Proposing a Low-Rank Approximation Method with Mathematical Guarantees for High-Dimensional Tensor Data

The analysis of high-dimensional tensor data presents significant challenges for traditional methods such as CANDECOMP/PARAFAC (CP) decomposition and Higher-Order Singular Value Decomposition (HOSVD) due to the size, complexity, and noise inherent in the data. Recent studies have emphasized the critical need for noise-robust approaches capable of reducing dimensionality while preserving data integrity, particularly for matrix-form data, underscoring the necessity of theoretical advancements in noise reduction. In this study, we propose a novel method, Noise-Reduced Higher-Order Singular Value Decomposition (NR-HOSVD), which integrates noise reduction techniques into the HOSVD framework to enhance dimensionality reduction for matrix data. In comparative simulations with CP decomposition and HOSVD, NR-HOSVD consistently demonstrated lower relative errors as the rank increased, particularly for high-dimensional tensors, surpassing the performance of conventional methods beyond a certain rank threshold. Furthermore, NR-HOSVD exhibited superior stability and faster convergence on large-scale datasets compared to existing approaches. These findings suggest that NR-HOSVD provides a more accurate and efficient approach to high-dimensional tensor data analysis, making it particularly well-suited for applications in big data and machine learning. In the future, NR-HOSVD is expected to serve as a pivotal method for enhancing the accuracy and efficiency of high-dimensional tensor data analysis, with broad applicability anticipated in fields such as large-scale data processing and precision medicine.