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Rethinking Multi-View Fusion for Urban Representation Learning

Sean Bin Yang, Zongyi Xu, Jilin Hu, Ying Sun, Kristian Torp, Bin Yang, Torben Bach Pedersen, Christian S. Jensen

arXiv:2610.31850v2cs.LGcs.AI

TL;DR

Urban signals vary in structure and coverage, and region-level view utility and view-induced objectives call for adaptive coordination. PRISM combines sparse region-level routing with PID-inspired discrepancy regulation; across three datasets and three tasks, it improves performance over state-of-the-art proposals while maintaining comparable model complexity.

  • Problem

    Urban signals vary in structure, granularity, noise, and spatial coverage, and region-level view utility and heterogeneous objectives challenge uniform integration.

    Source: §1 · pp. 1–2

  • Method

    PRISM combines sparse MoE routing to allocate views by region with PID-inspired discrepancy regularization for heterogeneous objectives.

    Source: §1 · p. 2

  • Results

    Across three large-scale urban datasets and three tasks, PRISM exhibits consistent performance improvements over state-of-the-art proposals with comparable model complexity.

    Source: §6 · p. 12

  • Takeaways & Limitations

    The results highlight adaptive fusion and feedback-aware optimization for stable training and preserving model robustness in heterogeneous urban representation learning.

    Source: §6 · p. 12

    At SF Crime with a mobility missing ratio of 0.7, HAFusion slightly outperforms PRISM under severe dominant-view incompleteness.

    Source: §5.8.1 · p. 11

Abstract

from arXiv · show

Urban region representation learning encodes heterogeneous urban signals, such as mobility flows, points of interest, and land-use patterns, into unified embeddings for downstream tasks, including check-in prediction, crime forecasting, and service demand estimation. Existing approaches rely on globally shared fusion and joint optimization, implicitly assuming uniform modality importance and naturally balanced multi-objective training. However, urban data is inherently heterogeneous. Our empirical analysis reveals two structural issues: (i) modality utility varies markedly across regions, causing static fusion to overemphasize signals from certain modalities; and (ii) heterogeneous objectives exhibit persistent gradient imbalance, causing representational bias and unstable training. We present PRISM, a self-adjusting framework for adaptive multi-view learning. PRISM adapts sparse Mixture-of-Experts (MoE) routing to learn region-specific view allocations, while a Proportional-Integral-Derivative (PID)-inspired regulation mechanism mitigates gradient conflicts during optimization. Experiments on three large-scale urban datasets show that PRISM achieves more competitive performance than strong multi-view baselines, while providing more stable optimization behavior and more balanced region representations.

1 INTRODUCTION

Urban signals vary across regions and tasks, while conventional fusion and joint optimization assume modality importance and objective contributions are uniform. PRISM addresses these challenges with region-adaptive view routing and PID-inspired optimization coordination.

1 INTRODUCTION

  • Task-dependent modality utility and persistent gradient disparities under conventional joint training can bias representations and degrade cross-region generalization.In San Francisco, mobility is most informative for crime prediction, while land-use and POI features are more useful for service calls.Source: §1 · p. 2
  • Globally shared fusion can under-represent regions whose useful modalities differ from those favored by shared parameters.The paper identifies commercial and residential regions as examples of potentially uneven representation.Source: §1 · p. 2
  • Jointly optimized modality objectives can dominate one another because their gradients differ in scale, noise, and convergence rate, destabilizing training.The stated challenge is to regulate competing objectives so informative modalities contribute proportionally.Source: §1 · p. 2
  • PRISM combines region-level sparse MoE view routing with PID-inspired discrepancy regularization for heterogeneous view-induced objectives.Its proportional, integral, and derivative terms represent current discrepancy, accumulated bias, and short-term variation; the terms act as loss-level penalties.Source: §1 · p. 2
  • The framework is evaluated on three large-scale urban datasets across multiple downstream tasks to assess effectiveness, robustness, and efficiency.The paper presents adaptive view allocation and PID-inspired optimization coordination as its two principal contributions.Source: §1 · p. 2

2 RELATED WORK

Prior urban representation work integrates heterogeneous signals, while existing routing and multi-task methods address different units or optimization settings. PRISM adapts these ideas to region-level view allocation and view-induced objectives.

