Fast algorithms for non-convex tensor completion problems

Abstract

Multi-dimensional image processing plays a pivotal role in diverse fields such as medicine, research, graphics, industry, remote sensing, virtual reality, and geospatial mapping, enabling advanced visualization and analysis. Traditional methods for processing multi-dimensional data, such as matrix-based singular value decomposition (SVD), often fail to capture the inherent multi-dimensional structure, leading to suboptimal performance in tasks like data recovery and feature extraction. To address these limitations, tensor singular value decomposition (t-SVD) has emerged as a powerful tool, specifically designed to handle the complex, multi-linear structures inherent in multi-dimensional data. Unlike matrix-based approaches, t-SVD operates directly on tensors, preserving the intrinsic relationships between dimensions and enabling more accurate representation and recovery of multi-dimensional data. Derived from t-SVD, surrogate functions such as the logDet function and Laplace function have been proposed to approximate the multi-rank of tensors, facilitating the recovery of the underlying structure of multi-dimensional images. However, real-world applications involving large-scale datasets (e.g., hyperspectral images, multispectral images, and grayscale videos) present significant computational challenges. Standard optimization algorithms, such as the alternating direction method of multipliers (ADMM), are often inefficient for solving the resulting large-scale non-convex optimization problems. To overcome these challenges, we propose efficient ADMM algorithms based on randomized singular value decomposition (r-SVD). These algorithms are specifically designed to handle the non-convexity and scalability issues associated with SVD-based optimization. We provide a detailed analysis of the computational complexity of the proposed algorithms, demonstrating their superiority over traditional methods in terms of efficiency and scalability. Extensive experiments on real-world datasets, including hyperspectral images, multispectral images, and grayscale videos, validate that our proposed algorithms achieve significant reductions in CPU time without compromising the quality of data recovery. By leveraging the strengths of r-SVD, our approach offers a robust and efficient solution for multi-dimensional image processing, addressing both theoretical and practical challenges in the field.