Objective Function of ICA with Smooth Estimation of Kurtosis

In this paper, a new objective function of ICA is proposed by a probabilistic approach to the quadratic terms. Many previous ICA methods are sensitive to the sign of kurtosis of source (sub-or super-Gaussian), where the change of the sign often causes a large discontinuity in the objective function. On the other hand, some other previous methods use continuous objective functions by using the squares of the 4th-order statistics. However, such squared statistics often lack the robustness because they magnify the outliers. In this paper, we solve this problem by introducing a new objective function which is given as a summation of weighted 4th-order statistics, where the kurtoses of sources are incorporated "smoothly" into the weights. Consequently, the function is always continuously differentiable with respect to both the kurtoses and the separating matrix to be estimated. In addition, we propose a new ICA method optimizing the objective function by the Givens rotations under the orthonormality constraint. Experimental results show that the proposed method is comparable to the other ICA methods and it outperforms them especially when sub-Gaussian sources are dominant.