Adaptive Learning in a World of Projections

Abstract

This article presents a general tool for convexly constrained parameter/function estimation both for classification and regression tasks, in a timeadaptive setting and in (infinite dimensional) reproducing kernel Hilbert spaces (RKHS). The thematical framework is that of the set theoretic estimation formulation and the classical projections onto convex sets (POCS) theory. However, in contrast to the classical POCS methodology, which assumes a finite number of convex sets, our method builds upon our recent extension of the theory, which considers an infinite number of convex sets. Such a context is necessary to cope with the adaptive setting rationale, where data arrive sequentially. This article's goal is to review the advances that have taken place in this area over the years and present them, in simple geometric arguments, as an integral part and natural evolution of the classical POCS methodology. The structure of the resulting algorithms is such that it allows extension to general RKHS. In this perspective, two very powerful techniques, convex optimization and (implicit) mapping to RKHS, are combined, which provide a framework for a unifying treatment of linear and nonlinear modeling of both classification and regression tasks. Typical signal processing problems, such as filtering, smoothing, equalization, and beamforming, fall under this common umbrella. The methodology allows for the incorporation of a set of convex constraints, which encode a priori information. Convexity, rather than differentiability, is the only prerequisite for adopting error measures that quantify the model's fit against a set of training data points. Moreover, the complexity per iteration step remains linear with respect to the number of unknown parameters. The potential of the theory is demonstrated via numerical simulations for two typical problems; adaptive equalization and adaptive robust beamforming.