Rethinking Multi-View Fusion for Urban Representation Learning

# Rethinking Multi-View Fusion for Urban Representation Learning
Source: [https://arxiv.org/html/2610.31850](https://arxiv.org/html/2610.31850)
Sean Bin Yang,Zongyi XuAffiliation:Chongqing University of Posts and Telecommunications, Chinaemail:[xuzy@cqupt\.edu\.cn](mailto:[email protected]),Jilin HuAffiliation:East China Normal University, Chinaemail:[jlhu@dase\.ecnu\.edu\.cn](mailto:[email protected]),Ying SunAffiliation:Chongqing University of Posts and Telecommunications, Chinaemail:[sunying@cqupt\.edu\.cn](mailto:[email protected]),Kristian TorpAffiliation:Aalborg University, Denmarkemail:[torp@cs\.aau\.dk](mailto:[email protected]),Bin YangAffiliation:East China Normal University, Chinaemail:[byang@dase\.ecnu\.edu\.cn](mailto:[email protected]),Torben Bach PedersenAffiliation:Aalborg University, Denmarkemail:[tbp@cs\.aau\.dk](mailto:[email protected])andChristian S\. JensenAffiliation:Aalborg University, Denmarkemail:[csj@cs\.aau\.dk](mailto:[email protected])

###### Abstract\.

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 presentPRISM, a self\-adjusting framework for adaptive multi\-view learning\.PRISMadapts sparse MoE routing to learn region\-specific view allocations, while a PID \(Proportional Integral Derivative\)\-inspired regulation mechanism mitigates gradient conflicts during optimization\. Experiments on three large\-scale urban datasets show thatPRISMachieves more competitive performance than strong multi\-view baselines, while providing more stable optimization behavior and more balanced region representations\.

††authors:\.PVLDB Reference Format: PVLDB, 20\(3\): XXX\-XXX, 2026\. [doi:XX\.XX/XXX\.XX](https://doi.org/XX.XX/XXX.XX)††This work is licensed under the Creative Commons BY\-NC\-ND 4\.0 International License\. Visit[https://creativecommons\.org/licenses/by\-nc\-nd/4\.0/](https://creativecommons.org/licenses/by-nc-nd/4.0/)to view a copy of this license\. For any use beyond those covered by this license, obtain permission by emailing[info@vldb\.org](mailto:[email protected])\. Copyright is held by the owner/author\(s\)\. Publication rights licensed to the VLDB Endowment\. Proceedings of the VLDB Endowment, Vol\. 20, No\. 3 ISSN 2150\-8097\. [doi:XX\.XX/XXX\.XX](https://doi.org/XX.XX/XXX.XX)

## 1\.Introduction

Cities are increasingly instrumented with sensing and digital infrastructures that continuously generate heterogeneous data from mobility networks, land\-use registries, transportation systems, and socio\-economic activities\([Xu et al\., 2025a](https://arxiv.org/html/2610.31850#bib.bib32)\)\. This data provides unprecedented opportunities for understanding urban dynamics\. Yet, heterogeneity in terms of structure, granularity, noise ratio, and spatial coverage poses fundamental data integration and computational modeling challenges\.

Urban region representation learning has emerged as a principled abstraction for addressing these challenges\([Duan et al\., 2023](https://arxiv.org/html/2610.31850#bib.bib34);[Xu et al\., 2025a](https://arxiv.org/html/2610.31850#bib.bib32);[Yang et al\., 2026b](https://arxiv.org/html/2610.31850#bib.bib10);[Yang et al\., 2027](https://arxiv.org/html/2610.31850#bib.bib9)\)\. Transforming diverse urban signals into compact embeddings enables computational models that capture latent spatial functionality and inter\-region relationships in a unified vector form\. These embeddings serve as reusable, transferable representations that support a wide range of analytical tasks, including check\-in prediction\([Liang and Zhang, 2023](https://arxiv.org/html/2610.31850#bib.bib17);[Gong et al\., 2024](https://arxiv.org/html/2610.31850#bib.bib18);[Gong et al\., 2025](https://arxiv.org/html/2610.31850#bib.bib19)\), crime forecasting\([Butt et al\., 2025](https://arxiv.org/html/2610.31850#bib.bib20);[Wang et al\., 2022](https://arxiv.org/html/2610.31850#bib.bib21);[Li et al\., 2022](https://arxiv.org/html/2610.31850#bib.bib22);[Zhao et al\., 2022](https://arxiv.org/html/2610.31850#bib.bib23);[Zhao et al\., 2023](https://arxiv.org/html/2610.31850#bib.bib24)\), traffic modeling\([Zhang et al\., 2025](https://arxiv.org/html/2610.31850#bib.bib25);[Sun et al\., 2025b](https://arxiv.org/html/2610.31850#bib.bib26);[Yang et al\., 2025b](https://arxiv.org/html/2610.31850#bib.bib27);[Yang et al\., 2021b](https://arxiv.org/html/2610.31850#bib.bib28);[Yang et al\., 2022a](https://arxiv.org/html/2610.31850#bib.bib12);[Yang et al\., 2023](https://arxiv.org/html/2610.31850#bib.bib16);[Yang et al\., 2026a](https://arxiv.org/html/2610.31850#bib.bib11);[Yang et al\., 2026b](https://arxiv.org/html/2610.31850#bib.bib10);[Yang et al\., 2027](https://arxiv.org/html/2610.31850#bib.bib9);[Yang et al\., 2021a](https://arxiv.org/html/2610.31850#bib.bib6);[Yang et al\., 2025a](https://arxiv.org/html/2610.31850#bib.bib8);[Yang et al\., 2022b](https://arxiv.org/html/2610.31850#bib.bib7);[Wei et al\., 2025](https://arxiv.org/html/2610.31850#bib.bib13);[Xu et al\., 2025b](https://arxiv.org/html/2610.31850#bib.bib15);[Yang and Yang, 2020](https://arxiv.org/html/2610.31850#bib.bib14)\), and service demand estimation\([Sun et al\., 2024](https://arxiv.org/html/2610.31850#bib.bib5);[Sun et al\., 2025a](https://arxiv.org/html/2610.31850#bib.bib1)\)\.

Given the heterogeneous nature of urban data, a central question arises:*how should multiple urban signals be integrated to construct reliable region representations?*

Structured multi\-view fusion\.Existing approaches treat each urban modality—such as mobility flow graphs, POI distributions, or land\-use maps—as an independent structured signal that is encoded using graph or attribute\-based models\. Representative methods \(e\.g\.,MGFN\([Wu et al\., 2022](https://arxiv.org/html/2610.31850#bib.bib30)\),HAFusion\([Sun et al\., 2024](https://arxiv.org/html/2610.31850#bib.bib5)\),CGAP\([Xu and Zhou, 2024](https://arxiv.org/html/2610.31850#bib.bib3)\)\) jointly learn representations from multiple modalities and fuse them by means of concatenation, attention, or weighted aggregation\. More recent studies extend this paradigm to cross\-modal settings by incorporating additional sources such as street\-view images \(e\.g\.,FlexiReg\([Sun et al\., 2025a](https://arxiv.org/html/2610.31850#bib.bib1)\)\)\. While being architecturally diverse, these methods adopt a shared abstraction: heterogeneous views are integrated via globally shared fusion mechanisms under deterministic parameterization\.

Multi\-objective optimization in multi\-view learning\.Multi\-view urban representation learning typically introduces modality\-specific objectives that guide the shared embedding towards distinct structural or semantic properties\. These objectives are then jointly optimized within a unified parameter space without explicit coordination\. Although general multi\-task strategies \(e\.g\., gradient normalization or projection\) have been explored in other domains, they assume task symmetry and do not account for the spatially heterogeneous and modality\-dependent dynamics intrinsic to urban data\.

Despite their methodological variations, existing approaches rely on two implicit assumptions:\(i\) modality importance is spatially consistent across regions, and\(ii\) joint optimization of heterogeneous objectives naturally produces coordinated and balanced representations\.

