Geometry and Symmetry in Short-and-Sparse Deconvolution
This addresses a signal processing problem for researchers in deconvolution, but it appears incremental as it builds on existing models with a focus on geometric analysis.
The paper tackles the Short-and-Sparse deconvolution problem by proposing a nonconvex optimization method that recovers short and sparse signals up to a signed shift symmetry, with provable success under conditions involving shift coherence and sparsity rates.
We study the $\textit{Short-and-Sparse (SaS) deconvolution}$ problem of recovering a short signal $\mathbf a_0$ and a sparse signal $\mathbf x_0$ from their convolution. We propose a method based on nonconvex optimization, which under certain conditions recovers the target short and sparse signals, up to a signed shift symmetry which is intrinsic to this model. This symmetry plays a central role in shaping the optimization landscape for deconvolution. We give a $\textit{regional analysis}$, which characterizes this landscape geometrically, on a union of subspaces. Our geometric characterization holds when the length-$p_0$ short signal $\mathbf a_0$ has shift coherence $μ$, and $\mathbf x_0$ follows a random sparsity model with sparsity rate $θ\in \Bigl[\frac{c_1}{p_0}, \frac{c_2}{p_0\sqrtμ+ \sqrt{p_0}}\Bigr]\cdot\frac{1}{\log^2p_0}$. Based on this geometry, we give a provable method that successfully solves SaS deconvolution with high probability.