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We study the Short-and-Sparse (SaS) deconvolution problem of recovering a short signal a0 and a sparse signal x0 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 regional analysis, which characterizes this landscape geometrically, on a union of subspaces. Our geometric characterization holds when the length-p0 short signal a0 has shift coherence µ, and x0 follows a random sparsity model with sparsity rate θ ∈ c 1 p 0 ,Based on this geometry, we give a provable method that successfully solves SaS deconvolution with high probability.Because of this symmetry, we only expect to recover a 0 and x 0 up to a signed shift (see Figure 1). Our problem of interest can be stated more formally as: Problem 1.1 (Short-and-Sparse Deconvolution). Given the cyclic convolution y = a 0 * x 0 ∈ R n of a 0 ∈ R p0 short (p 0 n), and x 0 ∈ R n sparse, recover a 0 and x 0 , up to a scaled shift.Despite a long history and many applications, until recently very little algorithmic theory was available for SaS deconvolution. Much of this difficulty can be attributed to the scale-shift symmetry: natural convex relaxations fail, and nonconvex formulations exhibit a complicated optimization landscape, with many 1 arXiv:1901.00256v2 [eess.SP] 11 Apr 2019 ϕ ρ is strongly convex in this region, and it has a minimizer very close to s [a 0 ].Geometry near a single shift. To gain intuition into the properties of ϕ ρ , we first visualize this function in the vicinity of a single shift s [a 0 ] of the ground truth a 0 . In Figure 3, we plot the function value of ϕ ρ overwhere B 2 ,r (a) is a ball of radius r around a. We make two observations:• The objective function ϕ ρ is strongly convex on this neighborhood of s [a 0 ].• There is a local minimizer very close to s [a 0 ].Write x 0J = ι * J x 0 and (x 0 ) I\J = P I\J x 0 . We can further notice that

Blind deconvolution is the problem of recovering a convolutional kernel a0 and an activation signal x0 from their convolution y = a0 x0. This problem is ill-posed without further constraints or priors. This paper studies the situation where the nonzero entries in the activation signal are sparsely and randomly populated. We normalize the convolution kernel to have unit Frobenius norm and cast the sparse blind deconvolution problem as a nonconvex optimization problem over the sphere. With this spherical constraint, every spurious local minimum turns out to be close to some signed shift truncation of the ground truth, under certain hypotheses. This benign property motivates an effective two stage algorithm that recovers the ground truth from the partial information offered by a suboptimal local minimum. This geometry-inspired algorithm recovers the ground truth for certain microscopy problems, also exhibits promising performance in the more challenging image deblurring problem. Our insights into the global geometry and the two stage algorithm extend to the convolutional dictionary learning problem, where a superposition of multiple convolution signals is observed.

a These authors contributed equally to this work 1 arXiv:1807.10752v1 [physics.comp-ph] a These authors contributed equally to this work

Short-and-sparse deconvolution (SaSD) is the problem of extracting localized, recurring motifs in signals with spatial or temporal structure. Variants of this problem arise in applications such as image deblurring, microscopy, neural spike sorting, and more. The problem is challenging in both theory and practice, as natural optimization formulations are nonconvex. Moreover, practical deconvolution problems involve smooth motifs (kernels) whose spectra decay rapidly, resulting in poor conditioning and numerical challenges. This paper is motivated by recent theoretical advances [ZLK `17, KZLW19], which characterize the optimization landscape of a particular nonconvex formulation of SaSD. This is used to derive a provable algorithm which exactly solves certain non-practical instances of the SaSD problem. We leverage the key ideas from this theory (sphere constraints, data-driven initialization) to develop a practical algorithm, which performs well on data arising from a range of application areas. We highlight key additional challenges posed by the ill-conditioning of real SaSD problems, and suggest heuristics (acceleration, continuation, reweighting) to mitigate them. Experiments demonstrate both the performance and generality of the proposed method.