In this paper, we study the problem of recovering two unknown signals from their convolution, which is commonly referred to as blind deconvolution. Reformulation of blind deconvolution as a low-rank recovery problem has led to multiple theoretical recovery guarantees in the past decade due to the success of the nuclear norm minimization heuristic. In particular, in the absence of noise, exact recovery has been established for sufficiently incoherent signals contained in lower-dimensional subspaces. However, if the convolution is corrupted by additive bounded noise, the stability of the recovery problem remains much less understood. In particular, existing reconstruction bounds involve large dimension factors and therefore fail to explain the empirical evidence for dimension-independent robustness of nuclear norm minimization. Recently, theoretical evidence has emerged for ill-posed behavior of low-rank matrix recovery for sufficiently small noise levels. In this work, we develop improved recovery guarantees for blind deconvolution with adversarial noise which exhibit square-root scaling in the noise level. Hence, our results are consistent with existing counterexamples which speak against linear scaling in the noise level as demonstrated for related low-rank matrix recovery problems.

翻译：本文研究从卷积结果中恢复两个未知信号的问题（即盲反卷积）。过去十年间，由于核范数最小化启发式方法的成功，盲反卷积的低秩恢复重构为多个理论恢复保证提供了依据。特别地，在无噪声条件下，对包含于低维子空间且充分非相干的信号已建立精确恢复理论。然而，当卷积被加性有界噪声污染时，恢复问题的稳定性仍缺乏深入理解。现有重构误差界因包含大维度因子，无法解释核范数最小化具有维度无关鲁棒性的实验现象。近期研究揭示了低秩矩阵恢复在噪声水平极低时可能出现病态行为的理论证据。本文针对对抗性噪声下的盲反卷积，提出了改进的恢复保证，其误差界呈现噪声水平的平方根标度。该结果与现有反例一致——这些反例表明，与相关低秩矩阵恢复问题不同，盲反卷积的误差不随噪声水平线性增长。