Exact Recovery of Non-Random Missing Multidimensional Time Series via Temporal Isometric Delay-Embedding Transform

Non-random missing data is a ubiquitous yet undertreated flaw in multidimensional time series, fundamentally threatening the reliability of data-driven analysis and decision-making. Pure low-rank tensor completion, as a classical data recovery method, falls short in handling non-random missingness, both methodologically and theoretically. Hankel-structured tensor completion models provide a feasible approach for recovering multidimensional time series with non-random missing patterns. However, most Hankel-based multidimensional data recovery methods both suffer from unclear sources of Hankel tensor low-rankness and lack an exact recovery theory for non-random missing data. To address these issues, we propose the temporal isometric delay-embedding transform, which constructs a Hankel tensor whose low-rankness is naturally induced by the smoothness and periodicity of the underlying time series. Leveraging this property, we develop the \textit{Low-Rank Tensor Completion with Temporal Isometric Delay-embedding Transform} (LRTC-TIDT) model, which characterizes the low-rank structure under the \textit{Tensor Singular Value Decomposition} (t-SVD) framework. Once the prescribed non-random sampling conditions and mild incoherence assumptions are satisfied, the proposed LRTC-TIDT model achieves exact recovery, as confirmed by simulation experiments under various non-random missing patterns. Furthermore, LRTC-TIDT consistently outperforms existing tensor-based methods across multiple real-world tasks, including network flow reconstruction, urban traffic estimation, and temperature field prediction. Our implementation is publicly available at https://github.com/HaoShu2000/LRTC-TIDT.

💡 Research Summary

This paper addresses the critical challenge of recovering multidimensional time series data with non-random (deterministic) missing patterns, such as block-wise missing segments caused by sensor failures. The authors identify key limitations in existing approaches: pure low-rank tensor completion methods fail under non-random missingness, while existing Hankel-structured methods lack a clear theoretical link between the low-rankness of the Hankelized tensor and the intrinsic properties of the original data, and further lack rigorous recovery guarantees for non-random missing scenarios.

To overcome these limitations, the authors propose a novel Temporal Isometric Delay-embedding Transform (TIDT). Unlike the conventional Multiway Delay-embedding Transform (MDT), TIDT is an isometric mapping that preserves consistency between the original time series tensor and its Hankelized version. Crucially, the authors theoretically demonstrate that the low-rank structure of the resulting Hankel tensor is naturally induced by the inherent smoothness and periodicity of the underlying time series data, providing a principled foundation for the method.

Leveraging this transform, the authors develop the Low-Rank Tensor Completion with TIDT (LRTC-TIDT) model. The model operates within the tensor Singular Value Decomposition (t-SVD) framework, seeking to recover the complete data by minimizing the tensor nuclear norm (TNN) of the TIDT-transformed tensor, subject to observation constraints.

The paper’s major theoretical contribution is establishing exact recovery guarantees for the LRTC-TIDT model under non-random missing patterns. By introducing the concepts of a minimum temporal sampling rate and the incoherence of the transformed tensor, the authors prove that once these mild conditions are satisfied, the model can exactly recover the original multidimensional time series from noiseless observations. This theoretical analysis is also extended to noisy observation settings and naturally incorporates forecasting tasks.

For optimization, an efficient Alternating Direction Method of Multipliers (ADMM) algorithm is designed to solve the LRTC-TIDT model, with each subproblem having a closed-form solution.

Comprehensive experiments validate the model’s superiority. On synthetic data, LRTC-TIDT achieves exact recovery under various non-random missing patterns, confirming the theory. On real-world tasks—including network traffic reconstruction, urban traffic speed estimation, and sea surface temperature field prediction—LRTC-TIDT consistently and significantly outperforms a range of state-of-the-art tensor completion baselines, such as pure TNN, MDT-based methods (MDT-Tucker, MDT-TT), and other Hankel-based methods (STH-LRTC). These results demonstrate the practical effectiveness of the proposed TIDT transform and the robustness of the LRTC-TIDT model in handling challenging real-world missing data problems.

In summary, this work provides a principled and theoretically grounded solution for recovering multidimensional time series with non-random missing values, bridging the gap between the intrinsic properties of time series data and structured low-rank completion theory.