Blind gain and phase calibration (BGPC) is a bilinear inverse problem
involving the determination of unknown gains and phases of the sensing system,
and the unknown signal, jointly. BGPC arises in numerous applications, e.g.,
blind albedo estimation in inverse rendering, synthetic aperture radar
autofocus, and sensor array auto-calibration. In some cases, sparse structure
in the unknown signal alleviates the ill-posedness of BGPC. Recently there has
been renewed interest in solutions to BGPC with careful analysis of error
bounds. In this paper, we formulate BGPC as an eigenvalue/eigenvector problem,
and propose to solve it via power iteration, or in the sparsity or joint
sparsity case, via truncated power iteration. Under certain assumptions, the
unknown gains, phases, and the unknown signal can be recovered simultaneously.
Numerical experiments show that power iteration algorithms work not only in the
regime predicted by our main results, but also in regimes where theoretical
analysis is limited. We also show that our power iteration algorithms for BGPC
compare favorably with competing algorithms in adversarial conditions, e.g.,
with noisy measurement or with a bad initial estimate.