2.1 Urban Region Representation Learning

  • Multi-view urban methods integrate heterogeneous signals into unified embeddings, whereas single-source methods remain shaped by one modality’s information scope.Prior approaches use multi-graph architectures, cross-attention, shared latent spaces, or fusion frameworks.Source: §2.1 · p. 3

2.2 Mixture-of-Experts and Adaptive Routing

  • PRISM adapts standard sparse MoE routing to select among heterogeneous urban views for each region, rather than introducing a new routing operator.The routed unit is a region-view representation, unlike token-, modality-, temporal-, or task-level routing in other settings.Source: §2.2 · p. 3

2.3 Adaptive Multi-Task Learning and PID-Inspired Regulation

  • Unlike conventional MTL and PID approaches, PRISM regularizes discrepancies among urban view-induced objectives using current, accumulated, and short-term discrepancy signals.GradNorm, PCGrad [53], and MGDA [29] are included as controlled baselines under the same backbone.Source: §2.3 · p. 3

3 PRELIMINARIES

The preliminaries define the urban views and optimization concepts used by PRISM, then state the goal of learning region representations with region-specific view importance.

3.1 Definitions

  • The paper represents mobility as a directed flow graph and POI and land-use features as region-by-category matrices.The three views are mobility, POI, and land use.Source: §3.1 · p. 4
  • The gradient stability score captures modality-gradient magnitude imbalance and pairwise directional conflict; smaller values indicate more balanced, aligned optimization.When gradient norms are identical and pairwise cosine similarities are nonnegative, both terms vanish and the score is zero.Source: §3.1 · p. 4

3.2 Problem Statement

  • The task is to learn unified region embeddings from heterogeneous urban views for downstream prediction, allowing each region to assign different importance to each view.The view-specific contribution weights represent spatial heterogeneity in region function, mobility, and land-use characteristics.Source: §3.2 · p. 4

4 METHODOLOGY

PRISM combines shared multi-view encoding with region-specific sparse routing and PID-inspired loss regulation. Its training objective joins modality-specific tasks with routing regularization, adding a lightweight router to the shared backbone.

4.1 Multi-View Encoding

  • PRISM encodes urban views in a shared latent space, using intra-view attention to retain modality-specific regional structure.POI and land-use features are projected with MLPs, while mobility features already occupy the shared space.Source: §4.1 · p. 5
  • Inter-view attention captures cross-modal correlations, while learnable residual mixing balances those interactions with stable intra-view representations.The resulting mixed representations provide a basis for adaptive fusion.Source: §4.1 · p. 5

4.2 Region-Level Adaptive Routing

  • Sparse MoE routing selects views separately for each region, using learned logits and top-k selection to allocate view contributions.An optional entropy regularizer sharpens region-specific routing distributions.Source: §4.2 · p. 5

4.3 PID-Inspired Discrepancy Regularization

  • PID-MSE regularizes POI and land-use similarity discrepancies with separate proportional, accumulated, and differential loss components, rather than implementing a classical PID controller.These components target instantaneous discrepancy, persistent bias, and rapid discrepancy variation, respectively.Source: §4.3 · pp. 5–6

4.4 Training Objective

  • Training combines a mobility reconstruction loss with POI and land-use similarity-alignment losses, optional routing-entropy regularization, and uncertainty-based objective weighting.The final objective uses PID-MSE for attribute-based views.Source: §4.4 · p. 6

4.5 Model Complexity Analysis

  • The router adds O(VNdℎ) computation and O(dℎ) parameters, while PRISM retains HAFusion’s asymptotic time and space complexity.The routing overhead is described as negligible relative to backbone attention because ℎ≪N.Source: §4.5 · pp. 6–7

5 EXPERIMENTAL STUDY

Across check-in, crime, and service-call prediction in three cities, PRISM outperforms the strongest baseline, HAF, on reported evaluations. The study evaluates region embeddings on real-world city datasets using regression tasks and standard error and explanatory-power metrics.

