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

References in corpus (5)

• Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
• Templates for Convex Cone Problems with Applications to Sparse Signal Recovery
• Improved Recovery Guarantees for Phase Retrieval from Coded Diffraction Patterns
• Variational regularization of complex deautoconvolution and phase retrieval in ultrashort laser pulse characterization
• Non-convex regularization of bilinear and quadratic inverse problems by tensorial lifting

Cited by in corpus (3)

• Asymptotic linear convergence of fully-corrective generalized conditional gradient methods
• Total Variation-Based Reconstruction and Phase Retrieval for Diffraction Tomography with an Arbitrarily Moving Object
• Total variation-based reconstruction and phase retrieval for diffraction tomography