Framework for Identi ability Analysis in Bilinear Inverse Problems with Applications to Subspace and Sparsity Models ∗
Framework for Identi ability Analysis in Bilinear Inverse Problems with Applications to Subspace and Sparsity Models ∗
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双线性反问题中的辨识能力分析框架及其在子空间和稀疏模型中的应用*
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发表时间：
2015
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通讯作者：
Y. Bresler
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作者：
Yanjun Li;Kiryung Lee;Y. Bresler
双线性反问题（BIPs），即两个向量在双线性映射下的图像的解决，在许多应用中出现。如果没有进一步的限制，BIP通常是不适定的。在实践中，利用自然信号的性质来求解BIPs。例如，施加子空间约束或稀疏约束以减少搜索空间。这些办法在实践中取得了一些成功。然而，目前关于BIPs的唯一性问题的研究还很少。对于大多数BIP，在什么条件下问题允许唯一解的基本问题尚未得到回答。例如，盲增益和相位校准（BGPC）是结构化的双线性逆问题，其出现在许多应用中，包括计算重新照明中的逆渲染（具有未知照明的光估计）、传感器阵列处理中的盲相位和增益校准以及多通道盲解卷积（MBD）。研究这类问题的独特性是很有趣的。(BIP)nd（x，y），s.t. F（x，y）= z，x ∈ Ω X，y ∈ Ω Y.（BGPC）nd（λ，X），s.t. diag（λ）AX = Y，λ ∈ C，X ∈ Ω X.本文定义了一个BIP到一组变换的恒等性。我们推导出这种可识别性的充分必要条件，即，的条件下，解决方案可以唯一确定的变换群。将这些结果应用于BGPC，我们得到了几种情况下，包括子空间，联合稀疏，稀疏模型的唯一恢复的充分条件。对于联合稀疏或稀疏约束的BGPC，我们开发了一个程序来计算相关的变换群。我们还给出了样本复杂度的紧下界形式的必要条件，并通过数值实验证明了这些界的紧性。BGPC的结果不仅证明了所提出的识别性分析的一般框架的应用，但在自己的权利也感兴趣。索引项唯一性、变换群、等价类、模糊度、盲增益和相位校准、传感器阵列处理、逆绘制、SAR自聚焦、多通道盲反卷积参考文献[1] Y. Li，K. Lee和Y. Bresler，一个统一的艾德框架，用于双线性逆问题的可识别性分析，并应用于子空间和稀疏模型，arXiv预印本arXiv：1501.06120，2015年。这项工作得到了国家科学基金会（NSF）的部分支持，赠款CCF 10 - 18789和IIS 14 - 47879。†电气和计算机工程系和协调科学实验室，伊利诺伊大学，厄巴纳香槟，IL 61801，美国。伊利诺伊大学统计学系和协调科学实验室，厄巴纳-香槟，IL 61801，美国。
Bilinear inverse problems (BIPs), the resolution of two vectors given their image under a bilinear mapping, arise in many applications. Without further constraints, BIPs are usually ill-posed. In practice, properties of natural signals are exploited to solve BIPs. For example, subspace constraints or sparsity constraints are imposed to reduce the search space. These approaches have shown some success in practice. However, there are few results on uniqueness in BIPs. For most BIPs, the fundamental question of under what condition the problem admits a unique solution, is yet to be answered. For example, blind gain and phase calibration (BGPC) is a structured bilinear inverse problem, which arises in many applications, including inverse rendering in computational relighting (albedo estimation with unknown lighting), blind phase and gain calibration in sensor array processing, and multichannel blind deconvolution (MBD). It is interesting to study the uniqueness of such problems. (BIP) nd (x, y), s.t. F(x, y) = z, x ∈ ΩX , y ∈ ΩY . (BGPC) nd (λ,X), s.t. diag(λ)AX = Y, λ ∈ C, X ∈ ΩX . In this paper, we de ne identi ability of a BIP up to a group of transformations. We derive necessary and su cient conditions for such identi ability, i.e., the conditions under which the solutions can be uniquely determined up to the transformation group. Applying these results to BGPC, we derive su cient conditions for unique recovery under several scenarios, including subspace, joint sparsity, and sparsity models. For BGPC with joint sparsity or sparsity constraints, we develop a procedure to compute the relevant transformation groups. We also give necessary conditions in the form of tight lower bounds on sample complexities, and demonstrate the tightness of these bounds by numerical experiments. The results for BGPC not only demonstrate the application of the proposed general framework for identi ability analysis, but are also of interest in their own right. Index terms uniqueness, transformation group, equivalence class, ambiguity, blind gain and phase calibration, sensor array processing, inverse rendering, SAR autofocus, multichannel blind deconvolution References [1] Y. Li, K. Lee, and Y. Bresler, A uni ed framework for identi ability analysis in bilinear inverse problems with applications to subspace and sparsity models, arXiv preprint arXiv:1501.06120, 2015. ∗This work was supported in part by the National Science Foundation (NSF) under Grants CCF 10-18789 and IIS 14-47879. †Department of Electrical and Computer Engineering and Coordinated Science Laboratory, University of Illinois, UrbanaChampaign, IL 61801, USA. ‡Department of Statistics and Coordinated Science Laboratory, University of Illinois, Urbana-Champaign, IL 61801, USA.