5.1 Settings

  • Evaluation uses NYC, Chicago, and San Francisco data for check-in, crime, and service-call regression, comparing representation-learning, multitask-optimization, and spatiotemporal baselines with MAE, RMSE, and R2.Region-level features include mobility flows, POIs, and land use; Lasso regression predicts aggregated event counts.Source: §5.1 · p. 7

5.2 Overall Performance

  • PRISM outperforms HAF across reported metrics for check-in, crime, and service-call prediction in the evaluated cities.For check-in in NYC, MAE falls from 202.8 to 200.7 and R2 rises from 0.844 to 0.852; crime R2 rises from 0.734 to 0.756 in NYC and 0.631 to 0.668 in Chi; service-call R2 rises from 0.613 to 0.650 in Chi.Source: §5.2 · p. 7

5.3 Ablation Studies

Ablations show that region-aware routing, uncertainty weighting, and PID-MSE each contribute to PRISM's performance or optimization behavior. Static concatenation produces notable check-in degradation, while removing uncertainty weighting increases error and variance.

Effect of Region-Aware Routing (RAR)

  • Check-in R² falls from 0.852 to 0.836 in NYC and from 0.904 to 0.850 in Chi when region-aware routing is replaced by concatenation.The passage reports similar trends for crime and service-call prediction.Source: §5.3 · p. 7, Table 2 · p. 8

Effect of Uncertainty Weighting (UM)

  • For NYC check-in prediction, removing uncertainty weighting raises MAE to 210.3 and produces substantially larger variance than the full model.The ablation also reports lower accuracy and higher variance overall without uncertainty weighting.Source: §5.3 · pp. 8–9

Effect of PID-MSE

  • Replacing PID-MSE with standard MSE degrades performance in most cases, while PID-MSE's proportional, accumulated, and differential error components target bias and local oscillations.The reported improvements are consistent for crime and service-call prediction.Source: §5.3 · p. 9

5.4 Parameter Sensitivity Analysis

Embedding dimensionality has a non-monotonic relationship with performance, with a mid-range typically best across cities. Router temperature also matters: 1.0 yields the best reported results across tasks and cities.

Effects of Region Embedding Size

  • Embedding-size performance is non-monotonic: very small dimensions underperform, while very large dimensions can saturate or slightly reduce performance.The study varies dimensions from 32 to 288; the authors suggest excessive size may introduce redundancy, noise, and optimization difficulty.Source: §5.4 · p. 9
  • The best embedding size typically falls within 96–144 dimensions across cities, and service-call prediction is more sensitive to dimensionality than crime forecasting.The authors suggest service demand may reflect a broader set of socio-economic factors.Source: §5.4 · p. 9

Effect of Top-k Router Temperature

  • A router temperature of 1.0 gives the best reported results for all three tasks across all three cities.Lower temperatures produce sharper routing, while higher temperatures produce smoother routing distributions.Source: §5.4 · p. 9, Figure 4 · p. 9

5.5 Model Scalability Analysis

The Router model's performance generally improves from Top-k=1 to 2, then stabilizes or declines slightly at 3. Its computational cost remains comparable to Concat across the tested settings.

Model Scalability

  • The comparison evaluates Router against PRISM-Concat on prediction R² and computational cost in GFLOPs across Top-k settings.Source: §5.5 · p. 9

Performance vs. Top-k

  • Router prediction performance often peaks at k=2, improving over k=1 before stabilizing or slightly declining at k=3.The passage attributes the pattern to a balance between expert diversity and noise control; Concat is relatively stable across k.Source: §5.5 · p. 10

Computational Cost

  • Increasing k slightly raises Router GFLOPs, but its overall computational cost remains comparable to Concat.On NYC and SF, Router sometimes has slightly lower GFLOPs than Concat.Source: §5.5 · p. 10

5.6 Model Efficiency Analysis

Across nine city–task settings, PRISM lies on or near the accuracy–efficiency Pareto frontier without noticeable increases in GPU memory. It improves R² over PRISM-Concat at marginally higher training cost, while HAFusion's higher efficiency score reflects shorter training time rather than better prediction.

