Estimating Functions and Superefficiency for ICA and Deconvolution
Andrzej S Cichocki, Шун-ичи Амари · 2002
Chapter 10 introduces the method of estimating functions to elucidate the common structures in most of the ICA/BSS and MBD algorithms. We use information geometry for this purpose, and define estimating functions in semiparametric statistical models, which include unknown functions as parameters. Differences in most existing algorithms are only in the choices of estimating functions. We then give error analysis and stability analysis in terms of estimating functions. This makes it possible to design various adaptive methods for choosing unknown parameters included in estimating functions, which control accuracy and stability. The Newton method is automatically derived by the standardized estimating functions. First the standard BSS/ICA problem is formulated in the framework of the semiparametric model and a family of estimating functions. Furthermore, the present chapter discusses and extends further convergence and efficiency of the batch estimator and natural gradient learning for blind separation/deconvolution via the semiparametric statistical model and estimating functions and standardized estimating functions derived by using efficient score functions elucidated recently by Amari et al. We present the geometrical properties of the manifold of the FIR filters based on the Lie group structure and formulate the multichannel blind deconvolution problem within the framework of the semiparametric model deriving a family of estimating functions for blind deconvolution. We then analyze the efficiency of the batch estimator based on estimating function, obtaining its convergence rate. Finally, we show that both batch learning and on-line natural gradient learning are superefficient under given nonsingular conditions.