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Guaranteed Multidimensional Time Series Prediction via Deterministic Tensor Completion Theory

Hao Shu, Jicheng Li, Yu Jin, Hailin Wang

TL;DR

This work tackles the problem of predicting multidimensional time series under deterministic missing patterns, where traditional tensor methods lack guarantees on exact recoverability. It introduces a deterministic tensor completion theory within the T-SVD framework and the Temporal Convolution Tensor Nuclear Norm (TCTNN) model, which applies temporal convolution before tensor-nuclear-norm based completion to guarantee exact predictions under suitable incoherence and sampling conditions. The method leverages the low-rankness induced by temporal smoothness and periodicity, and provides an exact-recovery bound linking horizon, kernel size, tubal rank, and incoherence. Empirically, TCTNN delivers superior prediction accuracy and computational efficiency across climate, network, and transport datasets, outperforming several baselines and demonstrating practical impact for real-time, few-shot multidimensional time-series forecasting.

Abstract

In recent years, the prediction of multidimensional time series data has become increasingly important due to its wide-ranging applications. Tensor-based prediction methods have gained attention for their ability to preserve the inherent structure of such data. However, existing approaches, such as tensor autoregression and tensor decomposition, often have consistently failed to provide clear assertions regarding the number of samples that can be exactly predicted. While matrix-based methods using nuclear norms address this limitation, their reliance on matrices limits accuracy and increases computational costs when handling multidimensional data. To overcome these challenges, we reformulate multidimensional time series prediction as a deterministic tensor completion problem and propose a novel theoretical framework. Specifically, we develop a deterministic tensor completion theory and introduce the Temporal Convolutional Tensor Nuclear Norm (TCTNN) model. By convolving the multidimensional time series along the temporal dimension and applying the tensor nuclear norm, our approach identifies the maximum forecast horizon for exact predictions. Additionally, TCTNN achieves superior performance in prediction accuracy and computational efficiency compared to existing methods across diverse real-world datasets, including climate temperature, network flow, and traffic ride data. Our implementation is publicly available at https://github.com/HaoShu2000/TCTNN.

Guaranteed Multidimensional Time Series Prediction via Deterministic Tensor Completion Theory

TL;DR

This work tackles the problem of predicting multidimensional time series under deterministic missing patterns, where traditional tensor methods lack guarantees on exact recoverability. It introduces a deterministic tensor completion theory within the T-SVD framework and the Temporal Convolution Tensor Nuclear Norm (TCTNN) model, which applies temporal convolution before tensor-nuclear-norm based completion to guarantee exact predictions under suitable incoherence and sampling conditions. The method leverages the low-rankness induced by temporal smoothness and periodicity, and provides an exact-recovery bound linking horizon, kernel size, tubal rank, and incoherence. Empirically, TCTNN delivers superior prediction accuracy and computational efficiency across climate, network, and transport datasets, outperforming several baselines and demonstrating practical impact for real-time, few-shot multidimensional time-series forecasting.

Abstract

In recent years, the prediction of multidimensional time series data has become increasingly important due to its wide-ranging applications. Tensor-based prediction methods have gained attention for their ability to preserve the inherent structure of such data. However, existing approaches, such as tensor autoregression and tensor decomposition, often have consistently failed to provide clear assertions regarding the number of samples that can be exactly predicted. While matrix-based methods using nuclear norms address this limitation, their reliance on matrices limits accuracy and increases computational costs when handling multidimensional data. To overcome these challenges, we reformulate multidimensional time series prediction as a deterministic tensor completion problem and propose a novel theoretical framework. Specifically, we develop a deterministic tensor completion theory and introduce the Temporal Convolutional Tensor Nuclear Norm (TCTNN) model. By convolving the multidimensional time series along the temporal dimension and applying the tensor nuclear norm, our approach identifies the maximum forecast horizon for exact predictions. Additionally, TCTNN achieves superior performance in prediction accuracy and computational efficiency compared to existing methods across diverse real-world datasets, including climate temperature, network flow, and traffic ride data. Our implementation is publicly available at https://github.com/HaoShu2000/TCTNN.
Paper Structure (29 sections, 8 theorems, 47 equations, 10 figures, 4 tables, 1 algorithm)

This paper contains 29 sections, 8 theorems, 47 equations, 10 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Deterministic completion theory
  4. Time series prediction via matrix/tensor completion
  5. Deterministic Tensor Completion
  6. T-SVD Framework
  7. Deterministic Tensor Completion
  8. Exact Recovery Guarantee
  9. Multidimensional Time Series Prediction
  10. Time series prediction via TNN
  11. Temporal Convolution Tensor
  12. Temporal Convolution Tensor Low-rankness from Smoothness and Periodicity
  13. Temporal Convolution Tensor Nulclear Norm
  14. Exact Prediction Guarantee
  15. Comparison with the CNNM model
  16. ...and 14 more sections

Key Result

Lemma 3.1

Let tensor $\mathcal{X}\in\mathbb{R}^{m_1\times m_2\times\cdots\times m_d}$, then it can be factorized as where $\mathcal{U}\in\mathbb{R}^{m_1\times m_1\times\cdots\times m_d}$, $\mathcal{V}\in\mathbb{R}^{m_2\times m_2\times\cdots\times m_d}$ are orthogonal tensors, and $\mathcal{S}\in\mathbb{R}^{m_1\times m_2\times\cdots\times m_d}$ is a f-diagonal tensor.

Figures (10)

  • Figure 1: Illustration of tensor completion for multidimensional time series prediction, where black cubes represent unsampled entries, and other cubes represent sampled entries.
  • Figure 2: Illustrations of horizontal/lateral mask sub-tensor sampling number in the three-dimensional case. (a): Arrangement of the mask tensor $\bar{\Omega}$, where the white cubes represent the value 1 (sampled entries) and the black cubes represent the value 0 (unsampled entries); (b):$\left|\bar{\Omega}_{3}\right|=10$ indicates that there are 10 sampled entries in the third horizontal mask sub-tensor; (c): $\left|\bar{\Omega}^{2}\right|=9$ indicates that there are 9 sampled entries in the second lateral mask sub-tensor.
  • Figure 3: Illustrations of temporal convolution tensor nulclear norm for multidimensional time series prediction. (a): Incomplete tensor, where black squares represent 0 values (not observed) and other color blocks represent non-zero values (observed); (b): incomplete temporal convolution tensor; (c): complete temporal convolution tensor; (d): complete tensor.
  • Figure 4: Illustrations of temporal convolution low-rankness from smoothness and periodicity using simulated data.
  • Figure 5: The predicted results of the TCTNN model and the corresponding true values under forecast horizon 4 on the NYC taxi dataset. Ground truth 1 and Ground truth 2 represent the true values at two distinct time points, while TCTNN 1 and TCTNN 2 denote the predicted values at the corresponding time points.
  • ...and 5 more figures

Theorems & Definitions (25)

  • Definition 3.1: T-product martin2013order
  • Definition 3.2: Circular convolution fahmy2012new
  • Definition 3.3: Identity tensor martin2013order
  • Definition 3.4: Transpose martin2013order
  • Definition 3.5: Orthogonal tensor martin2013order
  • Definition 3.6: F-diagonal tensor martin2013order
  • Lemma 3.1: T-SVD martin2013order
  • Definition 3.7: Tensor tubal rank) lu2019tensor qin2022low
  • Definition 3.8: Tensor multi-rank lu2019tensor qin2022low
  • Definition 3.9: Tensor spectral norm lu2019tensor qin2022low
  • ...and 15 more