are a great variety of applications in machine learning and computer science, which require distribution estimation and/or comparison.},
location = {25th International Conference on Machine Learning (ICML 2008)},
URL = {http://alex.smola.org/icml2008/}
}
@ author = {Gretton, A. },
title = {Hilbert Space Representations of Probability Distributions},
year = {2007},
month = {10},
institution like {%Institute of Statistical Mathematics%},
abstract = {Many problems in unsupervised learning require the analysis of features of probability distributions. At the most fundamental level, we might wish to determine whether two distributions are the same, based on samples from each - this is known as the two-sample or homogeneity problem. We use kernel methods to address this problem, by mapping probability distributions to elements in a reproducing kernel Hilbert space (RKHS). Given a sufficiently rich RKHS, these representations are unique: thus comparing feature space representations allows us to compare distributions without ambiguity. Applications include testing whether cancer subtypes are distinguishable on the basis of DNA microarray data, and whether low frequency oscillations measured at an electrode in the cortex have a different distribution during a neural spike.
A more difficult problem is to discover whether two random variables drawn from a joint distribution are independent. It turns out that any dependence between pairs of random variables can be encoded in a cross-covariance operator between appropriate RKHS representations of the variables, and we may test independence by looking at a norm of the operator. We demonstrate this independence test by establishing dependence between an English text and its French translation, as opposed to French text on the same topic but otherwise unrelated. Finally, we show that this operator norm is itself a difference in feature means.},
location = {2nd Workshop on Machine Learning and Optimization at the ISM},
URL = {http://www.ism.ac.jp/~tmatsui/kinou2_p4/workshop_OCT07.html}
}
@ author = {Gretton, A. and K. Borgwardt and M. Rasch and B. Schölkopf and A. Smola},
title = {A Kernel Method for the Two-Sample-Problem},
year = {2006},
month = {12},
institution like {%MPI for Biological Cybernetics%},
abstract = {We propose two statistical tests to determine if two samples are
from different distributions. Our test statistic is in both cases
the distance between the means of the two samples mapped into a
reproducing kernel Hilbert space (RKHS). The first test is based on
a large deviation bound for the test statistic, while the second is
based on the asymptotic distribution of this statistic. We show that
the test statistic can be computed in $O(m^2)$ time. We apply our
approach to a variety of problems, including attribute matching for
databases using the Hungarian marriage method, where our test performs strongly.
We also demonstrate excellent
performance when comparing distributions over graphs, for which no
alternative tests currently exist.},
location = {Twentieth Annual Conference on Neural Information Processing Systems : NIPS 2006}
}
@ author = {Gretton, A. and A. Smola and O. Bousquet and R. Herbrich and A. Belitski and M. Augath and Y. Murayama and B. Schölkopf and N. K. Logothetis},
title = {Kernel Constrained Covariance for Dependence Measurement},
year = {2005},
month = {01},
institution like {%MPI for Biological Cybernetics%},
abstract = {We discuss reproducing kernel Hilbert space (RKHS)-based measures of statistical dependence,
with emphasis on constrained covariance (COCO), a novel criterion to
test dependence of random variables. We show that COCO is a test for independence if and only if the associated RKHSs
are universal.
That said, no independence
test exists that can distinguish dependent and independent random variables in all circumstances. Dependent random variables can result in a COCO which is arbitrarily close to zero when the source densities are highly non-smooth. All current kernel-based independence tests share this behaviour. We demonstrate exponential convergence between the population and empirical COCO. Finally, we use COCO as a measure of joint neural activity between voxels in MRI recordings of the macaque monkey, and compare the results to the mutual information and the correlation. We also show the effect of removing breathing artefacts from the MRI recording.},
location = {AISTATS 2005}
}
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