A Greedy Version of the Frame Algorithm

TL;DR

This paper introduces an adaptive, greedy version of the frame algorithm that improves convergence and robustness without needing frame bounds, demonstrated through numerical examples.

Contribution

It presents a novel greedy, adaptive frame algorithm that guarantees convergence and robustness, unlike traditional methods that require frame bounds.

Findings

Achieves the same convergence rate as the classical frame algorithm.

Does not require prior knowledge of frame bounds.

Demonstrates improved performance in noisy measurement scenarios.

Abstract

The frame algorithm uses a simple recursive formula to approximate an unknown vector from its frame coefficients. This note introduces an adaptive version of the frame algorithm that maximizes the error reduction between steps in terms of an equivalent norm. This greedy version of the frame algorithm is proven to achieve the same guaranteed convergence rate as the traditional frame algorithm, yet, unlike its classical counterpart, does not require knowledge of the frame bounds. The robustness of the greedy frame algorithm with respect to noisy measurements is also established. Two numerical examples are included to demonstrate the benefit of the greedy algorithm in applications.