Tensor-Free Proximal Methods for Lifted Bilinear/Quadratic Inverse Problems with Applications to Phase Retrieval

Abstract We propose and study a class of novel algorithms that aim at solving bilinear quadratic inverse problems. Using convex relaxation based on tensorial lifting, applying first-order proximal algorithms, these problems could be solved numerically by singular value thresholding methods. However, direct realization for, e.g., image recovery is often impracticable, since computations have to performed the tensor-product space, whose dimension usually tremendous. To overcome this limitation, we derive tensor-free versions common methods exploiting low-rank representations incorporating an augmented Lanczos process. reweighting technique, further improve convergence behavior rank evolution iterative algorithms. Applying method two-dimensional masked Fourier phase retrieval problem, obtain efficient method. Moreover, are flexible enough incorporate priori smoothness constraints greatly results.