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A Multi-View Coupled Tensor Decomposition for Lightweight Online Adaptive Traffic Prediction

Quan Yu, Jie Ni, Yu-Hong Dai, Xiongjun Zhang

arXiv:2608.25498v1math.OCcs.LG

TL;DR

Online traffic prediction must cope with incomplete, anomalous multi-view observations while adapting continuously. MVCTD builds a coupled latent forecasting space with shared spatial and view-specific temporal structure, then performs lightweight current-state refinement and closed-form updates; experiments report accurate forecasts with favorable runtime under severe missingness.

  • Problem

    Online traffic prediction requires continuous forecasting from multi-view traffic streams despite missing observations, anomalous disturbances, and computational constraints.

  • Method

    MVCTD jointly models shared spatial structure, view-dependent latent temporal dynamics, group-sparse anomalies, and temporal regularity in a coupled latent tensor formulation with lightweight online updates.

  • Results

    MVCTD achieves accurate forecasts with favorable runtime under severe missingness on real-world traffic datasets.

  • Takeaways & Limitations

    MVCTD is suitable for online traffic prediction from imperfect multi-view traffic streams.

Abstract

from arXiv · show

Accurate online traffic prediction is essential for intelligent transportation systems, where forecasting must be performed continuously under imperfect sensing conditions. Missing observations and anomalous disturbances make this task challenging, particularly when prediction relies on a single traffic view. This paper proposes a Multi-View Coupled Tensor Decomposition (MVCTD) model for online traffic prediction from imperfect multi-view observations, such as speed, flow, and occupancy. The proposed model uses coupled tensor decomposition to build a structured latent forecasting space, in which shared spatial structures across traffic views and view-specific temporal dynamics are jointly modeled. A group sparse regularization is further introduced to capture correlated abnormal responses induced by real traffic anomalies and thus reduce their influence on forecasts. For streaming deployment, MVCTD performs iterative refinement only on the current latent tensor, while the remaining model variables are updated by lightweight closed-form steps based on summarized historical information, thereby avoiding repeated optimization over the full historical sequence. Experiments on real-world traffic datasets demonstrate that MVCTD achieves accurate forecasts with favorable runtime under severe missingness, confirming its suitability for online traffic prediction.

I. INTRODUCTION

Online traffic prediction must handle high-dimensional, multi-view streams that are incomplete and corrupted while adapting efficiently as new observations arrive. MVCTD addresses these challenges with coupled latent modeling, anomaly separation, and lightweight streaming updates.

  • Traffic prediction supports proactive traffic control, congestion mitigation, and route guidance, but real-world measurements are high-dimensional, spatio-temporal, incomplete, and sometimes abnormal.
  • Separate forecasting of speed, flow, and occupancy can miss complementary cross-view information, while naive stacking can obscure view-specific temporal behavior.
  • Repeatedly re-optimizing the entire historical sequence becomes computationally unattractive as the history grows during online deployment.
  • MVCTD constructs a structured latent forecasting space using a shared spatial dictionary and view-dependent latent temporal factors.
  • The robust formulation jointly models latent autoregression, cross-view group-sparse anomalies, temporal regularization, missing-data recovery, and prediction.
  • Its two-stage streaming scheme refines only the current latent tensor online and updates remaining variables through closed-form steps, avoiding full-history re-optimization.

III. MVCTD FOR LIGHTWEIGHT ONLINE ADAPTIVE TRAFFIC FORECASTING

MVCTD formulates incomplete multi-view traffic forecasting as a unified latent tensor problem. It combines coupled factorization, sparse anomaly modeling, latent temporal regularization, and parsimonious autoregression for offline initialization.

  • A. Problem Formulation: MVCTD receives partial traffic tensors and predicts the next complete tensor in a rolling forecasting procedure.
  • A. Problem Formulation: Completion, anomaly suppression, and prediction are coupled in one latent tensor formulation rather than treated as separate preprocessing and forecasting stages.
  • B. Stage I: Offline Initialization: The regular component uses a shared spatial dictionary and day-specific latent temporal factors to encode cross-view complementarities and compact recurrent patterns.
  • B. Stage I: Offline Initialization: Group sparsity across views separates correlated abnormal responses from regular traffic dynamics, while low-rank regularization promotes a compact regular component.
  • B. Stage I: Offline Initialization: Subset autoregression selects sparse physically meaningful lags, such as daily or weekly cycles, to capture multi-scale periodicities with fewer parameters.
  • B. Stage I: Offline Initialization: Temporal priors are imposed on latent tensors, encouraging extrapolatable recurrent patterns without smoothing anomalies embedded in raw traffic states.

