Revisiting a Fast Newton Solver for a 2-D Spectral Estimation Problem: Computations with the Full Hessian

TL;DR

This paper improves a 2-D spectral estimation method by exploiting the Toeplitz-block Toeplitz structure of the Hessian, enabling faster Newton-based optimization with demonstrated efficiency in simulations.

Contribution

It reveals the Toeplitz-block Toeplitz structure of the full Hessian in the dual problem, facilitating a rapid Newton solver for 2-D spectral estimation.

Findings

Hessian has Toeplitz-block Toeplitz structure

Fast inversion algorithm improves Newton method convergence

Simulation results show superior speed of the proposed approach

Abstract

Spectral estimation plays a fundamental role in frequency-domain identification and related signal processing problems. This paper revisits a 2-D spectral estimation problem formulated in terms of convex optimization. More precisely, we work with the dual optimization problem and show that the full Hessian of the dual function admits a Toeplitz-block Toeplitz structure which is consistent with our finding in a previous work. This particular structure of the Hessian enables a fast inversion algorithm in the solution of the dual optimization problem via Newton's method whose superior speed of convergence is illustrated via simulations.