Optimal frames and Newton's method

TL;DR

This paper develops an efficient Newton's method approach for optimizing over parametrized finite frames to best approximate data vectors, analyzing sensitivity to noise and conditions for uniqueness.

Contribution

It derives explicit derivatives for the optimization problem and explores conditions for the uniqueness of the minimizer, enhancing frame-based data approximation methods.

Findings

Analytic expressions for derivatives enable efficient optimization.

Sensitivity analysis of the minimizer to data noise.

Conditions for the uniqueness of the optimal frame member.

Abstract

Given a parametrized family of finite frames, we consider the optimization problem of finding the member of this family whose coefficient space most closely contains a given data vector. This nonlinear least squares problem arises naturally in the context of a certain type of radar system. We derive analytic expressions for the first and second partial derivatives of the objective function in question, permitting this optimization problem to be efficiently solved using Newton's method. We also consider how sensitive the location of this minimizer is to noise in the data vector. We further provide conditions under which one should expect the minimizer of this objective function to be unique. We conclude by discussing a related variational-calculus-based approach for solving this frame optimization problem over an interval of time.