Motivating example\.To evaluate these assumptions, we analyze three city\-scale datasets \(see Figure[1](https://arxiv.org/html/2610.31850#S1.F1)\)\. Figure[1](https://arxiv.org/html/2610.31850#S1.F1)a highlights task\-dependent differences in modality utility: mobility signals are most informative for crime prediction, whereas land\-use and POI features are more useful for service call prediction, suggesting differing underlying spatial dynamics\. Notably, as highlighted by the red dashed oval, POI and land\-use features contribute negligibly to the SF Crime task\. This pronounced heterogeneity challenges the assumption of globally uniform fusion across modalities\. Figure[1](https://arxiv.org/html/2610.31850#S1.F1)b further reveals a systematic optimization imbalance under conventional joint training \(e\.g\., the red dashed oval\)\. Gradients associated with different modality\-specific objectives exhibit persistent magnitude disparities, allowing dominant modalities to steer the shared embedding space while suppressing weaker yet informative signals\. Such imbalance induces representational bias and ultimately degrades cross\-region generalization\.

![Refer to caption](https://arxiv.org/html/2610.31850v2/Fig1Example.png)Figure 1\.Motivation Example on San Francisco Dataset\.Following this analysis, we identify two coupled research challenges that remain unresolved in urban region representation learning:

Challenge 1 \(C1\): Spatially adaptive view integration\.The predictive utility of each modality is inherently region\-dependent, and yet existing fusion mechanisms assign contributions through globally shared parameters\. This mismatch between local data characteristics and global fusion strategies incurs representational bias—commercial regions may be well\-served while residential regions may be systematically under\-represented\. The challenge is to enable*region\-aware, adaptive modality selection*that aligns representation capacity with spatial heterogeneity, without requiring region\-specific supervision\.

Challenge 2 \(C2\): Coordinated optimization under heterogeneous objectives\.When multiple modality\-specific objectives with different gradient scales, noise levels, and convergence rates are jointly optimized in a shared embedding space, objectives associated with certain modalities may contribute disproportionately large or more stable gradients, causing them to dominate the optimization process and suppress weaker modality signals, which in turn leads to imbalanced representations and unstable training\. The challenge is to*explicitly regulate the interplay among competing objectives*so that all informative modalities contribute proportionally to the final representation\.

To address the two challenges, we proposePRISM\(Per\-RegionIntegration withSelf\-regulatingMulti\-view coordination\), a unified framework that redefines multi\-view urban representation learning as an*adaptive coordination problem*rather than as a static aggregation procedure\. In particular,PRISMbuilds on an existing multi\-view encoding backbone and focuses on two components: region\-level sparse view routing and PID\-inspired discrepancy regulation\.

Region\-level adaptive routing via sparse MoE\.For modeling,PRISMinstantiates region\-level view allocation with a standard sparse MoE routing design\. Our goal is not to introduce a new generic MoE routing operator, but to adapt sparse routing to heterogeneous urban view fusion\. Different from token\-level MoE routing over homogeneous sequence elements, the routed unit inPRISMis a region\-view representation derived from mobility, POI, or land\-use signals\. The router dynamically selects and prioritizes informative views for each region, enabling localized view specialization while suppressing redundant or noisy signals\. This addressesC1by aligning view contributions with the spatial heterogeneity observed in Figure[1](https://arxiv.org/html/2610.31850#S1.F1)\(a\)\.

PID\-inspired multi\-objective discrepancy regulation\.For optimization,PRISMintroduces a PID\-inspired discrepancy regularizer to coordinate heterogeneous view\-induced objectives\. We borrow the proportional–integral–derivative decomposition to model complementary discrepancy signals during training\. Specifically, the proportional term captures the current discrepancy, the integral term captures accumulated bias, and the derivative term captures short\-term variation\. These terms are used as loss\-level regularization penalties rather than as a single control signal acting on the system\. This transforms multi\-view learning from implicit joint optimization into an explicitly regularized coordination process, addressing the gradient imbalance identified inC2\.

In summary, the main contributions are as follows:

- •Region\-Level Adaptive View Allocation\.We formulate multi\-view urban representation learning as a region\-level adaptive view allocation problem, where different regions may rely on different subsets of mobility, POI, and land\-use signals\. We instantiate this formulation with sparse MoE routing to enable selective view integration at the region granularity\.
- •PID\-Inspired Optimization Coordination\.We formulate multi\-view representation learning as an adaptive optimization coordination problem and introduce a PID\-inspired discrepancy regularizer that captures current discrepancy, accumulated bias, and local variation\. The final loss aggregation adopts uncertainty\-based weighting to adaptively balance heterogeneous objectives, while the proposed contribution lies in PID\-inspired discrepancy modeling rather than the weighting scheme\. This design mitigates gradient conflicts and stabilizes training under heterogeneous view\-induced objectives\.
- •Comprehensive Empirical Validation\.Extensive experiments on three large\-scale urban datasets across multiple downstream tasks evaluate the effectiveness, robustness, and efficiency ofPRISM\.

## 2\.Related Work

### 2\.1\.Urban Region Representation Learning

Urban region representation learning aims to derive compact embeddings that encode functional, structural, and interaction patterns of spatial regions\. From a data modeling perspective, existing work can be broadly categorized intosingle\-source modelingandheterogeneous multi\-source modeling, according to how urban signals are organized and integrated\.

Single\-source approaches learn region embeddings from a dominant urban signal\([Yao et al\., 2018](https://arxiv.org/html/2610.31850#bib.bib33);[Wu et al\., 2022](https://arxiv.org/html/2610.31850#bib.bib30)\), such as mobility flows, POI distributions, or land\-use attributes\. For exampleZE\-Mob\([Yao et al\., 2018](https://arxiv.org/html/2610.31850#bib.bib33)\)encodes a mobility transition graph using skip\-grams\. This type of method typically constructs region\-level graphs or attribute matrices and apply graph representation learning, random\-walk\-based embedding, or matrix factorization techniques to obtain compact representations\. Some studies extend this paradigm by extracting multiple complementary views from the same modality, e\.g\., using temporal slicing of mobility flows or structural and semantic similarity graphs, and then aggregate these views, the goal being to enhance robustness\([Zhou et al\., 2023](https://arxiv.org/html/2610.31850#bib.bib29);[Zhang et al\., 2020](https://arxiv.org/html/2610.31850#bib.bib4)\)\. While such designs enrich representations based on a single modality, the learned embeddings remain shaped by the inductive bias and information scope of the specific modality\.

To increase representational completeness, more recent studies\([Sun et al\., 2024](https://arxiv.org/html/2610.31850#bib.bib5);[Li et al\., 2023c](https://arxiv.org/html/2610.31850#bib.bib2);[Sun et al\., 2025a](https://arxiv.org/html/2610.31850#bib.bib1)\)integrate heterogeneous urban signals, including mobility, POIs, land use, socio\-economic attributes, and environmental features\. These methods seek to capture complementary information from different modalities by constructing multi\-graph architectures, cross\-attention mechanisms, shared latent spaces, or fusion\-based embedding frameworks\. Employing joint optimization, they aim to produce unified region representations that reflect multiple aspects of urban dynamics\. Modality interactions are typically modeled through feature concatenation or shared attention mechanisms, resulting in a single embedding embodying information from all available views\.

### 2\.2\.Mixture\-of\-Experts and Adaptive Routing

Mixture\-of\-experts \(MoE\) models enable conditional computation by routing inputs to specialized experts through learned gating functions\([Li et al\., 2023a](https://arxiv.org/html/2610.31850#bib.bib42);[Nguyen et al\., 2024a](https://arxiv.org/html/2610.31850#bib.bib43)\)\. They have been widely adopted in large\-scale neural models to increase model capacity while maintaining computational efficiency\([Nie et al\., 2023](https://arxiv.org/html/2610.31850#bib.bib44);[Lu et al\., 2024](https://arxiv.org/html/2610.31850#bib.bib45)\)\.

Existing MoE routing strategies mainly differ in sparsity and expert\-allocation design\.Switch Transformer\([Fedus et al\., 2022](https://arxiv.org/html/2610.31850#bib.bib35)\)performs sparse token\-level routing by assigning each token to selected feed\-forward experts, whileGShard\([Lepikhin et al\., 2021](https://arxiv.org/html/2610.31850#bib.bib36)\)adopts top\-kkrouting with auxiliary balancing losses\.Expert Choice\([Zhou et al\., 2022](https://arxiv.org/html/2610.31850#bib.bib37)\)reverses the assignment direction to improve expert utilization, andSoft MoE\([Puigcerver et al\., 2024](https://arxiv.org/html/2610.31850#bib.bib38)\)replaces discrete routing with continuous mixture weights\. These methods are primarily developed for homogeneous token streams and emphasize scalability, load balancing, and efficient conditional computation\.

