Geometry and Symmetry in Short-and-Sparse Deconvolution

Understand

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.