Low-rank Tensor Autoregressive Predictor for Third-Order Time-Series Forecasting
Haoning Wang, Liping Zhang
TL;DR
The paper tackles forecasting of third-order tensor time series by tackling dimensionality reduction with a novel LOTAP framework that couples autoregression with truncated t-SVD. By transforming to the Fourier domain, LOTAP decomposes the problem into four subproblems with closed-form solutions within an alternating minimization scheme, achieving substantial speedups over Tucker-decomposition-based methods while preserving forecasting accuracy. It demonstrates complexity reduction to $O(T n_1 n_2 n_3 + (n_1+n_2) n_3 r^2)$ per iteration, validates low-rank structure across real and synthetic data, and shows robust performance even with limited training data. The work also discusses extensions to time-series imputation and multi-directional t-SVD, underscoring LOTAP’s broad applicability to high-dimensional tensor forecasting tasks.
Abstract
Recently, tensor time-series forecasting has gained increasing attention, whose core requirement is how to perform dimensionality reduction. In this paper, we establish a least square optimization model by combining tensor singular value decomposition (t-SVD) with autoregression (AR) to forecast third-order tensor time-series, which has great benefit in computational complexity and dimensionality reduction. We divide such an optimization problem using fast Fourier transformation and t-SVD into four decoupled subproblems, whose variables include regressive coefficient, f-diagonal tensor, left and right orthogonal tensors, and propose an efficient forecasting algorithm via alternating minimization strategy, called Low-rank Tensor Autoregressive Predictor (LOTAP), in which each subproblem has a closed-form solution. Numerical experiments indicate that, compared to Tucker-decomposition-based algorithms, LOTAP achieves a speed improvement ranging from $2$ to $6$ times while maintaining accurate forecasting performance in all four baseline tasks. In addition, this algorithm is applicable to a wider range of tensor forecasting tasks because of its more effective dimensionality reduction ability.