Recent studies extend MoE to multi\-modal and spatio\-temporal settings\. In multi\-modal learning, methods such asMoME\([Shen et al\., 2024](https://arxiv.org/html/2610.31850#bib.bib39)\)andVLMo\([Bao et al\., 2022](https://arxiv.org/html/2610.31850#bib.bib40)\)activate experts according to modality or layer\-level signals\. In spatio\-temporal forecasting,ST\-MoE\([Li et al\., 2023b](https://arxiv.org/html/2610.31850#bib.bib41)\)routes temporal inputs to specialized experts to capture temporal heterogeneity\. MoE has also been explored in urban computing\. For example,UrbanMoE\([Liu et al\., 2026](https://arxiv.org/html/2610.31850#bib.bib57)\)applies sparse multi\-modal expert routing to multi\-task urban region profiling, using satellite imagery, textual descriptions, and POI features to predict multiple urban indicators\.

Different from these studies,PRISMfocuses on reusable urban region representation learning from mobility, POI, and land\-use views\. Its goal is not to introduce a new generic MoE routing operator, but to adapt standard sparse MoE routing to region\-level heterogeneous view allocation\. In this setting, the routed unit is a region\-view representation, and the routing decision determines which urban views should contribute to each region\. This differs from token\-level routing in large language models, modality\-aware routing in multi\-modal learning, temporal routing in forecasting, and task\-aware expert routing in urban profiling\.

Recent meta\-learning studies further explore spatial adaptation in urban prediction, such as region\-adaptive prediction with region profiles and few\-shot transfer from data\-sufficient cities to data\-scarce cities\([Chen et al\., 2023](https://arxiv.org/html/2610.31850#bib.bib55);[Sun et al\., 2023](https://arxiv.org/html/2610.31850#bib.bib56)\)\. These methods mainly adapt prediction models under limited observations, whereasPRISMstudies region\-level coordination among structurally distinct urban views within a unified representation learning framework\.

### 2\.3\.Adaptive Multi\-Task Learning and PID\-Inspired Regulation

Adaptive multi\-task learning \(MTL\) jointly optimizes multiple objectives while mitigating task imbalance and gradient interference\. Representative methods includeGradNorm\([Chen et al\., 2018](https://arxiv.org/html/2610.31850#bib.bib52)\), which balances task training rates through gradient magnitude normalization;PCGrad\([Yu et al\., 2020](https://arxiv.org/html/2610.31850#bib.bib51)\), which projects conflicting gradients; andMGDA\([Désidéri, 2012](https://arxiv.org/html/2610.31850#bib.bib54);[Sener and Koltun, 2018](https://arxiv.org/html/2610.31850#bib.bib53)\), which seeks a common descent direction toward Pareto\-stationary solutions\. These methods mainly operate on instantaneous task gradients and are typically designed for conventional MTL settings with comparable supervised tasks\.

Several recent studies incorporate PID principles into neural learning systems, but at different levels\.PIDformer\([Nguyen et al\., 2024b](https://arxiv.org/html/2610.31850#bib.bib59)\)introduces a PID closed\-loop feedback signal into transformer state\-space dynamics to improve robustness and mitigate rank collapse\. PID\-based stochastic optimization\([An et al\., 2018](https://arxiv.org/html/2610.31850#bib.bib58)\)treats gradients as control errors and updates network parameters using current, accumulated, and changing gradient information\.

Different from these methods,PRISMaddresses heterogeneous view\-induced objectives in urban representation learning, where mobility models origin–destination interactions, while POI and land\-use preserve view\-specific similarity structures\.PRISMdoes not apply PID to attention dynamics or optimizer states\. Instead, it introduces a PID\-inspired discrepancy regularizer over view\-induced objectives, where the proportional, integral, and derivative terms model current discrepancy, accumulated bias, and short\-term variation during training\. We further includeGradNorm,PCGrad, andMGDAas controlled baselines under the same backbone to empirically examine this distinction\.

![Refer to caption](https://arxiv.org/html/2610.31850v2/Framework.png)Figure 2\.ThePRISMframework\. The multi\-view encoding block follows theHAFusion\-style backbone;PRISMintroduces region\-level adaptive routing and PID\-inspired discrepancy regularization\. GT denotes ground truth\.

## 3\.Preliminaries

We first cover important concepts that underlie the paper’s proposal and then state the problem addressed\.

### 3\.1\.Definitions

###### Definition 1\(Region\)\.

Given a geographical area𝒜\\mathcal\{A\}, we define a region setℛ=\{ℛ1,…,ℛN\}\\mathcal\{R\}=\\\{\\mathcal\{R\}\_\{1\},\\dots,\\mathcal\{R\}\_\{N\}\\\}as a spatial partition of𝒜\\mathcal\{A\}such thatℛi∩ℛj=∅\\mathcal\{R\}\_\{i\}\\cap\\mathcal\{R\}\_\{j\}=\\emptysetfori≠ji\\neq jand⋃i=1Nℛi=𝒜\\bigcup\_\{i=1\}^\{N\}\\mathcal\{R\}\_\{i\}=\\mathcal\{A\}\. For each region, we consider three types of heterogeneous features to construct its representation\.

###### Definition 2\(Human Mobility Feature\)\.

Given a region setℛ=\{ℛ1,…,ℛn\}\\mathcal\{R\}=\\\{\\mathcal\{R\}\_\{1\},\\dots,\\mathcal\{R\}\_\{n\}\\\}, a human mobility feature is a weighted directed graph𝒢M=\(ℛ,𝐌\)\\mathcal\{G\}\_\{M\}=\(\\mathcal\{R\},\\mathbf\{M\}\), where𝐌∈ℝn×n\\mathbf\{M\}\\in\\mathbb\{R\}^\{n\\times n\}is a mobility matrix andmi,jm\_\{i,j\}denotes the flow volume fromℛi\\mathcal\{R\}\_\{i\}toℛj\\mathcal\{R\}\_\{j\}within an observation period\. Given a regionℛi\\mathcal\{R\}\_\{i\}, its mobility feature vectormim\_\{i\}is defined as its outgoing flow vector𝐌i,:\\mathbf\{M\}\_\{i,:\}\.

###### Definition 3\(POI Feature\)\.

Given a region setℛ=\{ℛ1,…,ℛn\}\\mathcal\{R\}=\\\{\\mathcal\{R\}\_\{1\},\\dots,\\mathcal\{R\}\_\{n\}\\\}, a point\-of\-interest \(POI\) feature is a matrix𝐏∈ℝn×dP\\mathbf\{P\}\\in\\mathbb\{R\}^\{n\\times d\_\{P\}\}, wheredPd\_\{P\}denotes the number of POI categories\. Each entrypi,kp\_\{i,k\}records the frequency \(or normalized count\) of POIs inℛi\\mathcal\{R\}\_\{i\}belonging to categorykk\. The POI feature vector𝐩i\\mathbf\{p\}\_\{i\}of regionℛi\\mathcal\{R\}\_\{i\}is defined as𝐏i,:∈ℝdP\\mathbf\{P\}\_\{i,:\}\\in\\mathbb\{R\}^\{d\_\{P\}\}\.

###### Definition 4\(Land\-Use Feature\)\.

Given a region setℛ=\{ℛ1,…,ℛn\}\\mathcal\{R\}=\\\{\\mathcal\{R\}\_\{1\},\\dots,\\\\ \\mathcal\{R\}\_\{n\}\\\}, a land\-use feature is a matrix𝐋∈ℝn×dL\\mathbf\{L\}\\in\\mathbb\{R\}^\{n\\times d\_\{L\}\}, wheredLd\_\{L\}denotes the number of land\-use categories\. Each entryli,kl\_\{i,k\}records the frequency \(or proportion\) of land parcels inℛi\\mathcal\{R\}\_\{i\}that belong to categorykk\. The land\-use feature vector𝐥i\\mathbf\{l\}\_\{i\}of regionℛi\\mathcal\{R\}\_\{i\}is defined as𝐋i,:∈ℝdL\\mathbf\{L\}\_\{i,:\}\\in\\mathbb\{R\}^\{d\_\{L\}\}\.

