Total Least Squares Phase Retrieval

Abstract

We address the phase retrieval problem with errors in the sensing vectors. A number of recent methods for phase retrieval are based on least squares (LS) formulations which assume errors in the quadratic measurements. We extend this approach to handle errors in the sensing vectors by adopting the total least squares (TLS) framework that is used in linear inverse problems with operator errors. We show how gradient descent and the specific geometry of the phase retrieval problem can be used to obtain a simple and efficient TLS solution. Additionally, we derive the gradients of the TLS and LS solutions with respect to the sensing vectors and measurements which enables us to calculate the solution errors. By analyzing these error expressions we determine conditions under which each method should outperform the other. We run simulations to demonstrate that our method can lead to more accurate solutions. We further demonstrate the effectiveness of our approach by performing phase retrieval experiments on real optical hardware which naturally contains both sensing vector and measurement errors.

References in corpus (12)

• Phase Retrieval via Wirtinger Flow: Theory and Algorithms
• An overview of low-rank matrix recovery from incomplete observations
• Sparsity-Cognizant Total Least-Squares for Perturbed Compressive Sampling
• An algebraic characterization of injectivity in phase retrieval
• On Differentiating Parameterized Argmin and Argmax Problems with Application to Bi-level Optimization
• Random Projections through multiple optical scattering: Approximating kernels at the speed of light
• Sketchy Decisions: Convex Low-Rank Matrix Optimization with Optimal Storage
• Reconstructing Point Sets from Distance Distributions
• Accelerating ptychographic reconstructions using spectral initializations
• Robust phase retrieval with the swept approximate message passing (prSAMP) algorithm
• Solving Complex Quadratic Systems with Full-Rank Random Matrices
• Don't take it lightly: Phasing optical random projections with unknown operators