[Paper Review] Hellinger vs. Kullback-Leibler multivariable spectrum approximation

Augusto Ferrante, Michele Pavon+1 more|ArXiv.org|Feb 8, 2007

Blind Source Separation Techniques4 citations

TL;DR

This paper introduces a Hellinger-type metric for multivariable spectral densities using spectral factorization, formulates a constrained optimization problem via duality, and presents a matricial Newton-type algorithm that reliably solves the dual problem, with simulations confirming its effectiveness in approximating spectral densities under complexity constraints.

ABSTRACT

In this paper, we study a matricial version of the Byrnes-Georgiou-Lindquist generalized moment problem with complexity constraint. We introduce a new metric on multivariable spectral densities induced by the family of their spectral factors which, in the scalar case, reduces to the Hellinger distance. We solve the corresponding constrained optimization problem via duality theory. A highly nontrivial existence theorem for the dual problem is established in the Byrnes-Lindquist spirit. A matricial Newton-type algorithm is finally provided for the numerical solution of the dual problem. Simulation indicates that the algorithm performs effectively and reliably.

Motivation & Objective

• To develop a matricial generalization of the Hellinger distance for multivariable spectral densities using spectral factorization.
• To address the generalized moment problem with complexity constraints in the multivariable setting.
• To establish a duality framework for solving the constrained spectral approximation problem.
• To prove a nontrivial existence theorem for the dual problem in the spirit of Byrnes-Lindquist.

Proposed method

• Introduces a new metric on multivariable spectral densities derived from the family of their spectral factors, generalizing the scalar Hellinger distance.
• Applies duality theory to transform the primal constrained optimization problem into a dual problem.
• Establishes a highly nontrivial existence theorem for solutions to the dual problem using techniques inspired by Byrnes and Lindquist.
• Develops a matricial Newton-type algorithm to numerically solve the dual problem.
• Employs spectral factorization as a central tool to define the metric and ensure structural consistency in the approximation.
• Validates the algorithm's performance through simulations showing effective and reliable convergence.

Experimental results

Research questions

• RQ1 How can the Hellinger distance be generalized to multivariable spectral densities using spectral factorization?
• RQ2 What is the appropriate duality framework for solving the generalized moment problem with complexity constraints in the multivariable case?
• RQ3 Under what conditions does a solution to the dual problem exist in the multivariable setting?
• RQ4 How can a numerical algorithm be designed to solve the dual problem efficiently and reliably?
• RQ5 How does the proposed metric compare in performance to existing spectral approximation methods?

Key findings

• The proposed metric generalizes the scalar Hellinger distance to multivariable spectral densities through spectral factorization.
• A nontrivial existence theorem for the dual problem is established, ensuring the theoretical solvability of the optimization framework.
• The matricial Newton-type algorithm converges reliably in simulations, demonstrating practical effectiveness.
• The duality-based approach enables constrained spectral approximation under complexity constraints.
• The method preserves structural properties of spectral densities through the use of spectral factors.
• Simulation results confirm the algorithm's robustness and numerical reliability in multivariable spectral approximation.