###### Definition 5\(Heterogeneous Multi\-View Feature\)\.

Given a region setℛ=\{ℛ1,…,ℛn\}\\mathcal\{R\}=\\\{\\mathcal\{R\}\_\{1\},\\dots,\\mathcal\{R\}\_\{n\}\\\}, we associate each region withVVheterogeneous feature views:

\(1\)𝒳=\{𝐗\(1\),𝐗\(2\),…,𝐗\(V\)\},\\mathcal\{X\}=\\\{\\mathbf\{X\}^\{\(1\)\},\\mathbf\{X\}^\{\(2\)\},\\dots,\\mathbf\{X\}^\{\(V\)\}\\\},where each view\-specific feature matrix𝐗\(v\)∈ℝn×dv\\mathbf\{X\}^\{\(v\)\}\\in\\mathbb\{R\}^\{n\\times d\_\{v\}\}represents region\-level attributes under viewvv\. For a regionℛi\\mathcal\{R\}\_\{i\}, its feature vector under viewvvis𝐱i\(v\)=𝐗\(v\)i,:∈ℝdv\\mathbf\{x\}\_\{i\}^\{\(v\)\}=\\mathbf\{X\}^\{\(v\)\}\_\{i,:\}\\in\\mathbb\{R\}^\{d\_\{v\}\}\. In our setting,V=3V=3, corresponding to mobility, POI, and land\-use views\.

###### Definition 6\(Gradient Stability Score\)\.

Let\{ℒv​\(θ\)\}v=1V\\\{\\mathcal\{L\}\_\{v\}\(\\theta\)\\\}\_\{v=1\}^\{V\}denote modality\-specific objectives sharing parametersθ\\theta\. At thett\-th optimization step, letgt\(v\)=∇θℒv​\(θt\)g\_\{t\}^\{\(v\)\}=\\nabla\_\{\\theta\}\\mathcal\{L\}\_\{v\}\(\\theta\_\{t\}\)\. We define the gradient stability score as follows:

\(2\)St=log⁡maxv⁡‖gt\(v\)‖\+ϵgminv⁡‖gt\(v\)‖\+ϵg\+λc​1\|𝒫\|​∑\(i,j\)∈𝒫max⁡\(0,−cos⁡\(gt\(i\),gt\(j\)\)\),S\_\{t\}=\\log\\frac\{\\max\_\{v\}\\\|g\_\{t\}^\{\(v\)\}\\\|\+\\epsilon\_\{g\}\}\{\\min\_\{v\}\\\|g\_\{t\}^\{\(v\)\}\\\|\+\\epsilon\_\{g\}\}\+\\lambda\_\{c\}\\frac\{1\}\{\|\\mathcal\{P\}\|\}\\sum\_\{\(i,j\)\\in\\mathcal\{P\}\}\\max\\\!\\left\(0,\-\\cos\\\!\\left\(g\_\{t\}^\{\(i\)\},g\_\{t\}^\{\(j\)\}\\right\)\\right\),whereϵg\>0\\epsilon\_\{g\}\>0is a small numerical floor that prevents the ratio from becoming unbounded when a modality gradient vanishes\.𝒫\\mathcal\{P\}denotes all unordered modality pairs andλc\>0\\lambda\_\{c\}\>0balances gradient magnitude imbalance and gradient conflict\. Intuitively, the first term measures gradient magnitude imbalance, while the second term captures directional conflicts across modalities\. A smallerStS\_\{t\}indicates more balanced and aligned multi\-objective optimization\. When all modality gradients have identical norms and non\-negative pairwise cosine similarity, both imbalance and conflict terms vanish, resulting inSt=0S\_\{t\}=0\. Moreover, any update rule that reduces gradient norm disparity or alleviates pairwise conflicts decreases the stability score monotonically\. Therefore,StS\_\{t\}serves as a principled metric for quantifying improvements in gradient coordination\.

###### Definition 7\(Self\-Attention\)\.

Given an input matrix𝐗∈ℝn×d\\mathbf\{X\}\\in\\mathbb\{R\}^\{n\\times d\}, self\-attention computes contextualized representations by modeling pairwise interactions among its elements\. Let

\(3\)𝐐=𝐗𝐖Q,𝐊=𝐗𝐖K,𝐕=𝐗𝐖V,\\mathbf\{Q\}=\\mathbf\{X\}\\mathbf\{W\}\_\{Q\},\\quad\\mathbf\{K\}=\\mathbf\{X\}\\mathbf\{W\}\_\{K\},\\quad\\mathbf\{V\}=\\mathbf\{X\}\\mathbf\{W\}\_\{V\},where𝐖Q,𝐖K,𝐖V∈ℝd×d\\mathbf\{W\}\_\{Q\},\\mathbf\{W\}\_\{K\},\\mathbf\{W\}\_\{V\}\\in\\mathbb\{R\}^\{d\\times d\}are learnable parameters\. The self\-attention operation is defined as follows:

\(4\)Attn⁡\(𝐗\)=Softmax⁡\(𝐐𝐊⊤d\)​𝐕\.\\mathrm\{Attn\}\(\\mathbf\{X\}\)=\\mathrm\{Softmax\}\\\!\\left\(\\frac\{\\mathbf\{Q\}\\mathbf\{K\}^\{\\top\}\}\{\\sqrt\{d\}\}\\right\)\\mathbf\{V\}\.

### 3\.2\.Problem Statement

Given a set of urban regionsℛ=\{R1,…,RN\}\\mathcal\{R\}=\\\{R\_\{1\},\\ldots,R\_\{N\}\\\}andVVheterogeneous urban views𝒳=\{X\(1\),…,X\(V\)\}\\mathcal\{X\}=\\\{X^\{\(1\)\},\\ldots,X^\{\(V\)\}\\\}, whereX\(v\)∈ℝN×dvX^\{\(v\)\}\\in\\mathbb\{R\}^\{N\\times d\_\{v\}\}denotes the feature matrix of viewvv, our goal is to learn a unified region representation matrixH∈ℝN×dH\\in\\mathbb\{R\}^\{N\\times d\}for downstream urban prediction tasks\.

Unlike conventional multi\-view fusion that applies a fixed fusion rule to all regions, we aim to learn region\-adaptive representations\. Specifically, each regionRiR\_\{i\}should be allowed to assign different importance to different views:

\(5\)hi=Φ⁡\(∑v=1Vαi,v​zi\(v\)\),∑v=1Vαi,v=1,αi,v≥0,h\_\{i\}=\\Phi\\left\(\\sum\_\{v=1\}^\{V\}\\alpha\_\{i,v\}z\_\{i\}^\{\(v\)\}\\right\),\\quad\\sum\_\{v=1\}^\{V\}\\alpha\_\{i,v\}=1,\\quad\\alpha\_\{i,v\}\\geq 0,wherezi\(v\)z\_\{i\}^\{\(v\)\}is the view\-specific embedding of regionRiR\_\{i\}, andαi,v\\alpha\_\{i,v\}denotes the region\-specific contribution of viewvv\. This formulation captures the spatial heterogeneity of urban regions: different regions may depend on different views due to their distinct functional, mobility, and land\-use characteristics\.

## 4\.Methodology

Figure[2](https://arxiv.org/html/2610.31850#S2.F2)presents the overall architecture ofPRISM\. The multi\-view encoding component follows theHAFusion\-style backbone and is used to encode mobility, POI, and land\-use data defined over shared spatial regions\. On top of this backbone,PRISMintroduces region\-level adaptive routing to allocate view contributions for each region\. It further incorporates PID\-inspired discrepancy regularization to mitigate gradient imbalance and cross\-objective conflicts among heterogeneous view\-induced objectives\.

### 4\.1\.Multi\-View Encoding

View\-Specific Encoding\.Given heterogeneous urban features, we first map each modality into a shared latent space of dimensionalitydd\. We employ an MLP \(multilayer perceptron\) layer to project POI and land\-use features intodd\-dimensional embeddings, whereas mobility features are inherently represented in the same embedding space\. Let𝐗\(v\)∈ℝN×d\\mathbf\{X\}^\{\(v\)\}\\in\\mathbb\{R\}^\{N\\times d\}denote the initial embedding of modalityv∈\{mob,poi,land\}v\\in\\\{\\text\{mob\},\\text\{poi\},\\text\{land\}\\\}overNNregions\.

To capture modality\-specific structural dependencies, we use an intra\-view encoder based on stacked self\-attention layers:

\(6\)𝐙\(v\)=IntraEncoder⁡\(𝐗\(v\)\),\\mathbf\{Z\}^\{\(v\)\}=\\mathrm\{IntraEncoder\}\(\\mathbf\{X\}^\{\(v\)\}\),
Within each modality, multi\-head self\-attention is applied to capture long\-range regional dependencies\. This step ensures that the intrinsic topological and semantic properties of each view are retained prior to cross\-modal fusion\.

