Selecting Regularization Parameters for nuclear norm type minimization problems

TL;DR

This paper presents a method to automatically select regularization parameters for nuclear norm minimization problems, improving efficiency and accuracy in low-rank matrix reconstruction from noisy data.

Contribution

The authors derive a closed-form solution for the regularization parameter based on the discrepancy principle, enabling automatic and efficient parameter selection.

Findings

The proposed method achieves comparable or better results than SURE-based approaches.

It reduces computational cost by 11-18 times.

Validated on synthetic and real MRI data.

Abstract

The reconstruction of low-rank matrix from its noisy observation finds its usage in many applications. It can be reformulated into a constrained nuclear norm minimization problem, where the bound $\eta$ of the constraint is explicitly given or can be estimated by the probability distribution of the noise. When the Lagrangian method is applied to find the minimizer, the solution can be obtained by the singular value thresholding operator where the thresholding parameter $\lambda$ is related to the Lagrangian multiplier. In this paper, we first show that the Frobenius norm of the discrepancy between the minimizer and the observed matrix is a strictly increasing function of $\lambda$. From that we derive a closed-form solution for $\lambda$ in terms of $\eta$. The result can be used to solve the constrained nuclear-norm-type minimization problem when $\eta$ is given. For the unconstrained nuclear-norm-type regularized problems, our result allows us to automatically choose a suitable regularization parameter by using the discrepancy principle. The regularization parameters obtained are comparable to (and sometimes better than) those obtained by Stein's unbiased risk estimator (SURE) approach while the cost of solving the minimization problem can be reduced by 11--18 times. Numerical experiments with both synthetic data and real MRI data are performed to validate the proposed approach.