A Low-Rank Approach to Off-The-Grid Sparse Deconvolution
This work provides a computationally efficient method for sparse spike deconvolution in high dimensions, addressing a key bottleneck in signal processing and imaging.
The paper tackles the off-the-grid sparse deconvolution problem, which is intractable for large-scale settings due to its semidefinite relaxation scaling as $f_c^{2d}$. They propose a penalized low-rank formulation and a conditional gradient algorithm with $O(f_c^d \\log f_c)$ complexity per iteration, achieving convergence in exactly $r$ steps (where $r$ is the number of Diracs).
We propose a new solver for the sparse spikes deconvolution problem over the space of Radon measures. A common approach to off-the-grid deconvolution considers semidefinite (SDP) relaxations of the total variation (the total mass of the absolute value of the measure) minimization problem. The direct resolution of this SDP is however intractable for large scale settings, since the problem size grows as $f_c^{2d}$ where $f_c$ is the cutoff frequency of the filter and $d$ the ambient dimension. Our first contribution introduces a penalized formulation of this semidefinite lifting, which has low-rank solutions. Our second contribution is a conditional gradient optimization scheme with non-convex updates. This algorithm leverages both the low-rank and the convolutive structure of the problem, resulting in an $O(f_c^d \log f_c)$ complexity per iteration. Numerical simulations are promising and show that the algorithm converges in exactly $r$ steps, $r$ being the number of Diracs composing the solution.