C. Stage II: Lightweight Online Adaptive Prediction

After offline initialization, MVCTD forecasts each new day and corrects its latent state when partial observations arrive. It keeps historical latent tensors fixed while adapting global parameters through lightweight updates.

  • At each new day, MVCTD first generates a prior forecast and then corrects the current latent state using the newly received partial observation.
  • The online stage estimates the current latent tensor, updates SAR coefficients, and updates the shared spatial basis through three decoupled subproblems.
  • The updated parameters are used recursively to roll the model forward and generate the next forecast.
  • Previously inferred latent tensors remain fixed, while iterative optimization is applied only to the current day’s latent inference problem.
  • SAR coefficients and the shared spatial basis are updated through closed-form steps using summarized historical information, avoiding full-history re-optimization.

D. Theoretical Unification of Multi-View Coupled Tensor Decomposition

The t-product representation unifies independent-view and shared-basis multi-view decompositions. This lets MVCTD share spatial structure across traffic variables while retaining view-dependent temporal patterns.

  • Implication: The representation therefore covers fully view-specific and shared-basis decompositions, supporting common road-network structure with view-dependent temporal patterns.
  • Theorem 3.2: Theorem 3.2 embeds separate matrix decompositions of each traffic view into a coupled tensor factorization X = A ∗ Z.The construction uses a coupled spatial tensor A and latent tensor Z with p no greater than the sum of the view ranks.
  • Independent-view case: The independent-view case permits distinct spatial bases by setting p = Σ_v r_v and constructing separate Fourier-domain slices.
  • Shared-basis case: The shared-basis case sets A_v = A for every view and reduces the representation to p = r_1.
  • Shared-basis case: When spatial bases are shared, the frontal slices satisfy Z^(v) = Z_v, recovering the classical joint matrix decomposition form.

IV. OPTIMIZATION ALGORITHM

The optimization procedure uses ADMM for offline initialization and lightweight online adaptation. Online updates optimize only the newly arrived day's latent tensor, then refresh model parameters in closed form to avoid batch re-optimization.

  • Offline ADMM: ADMM introduces auxiliary variables Y_d = Z_d and C_d = Z_d to decouple latent temporal regularization and SAR fitting from the Z_d update.
  • Offline ADMM: The batch procedure updates primal blocks sequentially, followed by dual-variable and penalty-parameter updates, until a termination criterion is met.
  • Online adaptation: Online adaptation warm-starts from the prior latent tensor, runs ADMM only for the current day, and refreshes SAR coefficients and the spatial dictionary through closed-form updates.
  • Complexity: For small R and L̄, offline cost scales nearly linearly with historical data size, while each online step updates only the newly arrived day's latent tensor.
  • Offline initialization: The offline algorithm computes updates for X_d, A, Z_d, Y_d, C_d, E_d, and SAR coefficients w_l before producing initialized model variables.
  • Complexity: Compared with batch re-optimization, the online update removes the multiplicative factor D from the iterative component, supporting streaming prediction.

V. NUMERICAL EXPERIMENTS

The numerical experiments evaluate MVCTD on real-world traffic prediction tasks with incomplete multi-view observations using MATLAB 2025b on a specified workstation.

  • Experimental setting: Experiments use real-world traffic prediction tasks with incomplete multi-view observations.
  • Experimental setting: All experiments run in MATLAB 2025b on an Intel Core i5-12500H CPU at 2.50 GHz with 16 GB RAM.

1) Implementation Details:

The implementation fixes regularization, penalty, lag, and iteration settings for MVCTD and competing methods during evaluation.

  • Parameter settings: Competing methods use parameter settings recommended in their original papers, while MVCTD uses empirically selected regularization parameters.
  • Parameter settings: MVCTD sets λ1 = 2e-1, λ2 = 5e-1, λ3 = 1e-2, λ4 = 2e-1, and γ = 5e-1.
  • Parameter settings: The penalty growth factor is β = 1.15, and the SAR lag set is L = {7}, with L̄ = 1 and L = 7.
  • Stopping criteria: Optimization terminates when its criterion is satisfied for five consecutive iterations, with a maximum of 200 iterations.

