Contravariant Adaptation on the Manifold of Causal, FIR, Invertible Multivariable Matrix Systems

This letter extends previous work on contravariant adaptation by providing a formula for the contravariant (natural) gradient on the manifold of multivariable, causal, invertible, finite impulse response (FIR) systems. The right action on the manifold of multivariable causal Toeplitz systems is defined. Using this right action, a bilinear form defined on the tangent space at the identity is extended throughout the entire manifold by invariance. This results in a formula analogous to the natural gradient for matrices, but which preserves the causal, FIR nature of the system. The contravariant conversion factor is block Toeplitz structured, so that implementations may employ fast Fourier transform based convolutions to produce lower complexity than would a comparably-sized generic matrix natural gradient.