A Uni ed Framework for Identi ability Analysis in Bilinear Inverse Problems with Applications to Subspace and Sparsity Models∗

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 cali-bration, 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 problemswith 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, Urbana-Champaign, IL 61801, USA.‡Department of Statistics and Coordinated Science Laboratory, University of Illinois, Urbana-Champaign, IL 61801, USA.