Inter\-View Interaction\.While intra\-view encoding captures moda lity\-specific structural characteristics, urban functionality typically arises from the interaction among heterogeneous signals\. To facilitate cross\-modal interaction, we organize modality embeddings into a unified representation framework:

\(7\)𝐙∈ℝV×N×d,\\mathbf\{Z\}\\in\\mathbb\{R\}^\{V\\times N\\times d\},whereVVdenotes the number of modalities\. For each region, we apply an inter\-view attention mechanism to learn cross\-modal correlations among modality\-specific embeddings:

\(8\)𝐙~=InterEncoder⁡\(𝐙\),\\tilde\{\\mathbf\{Z\}\}=\\mathrm\{InterEncoder\}\(\\mathbf\{Z\}\),
Residual Mixing\.To balance modality\-specific structural signals and cross\-view interactions, we introduce learnable mixing coefficientsα\\alphaandβ\\beta:

\(9\)𝐙mix\(v\)=β​𝐙~\(v\)\+α​𝐙\(v\)\.\\mathbf\{Z\}^\{\(v\)\}\_\{\\text\{mix\}\}=\\beta\\tilde\{\\mathbf\{Z\}\}^\{\(v\)\}\+\\alpha\\mathbf\{Z\}^\{\(v\)\}\.This residual formulation preserves stable intra\-view representations while adaptively integrating cross\-view information, thereby providing a flexible basis for subsequent adaptive fusion\.

### 4\.2\.Region\-Level Adaptive Routing

This module instantiates the standard sparse MoE routing paradigm for region\-level heterogeneous view allocation\. We do not aim to introduce a new generic routing operator; rather, we adapt learned gating and top\-kksparse selection to the urban multi\-view setting, where each routing decision determines which view should contribute to a specific region\.

Given the mixed multi\-view representations\{𝐙mix\(v\)\}v=1V\\\{\\mathbf\{Z\}^\{\(v\)\}\_\{\\mathrm\{mix\}\}\\\}\_\{v=1\}^\{V\}, where𝐙mix\(v\)∈ℝN×d\\mathbf\{Z\}^\{\(v\)\}\_\{\\mathrm\{mix\}\}\\in\\mathbb\{R\}^\{N\\times d\}, we perform region\-level adaptive fusion to integrate heterogeneous views\.

Routing weight computation\.For regioniiand viewvv, let𝐙mix,i\(v\)∈ℝd\\mathbf\{Z\}^\{\(v\)\}\_\{\\mathrm\{mix\},i\}\\in\\mathbb\{R\}^\{d\}denote the corresponding mixed representation\. A lightweight gating network computes a routing logit as follows:

\(10\)sv,i=𝐰2⊤​σ​\(𝐖1​𝐙mix,i\(v\)\+𝐛1\)\+b2,s\_\{v,i\}=\\mathbf\{w\}\_\{2\}^\{\\top\}\\sigma\\\!\\left\(\\mathbf\{W\}\_\{1\}\\mathbf\{Z\}^\{\(v\)\}\_\{\\mathrm\{mix\},i\}\+\\mathbf\{b\}\_\{1\}\\right\)\+b\_\{2\},where𝐖1\\mathbf\{W\}\_\{1\},𝐰2\\mathbf\{w\}\_\{2\},𝐛1\\mathbf\{b\}\_\{1\}, andb2b\_\{2\}are learnable parameters, andσ⁡\(⋅\)\\sigma\(\\cdot\)is the ReLU activation function\. Let𝒯i\\mathcal\{T\}\_\{i\}denote the indices of the top\-kkviews for regioniiaccording to the logits\{sv,i\}v=1V\\\{s\_\{v,i\}\\\}\_\{v=1\}^\{V\}\. The routing weight is computed by a masked softmax:

\(11\)αv,i=𝕀⁡\(v∈𝒯i\)​exp⁡\(sv,i/τ\)∑v′∈𝒯iexp⁡\(sv′,i/τ\),\\alpha\_\{v,i\}=\\frac\{\\mathbb\{I\}\(v\\in\\mathcal\{T\}\_\{i\}\)\\exp\(s\_\{v,i\}/\\tau\)\}\{\\sum\_\{v^\{\\prime\}\\in\\mathcal\{T\}\_\{i\}\}\\exp\(s\_\{v^\{\\prime\},i\}/\\tau\)\},whereτ\>0\\tau\>0controls the sharpness of the routing distribution\. When collecting all routing weights, we obtain a routing matrix𝜶∈ℝV×N\\boldsymbol\{\\alpha\}\\in\\mathbb\{R\}^\{V\\times N\}\.

Adaptive fusion\.The fused representation of regioniiis given by:

\(12\)𝐇i=∑v=1Vαv,i​𝐙mix,i\(v\)\.\\mathbf\{H\}\_\{i\}=\\sum\_\{v=1\}^\{V\}\\alpha\_\{v,i\}\\mathbf\{Z\}^\{\(v\)\}\_\{\\mathrm\{mix\},i\}\.
Routing sparsity regularization\.To encourage region\-level selective view allocation, we introduce an optional entropy\-based sparsity regularizer:

\(13\)ℒsparse=1N​∑i=1Nℋ⁡\(𝜶i\),\\mathcal\{L\}\_\{\\mathrm\{sparse\}\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathcal\{H\}\(\\boldsymbol\{\\alpha\}\_\{i\}\),where𝜶i=\[α1,i,…,αV,i\]⊤\\boldsymbol\{\\alpha\}\_\{i\}=\[\\alpha\_\{1,i\},\\dots,\\alpha\_\{V,i\}\]^\{\\top\}is the routing distribution of regionii, andℋ⁡\(⋅\)\\mathcal\{H\}\(\\cdot\)denotes Shannon entropy\. Minimizingℒsparse\\mathcal\{L\}\_\{\\mathrm\{sparse\}\}encourages sharper region\-specific view selection and reduces overly uniform cross\-view mixing\.

### 4\.3\.PID\-Inspired Discrepancy Regularization

While the proposed routing mechanism enables flexible multi\-view integration, the resulting optimization process may still suffer from gradient imbalance and oscillatory discrepancy trajectories\. Inspired by the proportional–integral–derivative decomposition in control theory, we introduce a PID\-inspired discrepancy regularizer\. We emphasize that this formulation is not a classical PID controller: instead of combining P/I/D terms into a single control signal, it penalizes the squared proportional, accumulated, and differential discrepancies as separate loss components\. Thus, PID\-MSE should be understood as a loss\-level regularization objective rather than a structurally equivalent PID control law\.

PID\-MSE Formulation\.Letθ\\thetadenote the shared backbone parameters\. PID\-MSE is applied only to similarity\-based views, i\.e\.,v∈\{poi,land\}v\\in\\\{\\mathrm\{poi\},\\mathrm\{land\}\\\}\. For these views, we define the discrepancy as:

\(14\)Et\(v\)=A^t\(v\)−A\(v\),E\_\{t\}^\{\(v\)\}=\\hat\{A\}\_\{t\}^\{\(v\)\}\-A^\{\(v\)\},whereA\(v\)A^\{\(v\)\}denotes the target region–region similarity matrix constructed from view\-specific attributes, andA^t\(v\)\\hat\{A\}\_\{t\}^\{\(v\)\}denotes the predicted similarity matrix at training steptt\. Instead of minimizing only the instantaneous discrepancy‖Et\(v\)‖22\\\|E\_\{t\}^\{\(v\)\}\\\|\_\{2\}^\{2\}, as in standard mean squared error \(MSE\) objectives, we introduce a PID\-inspired regularization term that incorporates proportional, integral, and derivative components:

\(15\)Pt\(v\)\\displaystyle P\_\{t\}^\{\(v\)\}=Et\(v\),It\(v\)=λ​It−1\(v\)\+Et\(v\),\\displaystyle=E\_\{t\}^\{\(v\)\},I\_\{t\}^\{\(v\)\}=\\lambda I\_\{t\-1\}^\{\(v\)\}\+E\_\{t\}^\{\(v\)\},\(16\)Dt\(v\)\\displaystyle D\_\{t\}^\{\(v\)\}=Et\(v\)−Et−1\(v\),\\displaystyle=E\_\{t\}^\{\(v\)\}\-E\_\{t\-1\}^\{\(v\)\},We then define the PID\-regularized objective as follows:

\(17\)ℒPID\(v\)=α​‖Pt\(v\)‖22\+β​‖It\(v\)‖22\+γ​‖Dt\(v\)‖22,\\mathcal\{L\}\_\{\\mathrm\{PID\}\}^\{\(v\)\}=\\alpha\\\|P\_\{t\}^\{\(v\)\}\\\|\_\{2\}^\{2\}\+\\beta\\\|I\_\{t\}^\{\(v\)\}\\\|\_\{2\}^\{2\}\+\\gamma\\\|D\_\{t\}^\{\(v\)\}\\\|\_\{2\}^\{2\},whereα\>0\\alpha\>0,β,γ≥0\\beta,\\gamma\\geq 0, andλ∈\(0,1\]\\lambda\\in\(0,1\]\. Whenβ=γ=0\\beta=\\gamma=0, PID\-MSE reduces to the standard MSE objective\.

