Guaranteed Functional Tensor Singular Value Decomposition.

Area of Science:

• Multivariate Statistics
• Functional Data Analysis
• Tensor Decomposition

Background:

• High-order longitudinal data presents unique analytical challenges.
• Existing dimension reduction techniques may not adequately capture functional dependencies in such data.
• Analysis of longitudinal data requires methods that can handle complex, multi-modal structures.

Purpose of the Study:

• Introduce a novel dimension reduction framework for tensors with mixed functional and tabular modes.
• Develop a method suitable for analyzing high-order longitudinal data.
• Provide a robust estimation technique for low-rank functional tensor structures.

Main Methods:

• Functional Tensor Singular Value Decomposition (FTSVD) framework.
• Reproducing Kernel Hilbert Space (RKHS) theory.
• RKHS-based constrained power iteration with spectral initialization.

Main Results:

• Successful estimation of singular vectors and functions within low-rank functional tensors.
• Establishment of non-asymptotic contractive error bounds for the algorithm.
• Demonstrated superiority through extensive simulations and real-world data experiments.

Conclusions:

• FTSVD offers a powerful new approach for dimension reduction in high-order longitudinal data.
• The proposed RKHS-based method provides accurate estimation with theoretical guarantees.
• The framework shows significant potential for advancing the analysis of complex functional tensor data.