# Structured Local Optima in Sparse Blind Deconvolution

arXiv: https://arxiv.org/abs/1806.00338 · HTML: https://mlsanity.com/paper/1806.00338/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2018.

## Built on by (most cited first, 15 of 15)

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- [Efficient Sparse Coding using Hierarchical Riemannian Pursuit](https://mlsanity.com/paper/2104.10314/) (2021)
- [Convex Sparse Blind Deconvolution](https://mlsanity.com/paper/2106.07053/) (2021)
- [A Geometric Approach to k-means](https://mlsanity.com/paper/2201.04822/) (2022)
- [Optimal vintage factor analysis with deflation varimax](https://mlsanity.com/paper/2310.10545/) (2023)
- [IB-AdCSCNet:Adaptive Convolutional Sparse Coding Network Driven by Information Bottleneck](https://mlsanity.com/paper/2405.14192/) (2024)
- [Sharper Convergence Rates for Nonconvex Optimisation via Reduction Mappings](https://mlsanity.com/paper/2506.08428/) (2025)
- [Neural Fields for NV-Center Inverse Sensing](https://mlsanity.com/paper/2605.13988/) (2026)

## Abstract (arXiv)

Blind deconvolution is a ubiquitous problem of recovering two unknown signals from their convolution. Unfortunately, this is an ill-posed problem in general. This paper focuses on the em short and sparse blind deconvolution problem, where the one unknown signal is short and the other one is sparsely and randomly supported. This variant captures the structure of the unknown signals in several important applications. We assume the short signal to have unit ℓ2 norm and cast the blind deconvolution problem as a nonconvex optimization problem over the sphere. We demonstrate that (i) in a certain region of the sphere, every local optimum is close to some shift truncation of the ground truth, and (ii) for a generic short signal of length k, when the sparsity of activation signal θlesssim k-2/3 and number of measurements mgtrsim poly(k), a simple initialization method together with a descent algorithm which escapes strict saddle points recovers a near shift truncation of the ground truth kernel.