Minimizing Quotient Regularization Model

TL;DR

This paper introduces a gradient flow-based iterative algorithm for minimizing quotient regularization models with absolutely one-homogeneous functions, effectively solving a nonconvex problem with proven convergence and superior accuracy.

Contribution

It proposes a novel gradient flow method for QRM with one-homogeneous functions, including convergence analysis and improved numerical performance over existing solvers.

Findings

Converges to a stationary point in nonconvex QRM.

Outperforms state-of-the-art QRM solvers in accuracy.

Applicable to signal and image processing tasks.

Abstract

Quotient regularization models (QRMs) are a class of powerful regularization techniques that have gained considerable attention in recent years, due to their ability to handle complex and highly nonlinear data sets. However, the nonconvex nature of QRM poses a significant challenge in finding its optimal solution. We are interested in scenarios where both the numerator and the denominator of QRM are absolutely one-homogeneous functions, which is widely applicable in the fields of signal processing and image processing. In this paper, we utilize a gradient flow to minimize such QRM in combination with a quadratic data fidelity term. Our scheme involves solving a convex problem iteratively.The convergence analysis is conducted on a modified scheme in a continuous formulation, showing the convergence to a stationary point. Numerical experiments demonstrate the effectiveness of the proposed algorithm in terms of accuracy, outperforming the state-of-the-art QRM solvers.