Topological approaches to adaptive spectrum modeling

The authors present initial investigations into a novel approach for adaptive parametric spectrum modeling that is based on numerical continuation methods derived from homotopy theory, a branch of algebraic topology. This initial investigation focuses on homotopies applied to system identification using the linear finite-impulse-response filter structure, to illustrate the transition from homotopy theory to the adaptive filtering of time series. Homotopy-based continuation methods are globally convergent numerical procedures from which robust adaptive infinite-impulse-response algorithms may eventually be derived. Homotopy theory is reviewed, and the theoretical justification for its application to the adaptive filtering of time series is presented. The practical problems associated with the development of homotopy-based adaptive algorithms are discussed in detail