Multichannel Sparse Blind Deconvolution on the Sphere
This addresses the challenge of recovering unknown signals and channels from convolutions in signal processing, with incremental improvements in theoretical guarantees and algorithm efficiency.
The paper tackles the problem of multichannel blind deconvolution with sparse channels by proposing a nonconvex optimization formulation on the unit sphere, showing that all local minima correspond to the inverse filter up to ambiguities and enabling recovery via manifold gradient descent, with numerical experiments demonstrating superior performance over previous methods.
Multichannel blind deconvolution is the problem of recovering an unknown signal $f$ and multiple unknown channels $x_i$ from their circular convolution $y_i=x_i \circledast f$ ($i=1,2,\dots,N$). We consider the case where the $x_i$'s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse output $y_i\circledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent (MGD) algorithm. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.