Pareto Dominance

  • PRISM lies on or near the Pareto frontier across all nine city–task combinations, and no competitor combines higher accuracy with lower training time.HAFusion trains faster but has lower prediction performance; PRISM-Concat is marginally faster in some scenarios but less accurate than routing-based PRISM.Source: §5.6 · p. 10

Quantitative Efficiency Comparison

  • Average efficiency scores are 6.76 for PRISM, 6.20 for PRISM-Concat, and 10.38 for HAFusion across nine settings.HAFusion's higher score primarily reflects shorter training time rather than improved prediction capability.Source: §5.6 · p. 10
  • PRISM achieves better R² than PRISM-Concat at typically 0.01–0.02 seconds of additional training time per epoch, with comparable GPU memory usage across methods.The passage characterizes this training-time overhead as marginal.Source: §5.6 · p. 10

5.7 Model Behavior Analysis

PRISM routes views differently across city–task settings and shows lower optimization instability than HAFusion. The analysis links adaptive routing and PID-based regulation to more coordinated multi-view learning.

Modality Heterogeneity

  • Routing preferences vary by city and task: Land dominates in NYC, Chi is more balanced, and Mobility dominates SF Crime.Figure 7a reports the Top-1 routed view ratio across city–task combinations.Source: §5.7 · p. 10, Figure 7 · p. 11

Optimization Instability

  • Across city–task combinations, PRISM (PID-MSE) consistently has lower optimization instability than HAFusion (MSE).The gap remains stable across epochs, with lower scores indicating better gradient alignment and reduced scale disparity.Source: §5.7 · p. 10, Figure 7 · p. 11
  • PRISM’s routing captures heterogeneous modality relevance, while PID-based regulation mitigates gradient competition and improves optimization stability.The authors connect these mechanisms to consistent performance gains across city–task settings.Source: §5.7 · p. 11

5.8 Robustness to View Corruption and Incompleteness

PRISM is tested under missing-view and noisy-view corruption of dominant city modalities. It generally retains stronger performance, with a noted exception under severe mobility masking for SF Crime.

5.8.1 Missing Views.

  • As missing-view ratios rise, both methods degrade, but PRISM performs best in nearly all tested ratios and city–task combinations.The selected dominant views are Land for NYC, POI for Chi, and Mobility for SF; PRISM also wins in most cases when the selected view is fully removed.Source: §5.8.1 · p. 11, Figure 8 · p. 11
  • At 0.7 mobility masking, HAFusion slightly outperforms PRISM on SF Crime, a limitation under severe dominant-view incompleteness.The authors note that dense fusion may be more stable at this corruption level; PRISM still outperforms the baseline in most cases at a missing-view ratio of 1.0.Source: §5.8.1 · p. 11

5.8.2 Noisy Views.

  • Under additive Gaussian noise, PRISM maintains higher R2 than HAFusion in NYC, decays more slowly in Chi, and preserves a stable margin in SF.The tested noise ratios are 0.1, 0.5, and 0.9, applied to each city’s dominant view.Source: §5.8.2 · p. 11, Figure 9 · p. 12
  • With mixture-of-Gaussians noise, PRISM generally leads, though HAFusion slightly surpasses it for Chi Check-in at ratio 0.1; under column-wise shuffling, PRISM leads in Chi and SF.For shuffling, the gap is marginal across corruption levels in NYC but grows at moderate-to-high ratios in Chi and SF, particularly for Crime and Service Call.Source: §5.8.2 · p. 12, Figure 10 · p. 12, Figure 11 · p. 12

6 CONCLUSION

PRISM combines region-wise sparse routing with PID-inspired discrepancy regulation for adaptive modality allocation and more stable multi-objective optimization. Experiments across three large-scale urban datasets and three tasks show consistent gains over state-of-the-art proposals at comparable model complexity.

6 CONCLUSION

  • Across three large-scale urban datasets and three tasks, PRISM improves performance over state-of-the-art proposals while maintaining comparable model complexity.The conclusion highlights adaptive fusion and feedback-aware optimization for stable training and robust urban representations.Source: §6 · p. 12
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