  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.
  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. 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
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
identifiability analysis, but are also of interest in their own right.
