Identifiability in Bilinear Inverse Problems With Applications to Subspace or Sparsity-Constrained Blind Gain and Phase Calibration

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, the properties of natural signals are exploited to solve BIPs. For example, subspace constrain...

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Vydáno v:	IEEE transactions on information theory Ročník 63; číslo 2; s. 822 - 842
Hlavní autoři:	Yanjun Li, Kiryung Lee, Bresler, Yoram
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York IEEE 01.02.2017 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	Albedo Ambiguity blind gain and phase calibration Calibration Deconvolution Discrete Fourier transforms equivalence class Ill posed problems Information theory Inverse problems inverse rendering Lower bounds Mapping Mathematical models multichannel blind deconvolution Rendering (computer graphics) SAR autofocus sensor array processing Sensor arrays Sparse matrices Sparsity Subspace constraints Subspaces Tightness transformation group Transformations Uniqueness Vector space
ISSN:	0018-9448, 1557-9654
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Abstract	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, the 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 BIP, 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. In this paper, we define identifiability of a BIP up to a group of transformations. We derive necessary and sufficient conditions for such identifiability, i.e., the conditions under which the solutions can be uniquely determined up to the transformation group. These conditions take the form of dividing the identifiability of the pair of unknown variables into the individual identifiability of each variable. Although verifying these individual conditions requires problem-specific procedures, this framework is universally applicable to all BIPs. Applying these results to BGPC, we derive sufficient 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 transformation groups corresponding to inherent ambiguities. 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 identifiability analysis, but are also of interest in their own right.
AbstractList	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, the 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 BIP, 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. In this paper, we define identifiability of a BIP up to a group of transformations. We derive necessary and sufficient conditions for such identifiability, i.e., the conditions under which the solutions can be uniquely determined up to the transformation group. These conditions take the form of dividing the identifiability of the pair of unknown variables into the individual identifiability of each variable. Although verifying these individual conditions requires problem-specific procedures, this framework is universally applicable to all BIPs. Applying these results to BGPC, we derive sufficient 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 transformation groups corresponding to inherent ambiguities. 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 identifiability analysis, but are also of interest in their own right.
Author	Kiryung Lee Bresler, Yoram Yanjun Li
Author_xml	– sequence: 1 surname: Yanjun Li fullname: Yanjun Li email: yli145@illinois.edu organization: Dept. of Electr. & Comput. Eng., Univ. of Illinois at Urbana-Champaign, Champaign, IL, USA – sequence: 2 surname: Kiryung Lee fullname: Kiryung Lee email: kiryung@ece.gatech.edu organization: Sch. of Electr. & Comput. Eng., Georgia Inst. of Technol., Atlanta, GA, USA – sequence: 3 givenname: Yoram surname: Bresler fullname: Bresler, Yoram email: ybresler@illinois.edu organization: Dept. of Electr. & Comput. Eng., Univ. of Illinois at Urbana-Champaign, Champaign, IL, USA
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SubjectTerms	Albedo Ambiguity blind gain and phase calibration Calibration Deconvolution Discrete Fourier transforms equivalence class Ill posed problems Information theory Inverse problems inverse rendering Lower bounds Mapping Mathematical models multichannel blind deconvolution Rendering (computer graphics) SAR autofocus sensor array processing Sensor arrays Sparse matrices Sparsity Subspace constraints Subspaces Tightness transformation group Transformations Uniqueness Vector space
Title	Identifiability in Bilinear Inverse Problems With Applications to Subspace or Sparsity-Constrained Blind Gain and Phase Calibration
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