Two product-space formulations for unifying multiple metrics in set-theoretic adaptive filtering

In this paper, we present two novel approaches to the issue of exploiting multiple metrics jointly for efficient adaptive filtering. The key is the introduction of product-space formulation for taking into account multiple metrics in a single Hilbert space. The first approach is based on the Pierra's idea of reformulating the problem of finding a common point of multiple closed convex sets as a problem of finding a common point of two closed convex sets in a product space. The second approach is a slight modification of the first one along the idea of constraint-embedding. An interesting relation between our approaches and the improved proportionate normalized least mean square (IPNLMS) algorithm is provided. The monotone approximation properties of the two algorithms are also presented. A numerical example suggests the efficacy of the presented multi-metric strategy.