Spurious minimizers in non uniform Fourier sampling optimization

TL;DR

This paper investigates the challenges in optimizing non-Cartesian Fourier sampling patterns, highlighting issues like spurious minimizers and vanishing gradients, and proposes solutions involving large datasets and stochastic algorithms.

Contribution

It identifies two key optimization issues in non-uniform Fourier sampling and demonstrates how large datasets and stochastic methods can address them.

Findings

Spurious minimizers are prevalent in non-uniform Fourier sampling optimization.

Large datasets help mitigate the problem of spurious minimizers.

Stochastic gradient algorithms with variable metrics improve optimization outcomes.

Abstract

A recent trend in the signal/image processing literature is the optimization of Fourier sampling schemes for specific datasets of signals. In this paper, we explain why choosing optimal non Cartesian Fourier sampling patterns is a difficult nonconvex problem by bringing to light two optimization issues. The first one is the existence of a combinatorial number of spurious minimizers for a generic class of signals. The second one is a vanishing gradient effect for the high frequencies. We conclude the paper by showing how using large datasets can mitigate first effect and illustrate experimentally the benefits of using stochastic gradient algorithms with a variable metric.