Manifold Gradient Descent Solves Multi-Channel Sparse Blind Deconvolution Provably and Efficiently
This addresses a challenging inverse problem in signal processing and computer vision, offering a provable solution with potential applications in those domains.
The paper tackles the problem of multi-channel sparse blind deconvolution, which involves learning an unknown filter from sparse input signals, and demonstrates that manifold gradient descent with random initializations provably recovers the filter under sufficient observations.
Multi-channel sparse blind deconvolution, or convolutional sparse coding, refers to the problem of learning an unknown filter by observing its circulant convolutions with multiple input signals that are sparse. This problem finds numerous applications in signal processing, computer vision, and inverse problems. However, it is challenging to learn the filter efficiently due to the bilinear structure of the observations with the respect to the unknown filter and inputs, as well as the sparsity constraint. In this paper, we propose a novel approach based on nonconvex optimization over the sphere manifold by minimizing a smooth surrogate of the sparsity-promoting loss function. It is demonstrated that manifold gradient descent with random initializations will provably recover the filter, up to scaling and shift ambiguity, as soon as the number of observations is sufficiently large under an appropriate random data model. Numerical experiments are provided to illustrate the performance of the proposed method with comparisons to existing ones.