Guaranteed Functional Tensor Singular Value Decomposition

TL;DR

This paper presents a new tensor decomposition method for high-dimensional longitudinal data with one functional and multiple tabular modes, combining tensor algebra and RKHS theory for improved dimension reduction.

Contribution

It introduces the functional tensor SVD framework with an RKHS-based constrained power iteration algorithm and provides theoretical error bounds for its performance.

Findings

The method accurately estimates singular vectors and functions in functional tensor data.

Theoretical error bounds demonstrate the algorithm's effectiveness under mild assumptions.

Experiments show superior performance on simulated and real datasets.

Abstract

This paper 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.