[论文解读] From Blind deconvolution to Blind Super-Resolution through convex programming

Augustin Cosse|arXiv (Cornell University)|Sep 26, 2017

Sparse and Compressive Sensing Techniques参考文献 40被引用 3

一句话总结

本文证明，在已知低维子空间中存在少量输入信号的条件下，通过凸优化的核范数最小化，可求解盲反卷积与盲超分辨率问题。关键结果表明，当环境维数 $ L \gtrsim K^{3/2} \mu_m^2 $ 且输入信号数量 $ N \gtrsim K^{1/2} \mu_h^2 $ 时（忽略对数因子），可通过构造 Neumann 级数并应用浓度不等式控制其各项，以高概率实现精确恢复。

ABSTRACT

This paper discusses the recovery of an unknown signal $x\in \mathbb{R}^L$ through the result of its convolution with an unknown filter $h \in \mathbb{R}^L$. This problem, also known as blind deconvolution, has been studied extensively by the signal processing and applied mathematics communities, leading to a diversity of proofs and algorithms based on various assumptions on the filter and its input. Sparsity of this filter, or in contrast, non vanishing of its Fourier transform are instances of such assumptions. The main result of this paper shows that blind deconvolution can be solved through nuclear norm relaxation in the case of a fully unknown channel, as soon as this channel is probed through a few $N \gtrsim μ^2_m K^{1/2}$ input signals $x_n = C_n m_n$, $n=1,\ldots,N,$ that are living in known $K$-dimensional subspaces $C_n$ of $\mathbb{R}^L$. This result holds with high probability on the genericity of the subspaces $C_n$ as soon as $L\gtrsim K^{3/2}$ and $N\gtrsim K^{1/2}$ up to log factors. Our proof system relies on the construction of a certificate of optimality for the underlying convex program. This certificate expands as a Neumann series and is shown to satisfy the conditions for the recovery of the matrix encoding the unknowns by controlling the terms in this series. An incidental consequence of the result of this paper, following from the lack of assumptions on the filter, is that nuclear norm relaxation can be extended from blind deconvolution to blind super-resolution, as soon as the unknown ideal low pass filter has a sufficiently large support compared to the ambient dimension $L$. Numerical experiments supporting the theory as well as its application to blind super-resolution are provided.

研究动机与目标

• 在不假设滤波器稀疏性或非相干性的前提下，建立通过凸规划求解盲反卷积的条件。
• 通过解除对滤波器的严格假设，将核范数最小化的适用范围从盲反卷积扩展至盲超分辨率。
• 为在低维子空间中使用最少数量的探测信号，精确恢复未知信号与滤波器，提供理论保证。
• 通过 Neumann 级数展开构建基于证明框架的最优性证书，以验证凸松弛的最优性。

提出的方法

• 通过将信号与滤波器升维为矩阵变量，将盲反卷积建模为低秩矩阵恢复问题，再通过最小化核范数以促进低秩性。
• 通过 Neumann 级数展开构造最优性证书，并证明其满足精确恢复的次梯度条件。
• 利用次指数 Bernstein 不等式与 U-统计量的 Rosenthal-Pinelis 不等式矩阵版本，对 Neumann 级数的前两项进行有界控制。
• 通过基于浓度与解耦技术的通用论证方法，控制其余高阶项。
• 证明依赖于相干性参数 $ \mu_m $ 与 $ \mu_h $，分别量化输入信号的能量分布与滤波器傅里叶变换的能量扩散程度。
• 通过观察到足够分散的低通滤波器可在相同凸松弛下支持恢复，将该框架扩展至盲超分辨率。

实验结果

研究问题

• RQ1 在不假设滤波器稀疏性或非相干性的前提下，能否通过凸规划求解盲反卷积？
• RQ2 在盲反卷积设置中，恢复信号与滤波器所需的最少输入信号数量是多少？
• RQ3 在滤波器的一般条件下，核范数最小化能否扩展至盲超分辨率？
• RQ4 在缺乏对滤波器或输入信号先验知识的情况下，如何构建并验证最优性证书？
• RQ5 相干性参数 $ \mu_m $ 与 $ \mu_h $ 在决定恢复的样本复杂度方面起什么作用？

主要发现

• 当 $ L \gtrsim K^{3/2} \mu_m^2 $ 且 $ N \gtrsim K^{1/2} \mu_h^2 $ 时（忽略对数因子），以高概率可实现对未知信号与滤波器的精确恢复。
• 该方法仅需少量位于已知 $ K $-维子空间中的输入信号，显著减少了所需测量数。
• 证明通过 Neumann 级数构造证书，前两项利用次指数与 Rosenthal 型不等式进行有界控制。
• 二阶项被分解为单变量项与交叉项，分别通过解耦与 U-统计量技术进行有界控制。
• 该方法无需依赖稀疏性或非相干性假设，与先前工作不同，且在滤波器傅里叶支撑足够大时可适用于盲超分辨率。
• 数值实验验证了理论预测，并展示了该方法在盲超分辨率应用中的有效性。