Recursive eigendecomposition via autoregressive analysis and ago-antagonistic regularization

A new recursive eigendecomposition algorithm of Complex Hermitian Tœplitz matrices is studied. Based on Trench’s inversion of Tœplitz matrices from their autoregressive analysis, we have developed a fast recursive iterative algorithm that takes into account the rank-one modification of successive order Toeplitz matrices. To speed up the computational time and to increase numerical stability of illconditioned eigendecomposition in case of very short data records analysis, we have extended this method by introducing an ago-antagonistic regularized reflection coefficient via Levinson equation. We provide a geometrical interpretation of this new recursive eigendecomposition. 2. PREAMBLE Let us remind you that Levinson algorithm provides Cholesky factorization of the inverse Tœplitz matrix. Rankone modification approach leads to the Gohberg-Semencul formula which is an integrated version of Trench algorithm [5]. Trench algorithm induces an order recursive structure of the inverse Tœplitz matrix. We propose to exploit this existing structure to achieve a fast and robust eigendecomposition. First, we obtain eigenvalues by finding the roots of an autoregressive parameters-based function [2]. At each order, a number of independent structurally identical nonlinear problems is solved in parallel. Derivative of this intermediate function is geometrically interpreted. In a second step, via Levinson equation, reflection coefficient i s used to decrease computational complexity and increase stability by an ago-antagonistic regularization [1][2]. Agoantagonism [6], conceived as Minimum Free Enthalpy concept in a thermodynamic analogy approach, extends regularization method and avoids over-regularization problems. Among research in the area of recursive eigenspace decomposition, other algorithms have been proposed taking advantage of direct Tœplitz matrix structure, like RISE [3][4], but they are not very well adapted to very short data records analysis. 3. RECURSIVE EIGENDECOMPOSITION VIA AUTOREGRESSIVE ANALYSIS 3.1 Yule-Walker and Levinson Equation Autoregressive analysis problem is solved by Yule-Walker equation. Order recursive structure of Tœplitz correlation matrix provides the recursive Levinson equation :