Visualizing data and finding patterns in data are ubiquitous problems in the sciences. Increasingly, applications seek signal and structure in a contrastive setting: a foreground dataset relative to a background dataset. For this purpose, we propose contrastive independent component analysis (cICA). This generalizes independent component analysis to independent latent variables across a foreground and background. We propose a hierarchical tensor decomposition algorithm for cICA. We study the identifiability of cICA and demonstrate its performance visualizing data and finding patterns in data, using synthetic and real-world datasets, comparing the approach to existing contrastive methods.

In the square linear blind source separation problem, one must find a linear unmixing operator which can detangle the result Xi(t) of mixing n unknown independent sources 8i(t) through an unknown n x n mixing matrix A( t) of causal linear filters: Xi E j aij * 8 j . We cast the problem as one of maximum likelihood density estima(cid:173) tion, and in that framework introduce an algorithm that searches for independent components using both temporal and spatial cues. We call the resulting algorithm "Contextual ICA," after the (Bell and Sejnowski 1995) Infomax algorithm, which we show to be a special case of cICA. Because cICA can make use of the temporal structure of its input, it is able separate in a number of situations where standard methods cannot, including sources with low kur(cid:173) tosis, colored Gaussian sources, and sources which have Gaussian histograms. Consider a set of n indepent sources 81 (t), .

The paper presents a novel technique of constrained independent component analysis (CICA) to introduce constraints into the clas(cid:173) sical ICA and solve the constrained optimization problem by using Lagrange multiplier methods. This paper shows that CICA can be used to order the resulted independent components in a specific manner and normalize the demixing matrix in the signal separation procedure. It can systematically eliminate the ICA's indeterminacy on permutation and dilation. The experiments demonstrate the use of CICA in ordering of independent components while providing normalized demixing processes.

The paper presents a novel technique of constrained independent component analysis (CICA) to introduce constraints into the classical ICAand solve the constrained optimization problem by using Lagrange multiplier methods. This paper shows that CICA can be used to order the resulted independent components in a specific manner and normalize the demixing matrix in the signal separation procedure. It can systematically eliminate the ICA's indeterminacy on permutation and dilation. The experiments demonstrate the use of CICA in ordering of independent components while providing normalized demixing processes. Keywords: Independent component analysis, constrained independent componentanalysis, constrained optimization, Lagrange multiplier methods 1 Introduction Independent component analysis (ICA) is a technique to transform a multivariate randomsignal into a signal with components that are mutually independent in complete statistical sense [1]. There has been a growing interest in research for efficient realization of ICA neural networks (ICNNs).