Tradeoffs between Convergence Speed and Reconstruction Accuracy in Inverse Problems

TL;DR

This paper explores how modifying iterative algorithms for inverse problems can achieve faster convergence within a limited number of iterations, balancing speed and accuracy by leveraging low-dimensional set approximations.

Contribution

It introduces a theoretical framework linking set approximation accuracy to convergence speed and reconstruction error, inspired by recent advances in sparse recovery and deep learning.

Findings

Faster convergence is possible with coarse set estimates.

Tradeoff exists between convergence speed and reconstruction accuracy.

The theory explains neural network approximations of iterative algorithms.

Abstract

Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is willing to tolerate, an important question is whether it is possible to modify the original iterations to obtain faster convergence to a minimizer achieving the allowed error without increasing the computational cost of each iteration considerably. Relying on recent recovery techniques developed for settings in which the desired signal belongs to some low-dimensional set, we show that using a coarse estimate of this set may lead to faster convergence at the cost of an additional reconstruction error related to the accuracy of the set approximation. Our theory ties to recent advances in sparse recovery, compressed sensing, and deep learning. Particularly, it may provide a possible explanation to the successful approximation of the l1-minimization solution by neural networks with layers representing iterations, as practiced in the learned iterative shrinkage-thresholding algorithm (LISTA).