Robust Rank-One Matrix Completion via Explicit Regularizer.

Area of Science:

• Robust Matrix Completion
• Robust Statistics
• Signal Processing

Background:

• The Welsch function, or maximum correntropy criterion, is common in robust matrix completion but down-weights normal data.
• Existing methods struggle with accurately handling both normal data and outliers simultaneously.

Purpose of the Study:

• To develop a novel robust function (t-Welsch) that assigns unity weight to normal data and improves outlier robustness.
• To apply the t-Welsch function to rank-one matching pursuit for accurate and robust low-rank matrix recovery.
• To analyze the convergence and computational complexity of the proposed matrix completion algorithm.

Main Methods:

• Derivation of the explicit regularizer (ER) for the Welsch function using half-quadratic (HQ) minimization.
• Development of the t-Welsch function with ER.
• Application of t-Welsch to rank-one matching pursuit and implementation via block coordinate descent (BCD).

Main Results:

• The t-Welsch function demonstrates superior robustness against outliers compared to Huber's weight.
• The proposed t-Welsch-based matrix completion algorithm achieves accurate low-rank recovery without prior rank information or SVD.
• Experimental results show outperformance against state-of-the-art methods on synthetic data, noisy images, and MIMO radar signals.

Conclusions:

• The t-Welsch function offers a significant advancement in robust matrix completion by effectively handling normal data and outliers.
• The developed algorithm provides a robust and accurate solution for low-rank matrix recovery across diverse applications.
• The study presents a novel approach with demonstrable improvements in recovery accuracy in challenging data scenarios.