MICA: multimodal independent component analysis

We extend the framework of ICA (independent component analysis) to the case that there is a pair of information sources. The goal of MICA is to extract statistically dependent pairs of features from the sources, where the components of feature vector extracted from each source are independent. Therefore, the cost function is constructed to maximize the degree of pairwise dependence as well as optimizing the cost function of ICA. We approximate the cost function by two dimensional Gram-Charlier expansion and propose a gradient descent algorithm derived by Amari's natural gradient. The relation between MICA and traditional CCA (canonical correlation analysis) is similar to the relation between ICA and PCA (principal component analysis). Introduction Recently, ICA (independent component analysis) has attracted a lot of attention because of its applications to brain science. One important application is data analysis for biomedical recordings (e.g. EEG, MEG, fMRI), in which artifactual noise components can be eliminated by ICA to extract pure neural brain activity[5]. Another application is extracting features in vision systems. Several kinds of image lters such as edge detector can be learned from natural images by ICA[4], which is also considered to be a model of early vision in the brain. In the present paper, we consider to extract common features from two sets of multidimensional inputs, whereas ICA deals with only one set of the inputs. It is the case, for example, that we have simultaneous recordings of EEG and MEG. Traditionally, CCA (canonical correlation analysis) is a widely known method to analyze the correlation structure between two sets of variables (see for example [1]). CCA maximizes the correlation coe cient between each pair of extracted features. Although this is equivalent to maximizing mutual information when given variables are jointly Gaussian-distributed, any nonsingular transformation in the canonical space does not change the amount of the mutual information. In CCA, the problem is partly solved by restricting the covariance matrices of feature vector to identity (sphering condition). However, the freedom of rotation transformation still remains for two axes in which the correlation coe cients are identical. Those assumptions and freedom can make di cult to analyze or to visualize each component of extracted features. The freedom of rotation also occurs in PCA (principal component analysis), and it has been solved by introducing the independence criterion in ICA instead of the sphering condition. We adopt that criterion in MICA, and furthermore the maximization of correlation coefcients is replaced by the maximization of mutual information itself to deal with non-Gaussian signals and nonlinear relation. In the following sections, rst we formulate the framework of MICA, next we derive the gradient descent algorithm and show some simulation results, and we conclude the paper by mentioning further applications and the relation to other methods. Formulation MICA can be formulated in a similar way to ICA except that MICA deals with a pair of sources ( g. 1): Let us consider unknown pairs of source signals (u i (t); v i (t)), i = 1; . . . ; n, where u i (t) and v i (t) are statistically related to each other at any xed time t, while vector components of fu i (t)g and fv i (t)g are mutually independent respectively. We assume (u (t);v (t)) is stationary and generated from unknown distribution p(u ;v ). The condition of the independence can be written by