Blind deconvolution is a ubiquitous problem aiming to recover a convolution kernel $\boldsymbol a_{0}\in \mathbb R ^{k}$ and an activation signal $\boldsymbol x_{0}\in \mathbb R ^{m}$ from their convolution $\boldsymbol y\in \mathbb R ^{m}$ . Unfortunately, this is an ill-posed problem in general. This paper focuses on the short and sparse blind deconvolution problem, where the convolution kernel is short ( $k\ll m$ ) and the activation signal is sparsely and randomly supported ( $\left \|{ \boldsymbol x_{0} }\right \|_{0}\ll m$ ). 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 $\boldsymbol a_{0}$ , when the sparsity of activation signal satisfies $\theta \lesssim k^{-2/3}$ and number of measurements $m\gtrsim \mathop {\mathrm {poly}}\nolimits \left ({k }\right) $ , the proposed initialization method together with a descent algorithm which escapes strict saddle points recovers some shift truncation of the ground truth kernel.

中文翻译：

---

稀疏盲解卷积中的结构化局部最优

盲解卷积是一个普遍存在的问题，旨在恢复卷积核 $\boldsymbol a_{0}\in \mathbb R ^{k}$ 和激活信号 $\boldsymbol x_{0}\in \mathbb R ^{m}$ 从他们的卷积 $\boldsymbol y\in \mathbb R ^{m}$ . 不幸的是，这通常是一个不适定的问题。本文着重于短而稀疏 盲反卷积问题，其中卷积核很短（ $k\ll m$ ) 并且激活信号是稀疏随机支持的 ​​( $\left \|{ \boldsymbol x_{0} }\right \|_{0}\ll m$ ）。该变体在几个重要的应用场景中捕获了卷积信号的结构。在本文中，我们将卷积核归一化为单位 Frobenius 范数，然后将盲反卷积问题转化为核球上的非凸优化问题。我们证明了 (i) 在球体的某个区域，每个局部最优都接近于基本事实的一些移位截断，以及 (ii) 对于通用单元核 $\boldsymbol a_{0}$ , 当激活信号的稀疏性满足 $\theta \lesssim k^{-2/3}$ 和测量次数 $m\gtrsim \mathop {\mathrm {poly}}\nolimits \left ({k }\right) $ ，所提出的初始化方法与逃避严格鞍点的下降算法一起恢复了地面实况内核的一些移位截断。