Fast off-the-grid sparse recovery with over-parametrized projected gradient descent

TL;DR

This paper introduces an off-the-grid over-parametrized initialization method for projected gradient descent, enabling faster 3D sparse recovery from Fourier measurements without relying on grids.

Contribution

It proposes a novel off-the-grid over-parametrized initialization based on OMP, improving the speed and efficiency of sparse recovery in 3D.

Findings

Faster sparse recovery in 3D using the proposed method.

Avoids the computational cost of grid-based methods.

Outperforms traditional iterative methods in speed.

Abstract

We consider the problem of recovering off-the-grid spikes from Fourier measurements. Successful methods such as sliding Frank-Wolfe and continuous orthogonal matching pursuit (OMP) iteratively add spikes to the solution then perform a costly (when the number of spikes is large) descent on all parameters at each iteration. In 2D, it was shown that performing a projected gradient descent (PGD) from a gridded over-parametrized initialization was faster than continuous orthogonal matching pursuit. In this paper, we propose an off-the-grid over-parametrized initialization of the PGD based on OMP that permits to fully avoid grids and gives faster results in 3D.