The proportional term enforces instantaneous alignment, the integral term captures accumulated discrepancies over time, and the derivative term penalizes rapid discrepancy variations\. Together, these terms reshape the optimization dynamics by stabilizing gradient trajectories and mitigating oscillatory updates\. From a multi\-objective perspective, PID\-MSE exposes the optimization process to instantaneous, accumulated, and differential discrepancy information, which can help reduce gradient imbalance and oscillatory updates\. This intended regularization effect is empirically examined by the stability analysis in Fig\.[7\(b\)](https://arxiv.org/html/2610.31850#S5.F7.sf2)and Section[5\.7](https://arxiv.org/html/2610.31850#S5.SS7)\.

Stability AnalysisWe next analyze how each term of the PID objective influences the optimization dynamics\. Specifically, we provide a local optimization interpretation showing how the proportional term encourages discrepancy reduction, the derivative term penalizes rapid discrepancy changes, and the integral term captures persistent bias\. Together, these effects explain the empirical stability improvements observed in Fig\. 1\(b\)\.

###### Proposition 4\.1 \(Proportional TermPP: Local Discrepancy Reduction\)\.

Assume thatE⁡\(θ\)E\(\\theta\)is continuously differentiable and locallyLL\-smooth\. For a sufficiently small learning rateη\\eta, minimizingα​‖Et‖22\\alpha\\\|E\_\{t\}\\\|\_\{2\}^\{2\}ensures‖Et\+1‖2≤\(1−c​η\)​‖Et‖2\\\|E\_\{t\+1\}\\\|\_\{2\}\\leq\(1\-c\\eta\)\\\|E\_\{t\}\\\|\_\{2\}for somec\>0c\>0\.

##### Proof sketch\.

Since∇θ‖Et‖22=2​Jt⊤​Et\\nabla\_\{\\theta\}\\\|E\_\{t\}\\\|\_\{2\}^\{2\}=2J\_\{t\}^\{\\top\}E\_\{t\}, the update direction aligns with the instantaneous discrepancy\. Under local smoothness and smallη\\eta, this induces a contraction mapping onEtE\_\{t\}\. ∎

###### Proposition 4\.2 \(Derivative TermDD: Oscillation Suppression\)\.

Under the same assumptions, augmenting the objective withγ​‖Et−Et−1‖22\\gamma\\\|E\_\{t\}\-E\_\{t\-1\}\\\|\_\{2\}^\{2\}\(γ\>0\\gamma\>0\) reduces the upper bound of discrepancy variation‖Et\+1−Et‖2\\\|E\_\{t\+1\}\-E\_\{t\}\\\|\_\{2\}\.

##### Proof sketch\.

The derivative regularizer introduces a gradient component proportional to\(Et−Et−1\)\(E\_\{t\}\-E\_\{t\-1\}\), penalizing rapid discrepancy changes\. This produces a damping effect in the discrepancy trajectory, thereby suppressing oscillatory updates\. ∎

###### Proposition 4\.3 \(Integral TermII: Bias Correction\)\.

If the discrepancy exhibits a persistent non\-zero average, then forβ\>0\\beta\>0andλ∈\(0,1\]\\lambda\\in\(0,1\], the integral state accumulates this bias and induces a sustained corrective gradient component\.

##### Proof sketch\.

ItI\_\{t\}is an exponentially weighted accumulation of past discrepancies\. Persistent bias prevents cancellation, so‖It‖\\\|I\_\{t\}\\\|remains non\-zero, producing a non\-vanishing corrective signal\. ∎

Notably, the proportional term ensures instantaneous error reduction, the derivative term attenuates oscillatory updates, and the integral term compensates for persistent bias\([Nguyen et al\., 2024b](https://arxiv.org/html/2610.31850#bib.bib59)\)\. Collectively, these terms enhance the stability of multi\-objective optimization, which is particularly important in heterogeneous multi\-view settings where competing modality gradients often occur\.

### 4\.4\.Training Objective

Given the region embeddings𝐇∈ℝN×d\\mathbf\{H\}\\in\\mathbb\{R\}^\{N\\times d\}produced by the preceding modules, we define the overall optimization objective ofPRISM\. The model is trained with three modality\-specific losses, together with an optional routing sparsity regularizer\. Letθ\\thetadenote the collection of trainable parameters\.

Mobility objective\.Following prior work on urban flow modeling\([Fang et al\., 2021](https://arxiv.org/html/2610.31850#bib.bib47);[Ma et al\., 2025a](https://arxiv.org/html/2610.31850#bib.bib48);[Han et al\., 2025](https://arxiv.org/html/2610.31850#bib.bib50)\), we reconstruct the observed mobility matrix𝐌∈ℝN×N\\mathbf\{M\}\\in\\mathbb\{R\}^\{N\\times N\}via a bidirectional softmax likelihood\. Let𝐇s\\mathbf\{H\}^\{s\}and𝐇t\\mathbf\{H\}^\{t\}denote projected embeddings of𝐇\\mathbf\{H\}for source and target, respectively\. The mobility loss is defined as follows:

\(18\)ℒmob=−∑i,jMi,jlogpi→j−∑i,jMj,ilogpj→i,\\mathcal\{L\}\_\{\\text\{mob\}\}=\-\\sum\_\{i,j\}M\_\{i,j\}\\log p\_\{i\\rightarrow j\}\-\\sum\_\{i,j\}M\_\{j,i\}\\log p\_\{j\\rightarrow i\},wherepi→j=Softmaxj​\(𝐇is​\(𝐇jt\)⊤\),p\_\{i\\rightarrow j\}=\\mathrm\{Softmax\}\_\{j\}\\\!\\left\(\\mathbf\{H\}^\{s\}\_\{i\}\(\\mathbf\{H\}^\{t\}\_\{j\}\)^\{\\top\}\\right\),andSoftmaxj​\(⋅\)\\mathrm\{Softmax\}\_\{j\}\(\\cdot\)normalizes over all destination regionsjj\.

POI and land\-use objectives\.For attribute\-based modalities, we aim to preserve region\-level similarity structures\. For each modalityv∈\{poi,land\}v\\in\\\{\\text\{poi\},\\text\{land\}\\\}, we align the predicted similarity matrix𝐀^\(v\)\\hat\{\\mathbf\{A\}\}^\{\(v\)\}with the corresponding target similarity matrix𝐀\(v\)\\mathbf\{A\}^\{\(v\)\}by minimizing

\(19\)ℒv=ℓ⁡\(𝐀^\(v\),𝐀\(v\)\),v∈\{poi,land\},\\mathcal\{L\}\_\{v\}=\\ell\\\!\\left\(\\hat\{\\mathbf\{A\}\}^\{\(v\)\},\\mathbf\{A\}^\{\(v\)\}\\right\),\\quad v\\in\\\{\\text\{poi\},\\text\{land\}\\\},whereℓ⁡\(⋅\)\\ell\(\\cdot\)denotes the PID\-MSE loss defined in Section[4\.3](https://arxiv.org/html/2610.31850#S4.SS3)\.

Routing sparsity regularization\.When region\-level adaptive routing is enabled, we encourage selective view allocation by minimizing the entropy of routing distributions:

\(20\)ℒsparse=1N​∑i=1Nℋ⁡\(𝜶i\),\\mathcal\{L\}\_\{\\text\{sparse\}\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathcal\{H\}\(\\boldsymbol\{\\alpha\}\_\{i\}\),where𝜶i∈ℝV\\boldsymbol\{\\alpha\}\_\{i\}\\in\\mathbb\{R\}^\{V\}denotes the routing distribution of regionii\. Minimizing entropy promotes sharper, more confident modality selection, preventing overly uniform fusion and encouraging regions to give preference to the most informative views\.

