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A deterministic analysis of variable-metric adaptive filtering algorithms under small metric-fluctuations

We present a rigorous deterministic analysis of the variable-metric adaptive filtering algorithms (including the transform-domain, LMS/Newton, and proportionate adaptive filters) by using the framework of variable-metric adaptive projected subgradient method (Yukawa et al. 2007). Under small metric-fluctuations, we present the useful properties of (i) monotone approximation — with respect to a certain constant metric — indicating the stability of the algorithm and (ii) convergence to an asymptotically optimal point. Numerical examples show the advantage of the variable-metric adaptive filtering algorithms and suggest the validity of the analysis.