# Guaranteed Multidimensional Time Series Prediction via Deterministic Tensor Completion Theory

arXiv: https://arxiv.org/abs/2501.15388 · HTML: https://mlsanity.com/paper/2501.15388/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 67 references that are in the map.
- First posted on arXiv 26 Jan 2025.

## Builds on (most cited first, 30 of 30)

- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [Bayesian Temporal Factorization for Multidimensional Time Series Prediction](https://mlsanity.com/paper/1910.06366/) (2019)
- [Regularized estimation in sparse high-dimensional time series models](https://mlsanity.com/paper/1311.4175/) (2013)
- [Guaranteed Tensor Recovery Fused Low-rankness and Smoothness](https://mlsanity.com/paper/2302.02155/) (2023)
- [Forecasting Time Series with VARMA Recursions on Graphs](https://mlsanity.com/paper/1810.08581/) (2018)
- [Bayesian Poisson Tucker Decomposition for Learning the Structure of International Relations](https://mlsanity.com/paper/1606.01855/) (2016)
- [Low-Rank Hankel Tensor Completion for Traffic Speed Estimation](https://mlsanity.com/paper/2105.11335/) (2021)
- [Factor Models for High-Dimensional Tensor Time Series](https://mlsanity.com/paper/1905.07530/) (2019)
- [Dictionary Learning with Low-rank Coding Coefficients for Tensor Completion](https://mlsanity.com/paper/2009.12507/) (2020)
- [Exact Decomposition of Joint Low Rankness and Local Smoothness Plus Sparse Matrices](https://mlsanity.com/paper/2201.12592/) (2022)
- [Autoregressive Models for Matrix-Valued Time Series](https://mlsanity.com/paper/1812.08916/) (2018)
- [Laplacian Convolutional Representation for Traffic Time Series Imputation](https://mlsanity.com/paper/2212.01529/) (2022)
- [Missing Slice Recovery for Tensors Using a Low-rank Model in Embedded Space](https://mlsanity.com/paper/1804.01736/) (2018)
- [High-dimensional vector autoregressive time series modeling via tensor decomposition](https://mlsanity.com/paper/1909.06624/) (2019)
- [Recovery of Future Data via Convolution Nuclear Norm Minimization](https://mlsanity.com/paper/1909.03889/) (2019)
- [Block Hankel Tensor ARIMA for Multiple Short Time Series Forecasting](https://mlsanity.com/paper/2002.12135/) (2020)
- [Time Series Forecasting via Learning Convolutionally Low-Rank Models](https://mlsanity.com/paper/2104.11510/) (2021)
- [Matrix completion with deterministic pattern - a geometric perspective](https://mlsanity.com/paper/1802.00047/) (2018)
- [Tensor Completion from Regular Sub-Nyquist Samples](https://mlsanity.com/paper/1903.00435/) (2019)
- [Structured low-rank matrix completion for forecasting in time series analysis](https://mlsanity.com/paper/1802.08242/) (2018)
- [Deterministic tensor completion with hypergraph expanders](https://mlsanity.com/paper/1910.10692/) (2019)
- [Discovering Dynamic Patterns from Spatiotemporal Data with Time-Varying Low-Rank Autoregression](https://mlsanity.com/paper/2211.15482/) (2022)
- [Fundamental Conditions for Low-CP-Rank Tensor Completion](https://mlsanity.com/paper/1703.10740/) (2017)
- [An Inexact Accelerated Stochastic ADMM for Separable Convex Optimization](https://mlsanity.com/paper/2010.12765/) (2020)
- [High-Dimensional Low-Rank Tensor Autoregressive Time Series Modeling](https://mlsanity.com/paper/2101.04276/) (2021)
- [Low-rank matrix completion theory via Plucker coordinates](https://mlsanity.com/paper/2004.12430/) (2020)
- [Generalized Symmetric ADMM for Separable Convex Optimization](https://mlsanity.com/paper/1812.03769/) (2018)
- [Deterministic and Probabilistic Conditions for Finite Completability of Low-Tucker-Rank Tensor](https://mlsanity.com/paper/1612.01597/) (2016)
- [An Efficient Forecasting Approach to Reduce Boundary Effects in Real-Time Time-Frequency Analysis](https://mlsanity.com/paper/2102.07226/) (2021)

## Built on by (most cited first, 2 of 2)

- [Exact Recovery of Non-Random Missing Multidimensional Time Series via Temporal Isometric Delay-Embedding Transform](https://mlsanity.com/paper/2512.10191/) (2025)
- [Comprehensive Robust Dynamic Mode Decomposition from Mode Extraction to Dimensional Reduction](https://mlsanity.com/paper/2601.11116/) (2026)

## Abstract (arXiv)

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.