Abstract. We describe a technique for comparing distributions without the need for density estimation as an intermediate step. Our approach relies on mapping the distributions into a reproducing kernel Hilbert space. Applications of this technique can be found in two-sample tests, which are used for determining whether two sets of observations arise from the same distribution, covariate shift correction, local learning, measures of independence, and density estimation. Kernel methods are widely used in supervised learning [1, 2, 3, 4], however they are much less established in the areas of testing, estimation, and analysis of probability distributions, where information theoretic approaches [5, 6] have long been dominant. Recent examples include [7] in the context of construction of graphical models, [8] in the context of feature extraction, and [9] in the context of independent component analysis. These methods have by and large a common issue: to compute quantities such as the mutual information, entropy, or Kullback-Leibler divergence, we require sophisticated space partitioning and/or

We introduce two new functionals, the constrained covariance and the kernel mutual information, to measure the degree of independence of random variables. These quantities are both based on the covariance between functions of the random variables in reproducing kernel Hilbert spaces (RKHSs). We prove that when the RKHSs are universal, both functionals are zero if and only if the random variables are pairwise independent. We also show that the kernel mutual information is an upper bound near independence on the Parzen window estimate of the mutual information. Analogous results apply for two correlation-based dependence functionals introduced earlier: we show the kernel canonical correlation and the kernel generalised variance to be independence measures for universal kernels, and prove the latter to be an upper bound on the mutual information near independence. The performance of the kernel dependence functionals in measuring independence is verified in the context of independent component analysis.

We introduce the blind subspace deconvolution (BSSD) problem, which is the extension of both the blind source deconvolution (BSD) and the independent subspace analysis (ISA) tasks. We examine the case of the undercomplete BSSD (uBSSD). Applying temporal concatenation we reduce this problem to ISA. The associated ‘high dimensional ’ ISA problem can be handled by a recent technique called joint f-decorrelation (JFD). Similar decorrelation methods have been used previously for kernel independent component analysis (kernel-ICA). More precisely, the kernel canonical correlation (KCCA) technique is a member of this family, and, as is shown in this paper, the kernel generalized variance (KGV) method can also be seen as a decorrelation method in the feature space. These kernel based algorithms will be adapted to the ISA task. In the numerical examples, we (i) examine how efficiently the emerging higher dimensional ISA tasks can be tackled, and (ii) explore the working and advantages of the derived kernel-ISA methods.

Recent approaches to Independent Component Analysis (ICA) have used kernel independence measures to obtain highly accurate solutions, particularly where classical methods experience difficulty (for instance, sources with near-zero kurtosis). FastKICA (Fast HSIC-based Kernel ICA) is a new optimisation method for one such kernel independence measure, the Hilbert-Schmidt Independence Criterion (HSIC). The high computational efficiency of this approach is achieved by combining geometric optimisation techniques, specifically an approximate Newton-like method on the orthogonal group, with accurate estimates of the gradient and Hessian based on an incomplete Cholesky decomposition. In contrast to other efficient kernel-based ICA algorithms, FastKICA is applicable to any twice differentiable kernel function. Experimental results for problems with large numbers of sources and observations indicate that FastKICA provides more accurate solutions at a given cost than gradient descent on HSIC. Comparing with other recently published ICA methods, FastKICA is competitive in terms of accuracy, relatively insensitive to local minima when initialised far from independence, and more robust towards outliers. An analysis of the local convergence properties of FastKICA is provided. 1

Abstract. Recently, several algorithms have been proposed for independent subspace analysis where hidden variables are i.i.d. processes. We show that these methods can be extended to certain AR, MA, ARMA and ARIMA tasks. Central to our paper is that we introduce a cascade of algorithms, which aims to solve these tasks without previous knowledge about the number and the dimensions of the hidden processes. Our claim is supported by numerical simulations. As an illustrative application where the dimensions of the hidden variables are unknown, we search for subspaces of facial components. 1

Recent approaches to independent component analysis have used kernel independence measures to obtain very good performance in ICA, particularly in areas where classical methods experience difficulty (for instance, sources with near-zero kurtosis). In this chapter, we compare two efficient extensions of these methods for large-scale problems: random subsampling of entries in the Gram matrices used in defining the independence measures, and incomplete Cholesky decomposition of these matrices. We derive closed-form, efficiently computable approximations for the gradients of these measures, and compare their performance on ICA using both artificial and music data. We show that kernel ICA can scale up to larger problems than yet attempted, and that incomplete Cholesky decomposition performs better than random sampling.

Abstract—In this paper we address the controlled complete

Abstract—An algorithm for the blind separation of mutually independent and/or temporally correlated sources is presented in this paper. The algorithm is closely related to the maximum likelihood approach based on entropy rate minimization but uses a simpler contrast function that can be accurately and efficiently estimated using nearest-neighbor distances. The advantages of the new algorithm are highlighted using simulations and real electroencephalographic data.

Abstract—Electrophysiological signals of the developing fetal brain and heart can be investigated by fetal magnetoencephalography (fMEG). During such investigations, the fetal heart activity and that of the mother should be monitored continuously to provide an important indication of current well-being. Due to physical constraints of an fMEG system, it is not possible to use clinically established heart monitors for this purpose. Considering this constraint, we developed a real-time heart monitoring system for biomagnetic measurements and showed its reliability and applicability in research and for clinical examinations. The developed system consists of real-time access to fMEG data, an algorithm based on Independent Component Analysis (ICA), and a graphical user interface (GUI). The algorithm extracts the current fetal and maternal heart signal from a noisy and artifact-contaminated data stream in real-time and is able to adapt automatically to continuously varying environmental parameters. This algorithm has been named Adaptive Real-time ICA (ARICA) and is applicable to real-time artifact removal as well as to related blind signal separation problems. Index Terms—Adaptive, clustering, fetal, independent component analysis, magnetoencephalography, real-time systems. I.

Abstract not found