Structured Local Optima in Sparse Blind Deconvolution
Blind deconvolution is a ubiquitous problem aiming to recover a convolution kernel <inline formula> <tex math notation="LaTeX"\boldsymbol a_{0}\in \mathbb R {k} /tex math>/inline formula> and an activation signal <inline formula> <tex math notation="LaTeX"\boldsymbol x_{0}\in \mathbb R {m} /tex math>/inline formula> from their convolution <inline formula> <tex math notation="LaTeX"\boldsymbol y\in \mathbb R {m} /tex math>/inline formula> Unfortunately, this is an ill posed problem in general. This paper focuses on the <italic>short and sparse</italic> blind deconvolution problem, where the convolution kernel is short <inline formula> <tex math notation="LaTeX"$k\ll m$ /tex math>/inline formula> and the activation signal is sparsely and randomly supported <inline formula> <tex math notation="LaTeX"\left \boldsymbol x_{0} \right |_{0}\ll m$ /tex math>/inline formula> This variant captures the structure of the convolutional signals in several important application scenarios. In this paper, we normalize the convolution kernel to have unit Frobenius norm and then cast the blind deconvolution problem as a nonconvex optimization problem over the kernel 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 unit kernel <inline formula> <tex math notation="LaTeX"\boldsymbol a_{0} /tex math>/inline formula> when the sparsity of activation signal satisfies <inline formula> <tex math notation="LaTeX"\theta \lesssim k^ 2/3} /tex math>/inline formula> and number of measurements <inline formula> <tex math notation="LaTeX"$m\gtrsim \mathop \mathrm {poly}\nolimits \left {k \right) /tex math>/inline formula> the proposed initialization method together with a descent algorithm which escapes strict saddle points recovers some shift truncation of the ground truth kernel.
IEEE Transactions on Information Theory