# Low-Rank Hankel Tensor Completion for Traffic Speed Estimation

arXiv: https://arxiv.org/abs/2105.11335 · HTML: https://mlsanity.com/paper/2105.11335/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 42 references that are in the map.
- First posted on arXiv 21 May 2021.

## Builds on (most cited first, 10 of 10)

- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Inductive Graph Neural Networks for Spatiotemporal Kriging](https://mlsanity.com/paper/2006.07527/) (2020)
- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery](https://mlsanity.com/paper/1307.5870/) (2013)
- [A Nonconvex Low-Rank Tensor Completion Model for Spatiotemporal Traffic Data Imputation](https://mlsanity.com/paper/2003.10271/) (2020)
- [Macroscopic Traffic Flow Modeling with Physics Regularized Gaussian Process: A New Insight into Machine Learning Applications](https://mlsanity.com/paper/2002.02374/) (2020)
- [Missing Slice Recovery for Tensors Using a Low-rank Model in Embedded Space](https://mlsanity.com/paper/1804.01736/) (2018)
- [Block Hankel Tensor ARIMA for Multiple Short Time Series Forecasting](https://mlsanity.com/paper/2002.12135/) (2020)
- [Incorporating Kinematic Wave Theory into a Deep Learning Method for High-Resolution Traffic Speed Estimation](https://mlsanity.com/paper/2102.02906/) (2021)
- [Large-Scale Low-Rank Matrix Learning with Nonconvex Regularizers](https://mlsanity.com/paper/1708.00146/) (2017)

## Built on by (most cited first, 22 of 22)

- [ImputeFormer: Low Rankness-Induced Transformers for Generalizable Spatiotemporal Imputation](https://mlsanity.com/paper/2312.01728/) (2023)
- [A Physics-Informed Deep Learning Paradigm for Traffic State and Fundamental Diagram Estimation](https://mlsanity.com/paper/2106.03142/) (2021)
- [Uncertainty Quantification of Sparse Travel Demand Prediction with Spatial-Temporal Graph Neural Networks](https://mlsanity.com/paper/2208.05908/) (2022)
- [Uncertainty Quantification via Spatial-Temporal Tweedie Model for Zero-inflated and Long-tail Travel Demand Prediction](https://mlsanity.com/paper/2306.09882/) (2023)
- [Laplacian Convolutional Representation for Traffic Time Series Imputation](https://mlsanity.com/paper/2212.01529/) (2022)
- [Correlating sparse sensing for large-scale traffic speed estimation: A Laplacian-enhanced low-rank tensor kriging approach](https://mlsanity.com/paper/2210.11780/) (2022)
- [ST-GIN: An Uncertainty Quantification Approach in Traffic Data Imputation with Spatio-temporal Graph Attention and Bidirectional Recurrent United Neural Networks](https://mlsanity.com/paper/2305.06480/) (2023)
- [Spatiotemporal Implicit Neural Representation as a Generalized Traffic Data Learner](https://mlsanity.com/paper/2405.03185/) (2024)
- [Traffic State Estimation from Vehicle Trajectories with Anisotropic Gaussian Processes](https://mlsanity.com/paper/2303.02311/) (2023)
- [Hankel-structured Tensor Robust PCA for Multivariate Traffic Time Series Anomaly Detection](https://mlsanity.com/paper/2110.04352/) (2021)
- [Fairness-Enhancing Vehicle Rebalancing in the Ride-hailing System](https://mlsanity.com/paper/2401.00093/) (2023)
- [Knowledge-data fusion oriented traffic state estimation: A stochastic physics-informed deep learning approach](https://mlsanity.com/paper/2409.00644/) (2024)
- [Spatiotemporal Regularized Tucker Decomposition Approach for Traffic Data Imputation](https://mlsanity.com/paper/2305.06563/) (2023)
- [Efficient and Robust Freeway Traffic Speed Estimation under Oblique Grid using Vehicle Trajectory Data](https://mlsanity.com/paper/2411.05842/) (2024)
- [Guaranteed Multidimensional Time Series Prediction via Deterministic Tensor Completion Theory](https://mlsanity.com/paper/2501.15388/) (2025)
- [PMA-Diffusion: A Physics-guided Mask-Aware Diffusion Framework for TSE from Sparse Observations](https://mlsanity.com/paper/2512.06183/) (2025)
- [Structured Low-Rank Tensor Learning](https://mlsanity.com/paper/2305.07967/) (2023)
- [Collaborative Imputation of Urban Time Series through Cross-city Meta-learning](https://mlsanity.com/paper/2501.11306/) (2025)
- [Robust Tensor Completion via Gradient Tensor Nulclear L1-L2 Norm for Traffic Data Recovery](https://mlsanity.com/paper/2506.22732/) (2025)
- [Exact Recovery of Non-Random Missing Multidimensional Time Series via Temporal Isometric Delay-Embedding Transform](https://mlsanity.com/paper/2512.10191/) (2025)
- [Uncertainty-Aware Graph Neural Reconstruction of Urban Temperature Fields from Sparse Sensors under Deployment Constraints](https://mlsanity.com/paper/2606.02038/) (2026)
- [Dual-Attention Convolution Experts for Sparse Tensor Completion](https://mlsanity.com/paper/2606.21427/) (2026)

## Abstract (arXiv)

This paper studies the traffic state estimation (TSE) problem using sparse observations from mobile sensors. Most existing TSE methods either rely on well-defined physical traffic flow models or require large amounts of simulation data as input to train machine learning models. Different from previous studies, we propose a purely data-driven and model-free solution in this paper. We consider the TSE as a spatiotemporal matrix completion/interpolation problem, and apply spatiotemporal delay embedding to transform the original incomplete matrix into a fourth-order Hankel structured tensor. By imposing a low-rank assumption on this tensor structure, we can approximate and characterize both global and local spatiotemporal patterns in a data-driven manner. We use the truncated nuclear norm of a balanced spatiotemporal unfolding -- in which each column represents the vectorization of a small patch in the original matrix -- to approximate the tensor rank. An efficient solution algorithm based on the Alternating Direction Method of Multipliers (ADMM) is developed for model learning. The proposed framework only involves two hyperparameters, spatial and temporal window lengths, which are easy to set given the degree of data sparsity. We conduct numerical experiments on real-world high-resolution trajectory data, and our results demonstrate the effectiveness and superiority of the proposed model in some challenging scenarios.