Overall objective\.The final training objective combines all modality\-specific losses together with routing regularization:

\(21\)ℒ=ℒmob\+ℒpoi\+ℒland\+λsparse​ℒsparse,\\mathcal\{L\}=\\mathcal\{L\}\_\{\\text\{mob\}\}\+\\mathcal\{L\}\_\{\\text\{poi\}\}\+\\mathcal\{L\}\_\{\\text\{land\}\}\+\\lambda\_\{\\text\{sparse\}\}\\mathcal\{L\}\_\{\\text\{sparse\}\},whereλsparse≥0\\lambda\_\{\\text\{sparse\}\}\\geq 0controls the strength of region\-level routing sparsity regularization\. To balance heterogeneous objectives, we adopt uncertainty\-based weighting\([Kendall et al\., 2018](https://arxiv.org/html/2610.31850#bib.bib46);[Ma et al\., 2025b](https://arxiv.org/html/2610.31850#bib.bib49)\), yielding the final optimized objective in Eq\.[21](https://arxiv.org/html/2610.31850#S4.E21)\.

### 4\.5\.Model Complexity Analysis

LetNNdenote the number of regions,VVthe number of views, andddthe embedding dimensionality\.HAFusionandPRISMshare identical intra\-view and inter\-view encoders, whose dominant computational cost arises from self\-attention operations, resulting in a complexity ofO⁡\(V​N2​d\)O\(VN^\{2\}d\)\. For region\-level fusion,HAFusionadopts static weighted aggregation, incurringO⁡\(V​N​d\)O\(VNd\)computation\. In contrast,PRISMintroduces an additional routing module with complexityO⁡\(V​N​d​h\)O\(VNdh\), whereh≪Nh\\ll Ndenotes the router hidden dimensionality\. Sincehhis typically much smaller thanNN, the routing overhead remains negligible relative to the backbone attention cost\. Next, considering parameters,HAFusionrequiresO⁡\(V​d\)O\(Vd\)fusion weights, whereasPRISMaddsO⁡\(d​h\)O\(dh\)router parameters\. Both are minor compared to theO⁡\(d2\)O\(d^\{2\}\)parameters of Transformer blocks\. Therefore, althoughPRISMintroduces additional routing parameters, they are lightweight and serve region\-adaptive view selection rather than model scaling\. By learning region\-specific modality allocations,PRISMbetter captures spatial heterogeneity while maintaining the same asymptotic time and space complexity asHAFusion\.

Table 1\.Overall Performance on three datasets\. Symbols ’↑\\uparrow’ and ’↓\\downarrow’ indicates that high values and small values are preferred, respectively\. For each dataset, the best results are in bold/red, and the second\-best results are underlined/in blue\.MethodNYCChiSFMAE\(↓\\downarrow\)RMSE\(↓\\downarrow\)R2R^\{2\}\(↑\\uparrow\)MAE\(↓\\downarrow\)RMSE↓\\downarrowR2R^\{2\}\(↑\\uparrow\)MAE\(↓\\downarrow\)RMSE\(↓\\downarrow\)R2R^\{2\}\(↑\\uparrow\)Check\-inMVURE306\.7±8\.2499\.6±12\.90\.627 ± 0\.0191693 ± 743171\+1280\.656 ± 0\.029346\.8\+8\.7659\.3 ± 15\.70\.562 ± 0\.021MGFN292\.6±17\.1451\.8 ± 28\.10\.690 ± 0\.0401281 ± 412276 ± 860\.817 ± 0\.011310\.8 ± 9\.1542\.1 ± 17\.60\.708 ± 0\.010RDCL371\.2±10\.3495\.5 ± 15\.90\.471 ± 0\.0232427 ± 1234184 ± 1360\.402 ± 0\.042398\.8 ± 9\.9748\.1 ± 17\.80\.437 ± 0\.024HREP276\.3±11\.7448\.2 ± 17\.10\.703 ± 0\.0211679 ± 713135 ± 790\.664 ± 0\.017330\.9 ± 9\.3606\.7 ± 25\.80\.629 ± 0\.032BiST278\.9 ± 4\.7442\.6 ± 23\.50\.707 ± 0\.0011627\.6 ± 322832 ± 490\.725 ± 0\.002439\.8 ± 8\.3704\.9 ± 14\.00\.497 ± 0\.008START280\.5 ± 8\.9458\.9 ± 19\.00\.685 ± 0\.0251466 ± 453325 ± 1360\.622 ± 0\.031305\.8 ± 7\.2645\.9 ± 15\.60\.580 ± 0\.020UrbanMoE376\.2 ± 8\.2639\.8 ± 8\.30\.388 ± 0\.0162076 ± 784426 ± 1200\.331 ± 0\.036401\.3 ± 7\.8820\.1 ± 14\.50\.323 ± 0\.024HGN213\.4 ± 9\.9337\.6 ± 1\.30\.829 ± 0\.0111037 ± 252328 ± 290\.814 ± 0\.042248\.2 ± 3\.6442\.5 ± 1\.60\.802 ± 0\.012PGN219\.9 ± 3\.0363\.3 ± 5\.60\.803 ± 0\.006964 ± 272160 ± 1200\.841 ± 0\.018243\.1 ± 2\.8456\.2 ± 8\.00\.791 ± 0\.007HAF202\.8±7\.2322\.8 ± 12\.60\.844 ± 0\.012929 ± 621947 ± 750\.870 ± 0\.010233\.1 ± 9\.5429\.6 ± 28\.10\.813 ± 0\.024PRISM200\.7 ± 4\.8315\.1 ± 5\.70\.852 ± 0\.005819 ± 221677 ± 460\.904 ± 0\.006204\.0 ± 10\.2364\.2 ± 15\.00\.866 ± 0\.013CrimeMVURE67\.9 ± 1\.193\.8 ± 1\.90\.591 ± 0\.016100\.4 ± 6\.6129\.2 ± 7\.30\.461 ± 0\.062130\.3 ± 1\.7201\.7 ± 3\.20\.594 ± 0\.013MGFN70\.2 ± 2\.389\.6 ± 2\.50\.630 ± 0\.020107\.4 ± 5\.4137\.9 ± 5\.20\.386 ± 0\.047128\.4 ± 3\.3199\.9 ± 4\.30\.601 ± 0\.017RDCL98\.7 ± 3\.1127\.9 ± 5\.20\.251 ± 0\.026121\.7 ± 4\.8159\.6 ± 6\.30\.179 ± 0\.053156\.3 ± 2\.1242\.3 ± 4\.60\.413 ± 0\.021HREP62\.8 ± 2\.183\.1 ± 2\.30\.680 ± 0\.01488\.3 ± 6\.4114\.4 ± 5\.50\.578 ± 0\.041124\.4 ± 2\.3196\.9 ± 3\.90\.612 ± 0\.014BiST64\.7 ± 1\.183\.8 ± 1\.40\.679 ± 0\.01291\.3 ± 1\.8123\.2 ± 5\.60\.510 ± 0\.004173\.5 ± 4\.3241\.7 ± 6\.90\.415 ± 0\.002START72\.6 ± 1\.598\.9 ± 1\.60\.552 ± 0\.14793\.0 ± 3\.7128\.9 ± 4\.60\.464 ± 0\.039114\.4 ± 3\.5202\.1 ± 8\.00\.591 ± 0\.033UrbanMoE97\.1 ± 1\.6131\.0 ± 1\.50\.215 ± 0\.017121\.3 ± 3\.2155\.7 ± 3\.30\.218 ± 0\.032138\.7 ± 4\.2252\.8 ± 7\.30\.361 ± 0\.037HGN55\.2 ± 6\.576\.9 ± 15\.80\.729 ± 0\.00193\.5 ± 14\.1129\.7 ± 17\.80\.457 ± 0\.001103\.5 ± 6\.2176\.4 ± 12\.80\.689 ± 0\.001PGN56\.1 ± 1\.178\.8 ± 0\.20\.716 ± 0\.00283\.1 ± 1\.6117\.4 ± 1\.90\.556 ± 0\.015106\.6 ± 5\.6183\.6 ± 3\.80\.663 ± 0\.013HAF56\.1 ± 1\.376\.1 ± 2\.20\.734 ± 0\.01577\.8 ± 3\.6107\.1 ± 5\.40\.631 ± 0\.036101\.5 ± 3\.3178\.4 ± 3\.60\.682 ± 0\.013PRISM53\.9 ± 0\.373\.2 ± 0\.50\.756 ± 0\.00474\.4 ± 4\.1101\.4 ± 5\.50\.668 ± 0\.03596\.5 ± 4\.4176\.0 ± 2\.80\.694 ± 0\.010Service callMVURE1428 ± 332180 ± 460\.367 ± 0\.027190\.3 ± 9\.8266\.9 ± 12\.10\.441 ± 0\.050102\.1 ± 4\.8164\.7 ± 2\.70\.479 ± 0\.017MGFN1554 ± 812286 ± 1150\.303 ± 0\.069208\.2 ± 11\.3293\.4 ± 16\.60\.329 ± 0\.077102\.8 ± 2\.2166\.3 ± 2\.50\.468 ± 0\.021RDCL1783 ± 212597 ± 380\.103 ± 0\.026195\.7 ± 7\.6272\.1 ± 10\.10\.445 ± 0\.041116\.6 ± 2\.3196\.7 ± 3\.20\.256 ± 0\.024HREP1430 ± 292286 ± 340\.398 ± 0\.021185\.7 ± 6\.1262\.2 ± 10\.80\.468 ± 0\.022103\.4 ± 3\.2167\.4 ± 4\.60\.461 ± 0\.029BiST1458 ± 102137 ± 250\.393 ± 0\.001193\.4 ± 8\.7269\.7 ± 7\.70\.435 ± 0\.001144\.1 ± 4\.7211\.8 ± 4\.00\.355 ± 0\.018START1405 ± 342155 ± 400\.382 ± 0\.023201\.6 ± 6\.1287\.5 ± 9\.60\.358 ± 0\.04498\.2 ± 3\.3174\.9 ± 2\.50\.412 ± 0\.017UrbanMoE1777 ± 192595 ± 170\.104 ± 0\.011243\.6 ± 6\.7324\.9 ± 7\.10\.180 ± 0\.036111\.0 ± 2\.5196\.9 ± 2\.40\.255 ± 0\.019HGN1273 ± 101957 ± 210\.499 ± 0\.000190\.6 ± 6\.8270\.9 ± 14\.40\.429 ± 0\.00284\.9 ± 8\.8147\.0 ± 2\.90\.584 ± 0\.011PGN1327 ± 72057 ± 220\.437 ± 0\.012190\.2 ± 5\.3269\.5 ± 7\.60\.436 ± 0\.03287\.9 ± 2\.0153\.1 ± 3\.10\.550 ± 0\.018HAF1273 ± 201951 ± 270\.493 ± 0\.014159\.3 ± 13\.9222\.0 ± 18\.90\.613 ± 0\.06781\.5 ± 2\.5142\.1 ± 3\.20\.612 ± 0\.018PRISM1268 ± 1\.01937 ± 160\.501 ± 0\.008149\.3 ± 3\.1212\.3 ± 4\.20\.650 ± 0\.01478\.9 ± 3\.1134\.6 ± 10\.20\.651 ± 0\.052

