Smooth optimization using global and local low-rank regularizers

Area of Science:

• Applied Mathematics
• Signal Processing
• Medical Imaging

Background:

• Low-rank regularizers using nuclear norm are common in inverse problems.
• Proximal gradient methods (PGM) are standard but fail with local low-rank models on overlapping patches.
• Existing heuristic methods for local low-rank models lack convergence guarantees.

Purpose of the Study:

• To develop a smooth, convex, and differentiable regularizer as an alternative to the nuclear norm.
• To enable the use of gradient-based optimization algorithms for local low-rank models.
• To improve dynamic magnetic resonance imaging (MRI) reconstruction using local low-rank models.

Main Methods:

• Replaced nuclear norm with a smooth approximation using a Huber-type function on singular values.
• Developed a theoretical framework based on singular value function theory.
• Derived a closed-form expression for the regularizer gradient.
• Introduced a novel step-size selection strategy using a quadratic majorizer.

Main Results:

• The proposed regularizer is convex, differentiable, and has a Lipschitz continuous gradient.
• Standard gradient-based algorithms, like nonlinear conjugate gradient, can be used.
• The framework effectively handles local low-rank models with overlapping patches.
• Empirical results demonstrate successful dynamic MRI reconstruction.

Conclusions:

• The proposed smooth regularizer overcomes limitations of the nuclear norm in local low-rank settings.
• The method provides theoretical convergence guarantees for gradient-based optimization.
• The framework offers a robust and efficient approach for dynamic MRI reconstruction.