Geometry and Symmetry in ShortandSparse Deconvolution We study the textitShortandSparse (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 textitregional analysis, which characterizes this landscape geometrically, on a union of subspaces. Our geometric characterization holds when the lengthp_0 short signal mathbf a_0 has shift coherence mu, and mathbf x_0 follows a random sparsity model with sparsity rate theta in Biglfracc_1p_0, fracc_2p_0sqrtmu  sqrtp_0Bigrcdotfrac1log2p_0. Based on this geometry, we give a provable method that successfully solves SaS deconvolution with high probability.

Variations on the Convolutional Sparse Coding Model Over the past decade, the celebrated sparse representation model has achieved impressive results in various signal and image processing tasks. A convolutional version of this model, termed convolutional sparse coding (CSC), has been recently reintroduced and extensively studied. CSC brings a natural remedy to the limitation of typical sparse enforcing approaches of handling global and highdimensional signals by local, patchbased, processing. While the classic field of sparse representations has been able to cater for the diverse challenges of different signal processing tasks by considering a wide range of problem formulations, almost all available algorithms that deploy the CSC model consider the same   ell _1  ell _2   problem form. As we argue in this paper, this CSC pursuit formulation is also too restrictive as it fails to explicitly exploit some local characteristics of the signal. This work expands the range of formulations for the CSC model by proposing two convex alternatives that merge global norms with local penalties and constraints. The main contribution of this work is the derivation of efficient and provably converging algorithms to solve these new sparse coding formulations.