Fast approximation of matrix exponential and its application to independent component analysis problem

This paper introduces a new low-computational method for approximating the skew-symmetric and skew-Hermitian matrix exponential. Our method belongs to the splitting methods, which we modify and combine with new low-cost analytic formula for sparse skew-symmetric and skew-Hermitian matrix ex- ponential, similar to the Euler-Rodrigues formula, well known for the skew-symmetric matrices in ℝ3 . Our new approximation procedure for skew-symmetric (skew-Hertmitian) matrix exponential that we use is computationally very cheap, which ensures high speed of algorithms using this operation in their structure. To evaluate this approximation method we used it for the optimization problem of Indepen- dent Component Analysis (ICA) type. The results are compared to other known ICA algorithms such as well known Infomax and JADE. The average increase in convergence speed in the studied range of the number of source images was approximately 7% compared to the second fastest ICA algorithm using the standard and universal matrix exponential formula. High quality of separation was also obtained, com- parable to well-known ICA algorithms such as Infomax or JADE. Obtained results confirm the effective- ness of the proposed method in technical applications and indicate potential use in on-line applications.