2) Traffic Datasets:

The study uses two real-world PeMS datasets containing three-view traffic measurements at 5-minute resolution. The datasets differ in sensor count, region, duration, and tensor size.

  • Dataset Overview: Both datasets contain volume, occupancy, and speed measurements at 5-minute resolution, forming three-view traffic tensors.Occupancy values are rescaled from [0, 1] to [0, 100] before evaluation.
  • PeMS-D4: PeMS-D4 contains 307 loop detectors in the San Francisco Bay Area over 59 days, yielding a 307 × 16, 992 × 3 tensor.The measurements span January 1 to February 28, 2018.
  • PeMS-D8: PeMS-D8 contains 170 sensors in the San Bernardino area over 62 days, yielding a 170 × 17, 856 × 3 tensor.The measurements span July 1 to August 31, 2016.

3) Evaluation Metrics:

Evaluation combines MAPE and RMSE with representative factorization, Bayesian, dynamical-system, and deep-learning baselines. MVCTD achieves the best reported forecasting accuracy on both complete and severely incomplete inputs while requiring substantially less runtime.

  • Evaluation Metrics: Forecasting accuracy is evaluated using Mean Absolute Percentage Error (MAPE) and Root Mean Square Error (RMSE).B is the number of evaluated entries, while Mijk and Xijk denote ground-truth and predicted values at index (i, j, k).
  • Baselines: MVCTD is compared with six baselines spanning low-rank factorization, Bayesian modeling, dynamical-system modeling, and deep learning.The baselines include HTMF, BTMF, circDMDsp, PatchTST, PreSTGNet, and AmTGBiM.
  • Fully Observed Results: MVCTD consistently obtains the lowest MAPE and RMSE on both datasets and across all three traffic views with fully observed inputs.On PeMS-D8, volume MAPE decreases from 21.65 to 11.20 and occupancy MAPE from 25.77 to 13.19 against the best competing values.
  • Incomplete-Input Results: With 80% missing entries, MVCTD ranks first on every reported metric in Table II.Volume MAPE increases by 1.06 on PeMS-D4 and 0.95 on PeMS-D8 relative to the fully observed case.
  • Runtime: MVCTD requires 101 seconds on PeMS-D4 and 49 seconds on PeMS-D8 with full observations, remaining faster than all competing methods.At 80% missingness, runtime decreases to 72 and 38 seconds, approximately 26 and more than 33 times faster than the fastest Table II baseline.

C. Model Analysis

Model analysis shows that anomaly sparsity, temporal regularization, multi-view coupling, and sparse autoregressive lags contribute to MVCTD’s forecasting performance, while the solver converges within 90 iterations.

  • Parameter Analysis: The model is most sensitive to λ1 and λ2, while λ3 and λ4 remain stable across a broad range and γ has a localized favorable region.The selected values are λ1 = 2e-1, λ2 = 5e-1, λ3 = 1e-2, λ4 = 2e-1, and γ = 5e-1.
  • Ablation Study: Removing anomaly sparsity increases volume MAPE from 11.31 to 22.94 under 40% missingness, the largest ablation degradation.Removing global periodicity or local smoothness also weakens accuracy across the three traffic views.
  • Effectiveness of Multi-View Modeling: MVCTD reduces speed MAPE from 4.67 to 4.49 on PeMS-D8 and volume MAPE from 15.92 to 15.48 on PeMS-D4 versus single-view modeling.On PeMS-D4, the single-view variant has slightly lower volume RMSE, but MVCTD achieves lower MAPE for all three views.
  • Effectiveness of Subset Autoregression: The sparse lag structure outperforms using only the previous day or a continuous seven-day window on most reported metrics.The results indicate that longer-term dependencies matter, while subset autoregression captures dominant periodic dependence without redundant intermediate lags.
  • Convergence Analysis: The error fluctuates during the first 20 iterations and then decreases rapidly, reaching the stopping criterion within 90 iterations under all tested missing rates.This supports the practicality of the batch solver for offline initialization.
  • Online Deployment: MVCTD combines offline initialization with online refinement of only the current latent tensor and lightweight updates of remaining model parameters.Experiments and ablations report accurate and efficient prediction under severe missingness, with contributions from anomaly modeling, temporal regularization, multi-view coupling, and subset autoregression.
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