# On the Global Geometry of Sphere-Constrained Sparse Blind Deconvolution

arXiv: https://arxiv.org/abs/1901.01913 · HTML: https://mlsanity.com/paper/1901.01913/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jan 2019.

## Built on by (most cited first, 5 of 5)

- [Blind Image Deconvolution Using Variational Deep Image Prior](https://mlsanity.com/paper/2202.00179/) (2022)
- [Escaping Saddle Points for Nonsmooth Weakly Convex Functions via Perturbed Proximal Algorithms](https://mlsanity.com/paper/2102.02837/) (2021)
- [Unique sparse decomposition of low rank matrices](https://mlsanity.com/paper/2106.07736/) (2021)
- [Convex Sparse Blind Deconvolution](https://mlsanity.com/paper/2106.07053/) (2021)
- [VDIP-TGV: Blind Image Deconvolution via Variational Deep Image Prior Empowered by Total Generalized Variation](https://mlsanity.com/paper/2310.19477/) (2023)

## Abstract (arXiv)

Blind deconvolution is the problem of recovering a convolutional kernel boldsymbol a0 and an activation signal boldsymbol x0 from their convolution boldsymbol y = boldsymbol a0 circledast boldsymbol 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.