Multi-Matrix Completion: A Novel Framework for Structurally Missing Elements

Area of Science:

• Data Science
• Machine Learning
• Computer Vision

Background:

• Traditional matrix completion (MC) and tensor completion (TC) assume random missing data, failing with real-world structural missingness where data is concentrated.
• Structural missingness, common in real-world datasets, significantly degrades the performance of existing MC and TC techniques.

Purpose of the Study:

• To introduce a novel framework, multi-matrix completion (MMC), designed to effectively handle structural missingness in data reconstruction.
• To develop a method that simultaneously reconstructs multiple matrices, capturing inter-matrix correlations for improved completion accuracy.

Main Methods:

• The proposed multi-matrix completion (MMC) framework utilizes tri-factorization across matrices to leverage correlations between them.
• Tikhonov regularization is applied to individual matrices within the MMC framework to exploit their inherent correlations.
• The optimization problem is solved using the alternating direction method of multipliers, with theoretical guarantees for global convergence.

Main Results:

• Multi-matrix completion (MMC) demonstrates superior performance in restoring data with structural missingness compared to traditional MC and TC methods.
• Experiments in image and video restoration validate the versatility and effectiveness of the MMC framework.
• MMC is robust to data smoothness variations, offering broader applicability than Fourier transform-based tensor completion methods.

Conclusions:

• The novel multi-matrix completion (MMC) framework effectively addresses the challenge of structural missingness in data reconstruction.
• MMC offers a versatile and high-performing solution for various data restoration tasks, including image and video processing.
• The method's ability to handle both random and structural missingness, along with its theoretical convergence guarantees, marks a significant advancement in completion techniques.