# A Nonconvex Approach for Exact and Efficient Multichannel Sparse Blind Deconvolution

arXiv: https://arxiv.org/abs/1908.10776 · HTML: https://mlsanity.com/paper/1908.10776/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Aug 2019.

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

- [The Power of Preconditioning in Overparameterized Low-Rank Matrix Sensing](https://mlsanity.com/paper/2302.01186/) (2023)
- [Convolutional Normalization: Improving Deep Convolutional Network Robustness and Training](https://mlsanity.com/paper/2103.00673/) (2021)
- [Efficient Sparse Coding using Hierarchical Riemannian Pursuit](https://mlsanity.com/paper/2104.10314/) (2021)
- [Resource-Efficient Invariant Networks: Exponential Gains by Unrolled Optimization](https://mlsanity.com/paper/2203.05006/) (2022)
- [Sharper Convergence Rates for Nonconvex Optimisation via Reduction Mappings](https://mlsanity.com/paper/2506.08428/) (2025)

## Abstract (arXiv)

We study the multi-channel sparse blind deconvolution (MCS-BD) problem, whose task is to simultaneously recover a kernel mathbf a and multiple sparse inputs mathbf xii=1p from their circulant convolution mathbf yi = mathbf a circledast mathbf xi  (i=1,cdots,p). We formulate the task as a nonconvex optimization problem over the sphere. Under mild statistical assumptions of the data, we prove that the vanilla Riemannian gradient descent (RGD) method, with random initializations, provably recovers both the kernel mathbf a and the signals mathbf xii=1p up to a signed shift ambiguity. In comparison with state-of-the-art results, our work shows significant improvements in terms of sample complexity and computational efficiency. Our theoretical results are corroborated by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods on both synthetic and real datasets.