Accurate estimation of ICA weight matrix by implicit constraint imposition using Lie group

This paper presents a new stochastic algorithm to optimize the independence criterion-mutual information-among multivariate data using local, global, and hybrid optimizers, in conjunction with techniques involving a Lie group and its corresponding Lie algebra, for implicit imposition of the orthonormality constraint among the estimated sources. The major advantage of the proposed algorithm is the increased accuracy with which the weight matrix in the independent component analysis (ICA) model is estimated, compared to conventional schemes. When the local optimizer with Lie group techniques and the fast fixed-point (fastICA) algorithm were experimented by inputting the same set of random vectors, the former method superseded the conventional one by producing accurate weight matrix estimates in a majority of the test cases. Importantly, in our approach, the use of a Lie group to "lock" the weight matrix estimates onto the constraint surface enabled easy realization of the hybrid optimizers to yield near-global-optimum solutions consistently in most of the test cases, compared to well-known global optimizers. The inherent computational overhead in the hybrid optimizers was lowered by preprocessing the input data and periodically integrating the local optimizers with the global one. The proposed algorithms were applied to six-dimensional multispectral satellite image data to emphasize their usefulness in terms of accurate ICA weight matrix estimation.