Guaranteed Functional Tensor Singular Value Decomposition

Abstract

This article introduces the functional tensor singular value decomposition (FTSVD), a novel dimension reduction framework for tensors with one functional mode and several tabular modes. The problem is motivated by high-order longitudinal data analysis. Our model assumes the observed data to be a random realization of an approximate CP low-rank functional tensor measured on a discrete time grid. Incorporating tensor algebra and the theory of reproducing kernel Hilbert space (RKHS), we propose a novel RKHS-based constrained power iteration with spectral initialization. Our method can successfully estimate both singular vectors and functions of the low-rank structure in the observed data. With mild assumptions, we establish the non-asymptotic contractive error bounds for the proposed algorithm. The superiority of the proposed framework is demonstrated via extensive experiments on both simulated and real data. Supplementary materials for this article are available online.

Keywords:

• Functional data analysis
• Low-rank tensor decomposition
• Reproducing Kernel Hilbert space
• Singular value decomposition

Supplementary Materials

The supplementary materials include additional simulation results, real data analysis on worldwide crop production, and proofs of technical results of this paper. The following references are used in the supplementary materials: Adamczak (2008); Adler and Taylor (2009); Bartlett, Bousquet, and Mendelson (2005); Bartlett and Mendelson (2002); Cai, Han, and Zhang (2020); Happ-Kurz (2020); Mendelson (2002); Pisier (1983); Rudelson and Vershynin (2013); Vershynin (2018); Wedin (1972); and Zhang and Zhou (2020).

Additional information

Funding

The research of R. Han and A. R. Zhang was supported in part by NSF grant CAREER-1944904 and NIH grant R01 GM131399.