A fast homotopy algorithm for gridless sparse recovery

Abstract

In this paper, we study the solving of the gridless sparse optimization problem and its application to 3D image deconvolution. Based on the recent works of [14] introducing the Sliding Frank-Wolfe algorithm to solve the Beurling LASSO problem, we introduce an accelerated algorithm, denoted BSFW, that preserves its convergence properties, while removing most of the costly local descents. Besides, as the solving of BLASSO still relies on a regularization parameter, we introduce a homotopy algorithm to solve the constrained BLASSO that allows to use a more practical parameter based on the image residual, e.g. its standard deviation. Both algorithms benefit from a finite termination property, i.e. they are guaranteed to find the solution in a finite number of step under mild conditions. These methods are then applied on the problem of 3D tomographic diffractive microscopy images, with the purpose of explaining the image by a small number of atoms in convolved observations. Numerical results on synthetic and real images illustrates the improvement provided by the BSFW method, the homotopy method and their combination.

On the uniqueness of solutions for the basis pursuit in the continuum

“…The study of optimization problems over the space of Radon measures can be traced back to the pioneering works of Beurling [22], where Fourier-domain measurements were also considered. In the early 2010s, the works of De Castro and Gamboa [2], Candès and Fernandez-Granda [3,23], and Bredies and Pikkarainen [4] considered optimization tasks of the form (3) (or its penalized version), with both theoretical analyses and novel algorithmic approaches to recover a sparse-measure solution, in the continuum [4,7,8,10,[24][25][26][27]. The existence of sparse-measure solutions, i.e., solutions of the form K k=1 a k δ x k , where K ∈ N * , a k ∈ R, and δ x k is the Dirac mass at the location x k , seems to have been proven for the first time in [22] and was later improved by Fisher and Jerome in [28].…”

“…They provide a general framework for the recovery of sparse continuous-domain signals (e.g., Dirac streams or splines) from possibly corrupted finite-dimensional measurements. Such techniques rely on solid theoretical foundations [2][3][4][5][6], but also on many algorithmic advances [4,7,8], and have found various data-science applications [7,[9][10][11]. It is well known that their discrete-domain counterparts (i.e., 1 regularization methods) lead to variational problems whose solutions are not necessarily unique [12].…”

Gridless three-dimensional compressive beamforming with the Sliding Frank-Wolfe algorithm

“…Setting the regularization parameter is known to be a difficult problem. To help this tuning, the homotopy approach can be used 33 : results of the algorithm for several parameters λ can be obtained by sequentially running the SFW algorithm for decreasing λ, initializing each run with the output of the previous run. In this case, the algorithm starts by solving (15) and checking the value of η, to avoid addition of a source if not necessary.…”

Proximal methods for point source localisation

“…The basic scheme of conditional gradient methods [5, 17,18,29,3,16] for (1.1) is to add a single Dirac measure (or spike) to the discrete measure 𝜇, and then optimise the weights of the spikes so far inserted. Repeat.…”