Multiplicative Algorithm for Orthgonal Groups and Independent Component Analysis

The multiplicative Newton-like method developed by the author et al. is extended to the situation where the dynamics is restricted to the orthogonal group. A general framework is constructed without specifying the cost function. Though the restriction to the orthogonal groups makes the problem somewhat complicated, an explicit expression for the amount of individual jumps is obtained. This algorithm is exactly second-order-convergent. The global instability inherent in the Newton method is remedied by a Levenberg-Marquardt-type variation. The method thus constructed can readily be applied to the independent component analysis. Its remarkable performance is illustrated by a numerical simulation. 1 Overview Many optimization problems take the form, “Find an optimal matrix under the constraints (1).. (2).. etc.” Some of these can be considered as optimizations on Lie groups. For groups, the fundamental manipulation is a multiplication whereas an addition is unnatural. In consideration of this fact, we have constructed a multiplicative Newton-like algorithm for maximizing the ∗[email protected]