Low-Cost High-Order Singular Value Decomposition for Tensor-Based Reconstruction from Sparse Sensor Measurements: Urban Flow and Air-Quality Applications

2026-06-23

AI Summary

This study addresses the challenge of reconstructing high-dimensional urban flow and air quality fields from sparse sensor measurements, where conventional matrix-based methods suffer from performance degradation due to structural loss incurred by vectorization. To overcome this limitation, the authors propose a low-computational-cost higher-order singular value decomposition (lcHOSVD) framework that, for the first time, integrates sparsity-aware mechanisms with tensor decomposition. This approach preserves the intrinsic high-order couplings among spatial, temporal, and physical variable dimensions while substantially reducing computational overhead. Experimental results demonstrate that the method achieves high-fidelity reconstruction of three-dimensional velocity and pollutant concentration fields using only 1–4% of spatial measurement points, yielding lower reconstruction errors than matrix-based alternatives and exhibiting enhanced robustness to non-uniform sensor distributions.

Abstract

Urban flow and air-quality simulations generate high-dimensional datasets describing velocity and pollutant transport across multiple spatial, temporal, and physical-variable dimensions. Reconstructing these fields from sparse sensor measurements is a fundamental challenge in environmental monitoring, digital twins, forecasting, and data assimilation. Existing low-cost reconstruction approaches are commonly based on matrix decompositions, which require multidimensional datasets to be flattened into two-dimensional snapshot matrices, thereby discarding important structural information. This work introduces the low-cost High-Order Singular Value Decomposition (lcHOSVD), a novel tensor-based sparse-sensing reconstruction framework for high-dimensional environmental fields. To the authors' knowledge, this is the first methodology that combines sparse sensing and HOSVD for field reconstruction. Unlike matrix-based approaches, lcHOSVD preserves the natural tensor structure of the data, enabling the exploitation of correlations across spatial, temporal, and physical-variable dimensions while substantially reducing the computational requirements of conventional HOSVD. The methodology is applied to urban flow and air-quality datasets, where three-dimensional velocity and pollutant concentration fields are reconstructed using only 1-4% of the available spatial locations. While lcSVD provides larger computational speed-ups, lcHOSVD consistently achieves lower reconstruction errors in configurations characterized by strong multidimensional coupling and heterogeneous dynamics across dimensions. Additional sensor-anisotropy analyses demonstrate that the tensor formulation is significantly more robust to uneven sensor distributions, a common situation in practical environmental monitoring networks.

Problem

• tensor reconstruction
• sparse sensing
• urban flow
• air quality
• high-dimensional data

Innovation

• tensor decomposition
• sparse sensing
• HOSVD
• environmental field reconstruction
• multidimensional data