Optimal Sparse Singular Value Decomposition for High-Dimensional High-Order Data.

Area of Science:

• Statistics
• Data Science
• Machine Learning

Background:

• High-dimensional, high-order data present challenges for traditional dimension reduction techniques.
• Sparsity is a common characteristic in many real-world datasets, requiring specialized methods.
• Existing tensor singular value decomposition (SVD) models may lack robustness under weaker assumptions.

Purpose of the Study:

• To propose a novel method for sparse tensor singular value decomposition (STAT-SVD) for effective dimension reduction.
• To develop a robust estimation procedure for high-dimensional data with sparsity.
• To provide theoretical guarantees and empirical validation for the proposed method.

Main Methods:

• A novel double projection and thresholding scheme is introduced for iterative refinement.
• The sparse tensor alternating thresholding for singular value decomposition (STAT-SVD) algorithm is detailed.
• Theoretical analysis includes deriving upper and lower bounds for estimation accuracy.

Main Results:

• STAT-SVD demonstrates robust estimation capabilities under weaker assumptions compared to standard tensor SVD.
• The method achieves minimax rate-optimality in a general class of situations.
• Simulation studies confirm the strong performance of STAT-SVD across various configurations.

Conclusions:

• STAT-SVD is an effective and robust method for dimension reduction in sparse, high-dimensional, high-order data.
• The proposed method offers theoretical guarantees and practical advantages over existing techniques.
• The utility of STAT-SVD is demonstrated on a real-world longitudinal mortality rates dataset.