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

## TL;DR

This paper introduces tensor-free proximal algorithms for solving bilinear and quadratic inverse problems, notably phase retrieval, by leveraging low-rank representations and an augmented Lanczos process to improve efficiency and convergence.

## Contribution

It develops tensor-free versions of singular value thresholding methods for inverse problems, enabling practical computation without high-dimensional tensor operations.

## Key findings

- Efficient phase retrieval via tensor-free algorithms
- Enhanced convergence with reweighting technique
- Incorporation of smoothness constraints improves recovery

## 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.