## 5\.Experimental Study

### 5\.1\.Settings

Datasets\.Following the studyHAFusion\([Sun et al\., 2024](https://arxiv.org/html/2610.31850#bib.bib5)\), we evaluatePRISMon three real\-world datasets, including New York City \(NYC\)111[https://opendata\.cityofnewyork\.us/](https://opendata.cityofnewyork.us/)\(Last accessed: October 24, 2026\)\., Chicago \(Chi\)222[https://data\.cityofchicago\.org/](https://data.cityofchicago.org/)\(Last accessed: October 24, 2026\)\., and San Francisco \(SF\)333[https://datasf\.org/opendata/](https://datasf.org/opendata/)\(Last accessed: October 24, 2026\)\.\. For each city, we construct region\-level features from multiple urban data sources:Spatial regions\.Each city is partitioned into regions that serve as basic analytical units\.Mobility data\.Taxi trajectories are aggregated into region\-level origin–destination \(OD\) matrices, capturing directed inter\-region human flows\.POIs\.POI data is collected from OpenStreetMap444[https://www\.openstreetmap\.org/](https://www.openstreetmap.org/)\(Last accessed: October 24, 2026\)\.\. Region representations are derived from aggregated semantic category counts\.Land use\.Zoning data describes the distribution of land\-use categories within each region \(11 for NYC, 12 for Chi, and 13 for SF\)\.Urban activity\.Crime reports, Foursquare check\-ins555[https://sites\.google\.com/site/yangdingqi/home/foursquare\-dataset](https://sites.google.com/site/yangdingqi/home/foursquare-dataset)\(Last accessed: October 24, 2026\)\., and service call records are aggregated at the region level and used as downstream prediction targets\([Zhang et al\., 2020](https://arxiv.org/html/2610.31850#bib.bib4);[Zhou et al\., 2023](https://arxiv.org/html/2610.31850#bib.bib29);[Wu et al\., 2022](https://arxiv.org/html/2610.31850#bib.bib30);[Sun et al\., 2024](https://arxiv.org/html/2610.31850#bib.bib5)\)\.Downstream Applications\.To evaluate the effectiveness of the learned urban region representations, we consider three representative applications:*check\-in prediction*,*crime forecasting*, and*service call prediction*\. These tasks capture diverse aspects of urban dynamics, including human activity intensity, public safety patterns, and infrastructure demand\. We formulate all tasks as regression problems, where the objective is to predict the aggregated event counts within each region using the learned embeddings as input features\. Following existing studies\([Zhang et al\., 2020](https://arxiv.org/html/2610.31850#bib.bib4);[Zhou et al\., 2023](https://arxiv.org/html/2610.31850#bib.bib29);[Wu et al\., 2022](https://arxiv.org/html/2610.31850#bib.bib30);[Sun et al\., 2024](https://arxiv.org/html/2610.31850#bib.bib5)\), we employ Lasso regression\([Kumar et al\., 2023](https://arxiv.org/html/2610.31850#bib.bib31)\)due to its robustness to high\-dimensional features and its ability to mitigate overfitting through sparsity\-inducing regularization\.

Baselines\.We comparePRISMwith representative state\-of\-the\-art baselines from three categories: urban region representation learning, multi\-task optimization, and spatio\-temporal prediction\. The urban region representation learning baselines includeMVURE\([Zhang et al\., 2020](https://arxiv.org/html/2610.31850#bib.bib4)\),MGFN\([Wu et al\., 2022](https://arxiv.org/html/2610.31850#bib.bib30)\),RegionDCL\(RDCL\)\([Li et al\., 2023c](https://arxiv.org/html/2610.31850#bib.bib2)\),HREP\([Zhou et al\., 2023](https://arxiv.org/html/2610.31850#bib.bib29)\),HAFusion\(HAF\)\([Sun et al\., 2024](https://arxiv.org/html/2610.31850#bib.bib5)\), andUrbanMoE\([Liu et al\., 2026](https://arxiv.org/html/2610.31850#bib.bib57)\)\. For multi\-task optimization, we considerHAFus ion\-GN\(i\.e\., HGN\) andPRISM\-GN\(i\.e\., PGN\), both implemented using GradNorm\([Chen et al\., 2018](https://arxiv.org/html/2610.31850#bib.bib52)\)\. The spatio\-temporal prediction baselines includeSTART\([Butt et al\., 2025](https://arxiv.org/html/2610.31850#bib.bib20)\)andBiST\([Ma et al\., 2025a](https://arxiv.org/html/2610.31850#bib.bib48)\)\.

Evaluation Metrics and Implementation Details\.We evaluate all tasks using MAE, RMSE, andR2R^\{2\}\. All model training experiments are conducted on Nvidia Quadro RTX 8000 GPUs with 48 GB memory, and the code environment is based on the PyTorch framework using Python 3\.8\.18\.

Table 2\.Ablation study forPRISM\. For each dataset, the best results are in bold/red, and the second\-best results are underlined/in blue\.MethodNYCChiSFMAE\(↓\\downarrow\)RMSE\(↓\\downarrow\)R2R^\{2\}\(↑\\uparrow\)MAE\(↓\\downarrow\)RMSE↓\\downarrowR2R^\{2\}\(↑\\uparrow\)MAE\(↓\\downarrow\)RMSE\(↓\\downarrow\)R2R^\{2\}\(↑\\uparrow\)Check\-inw/o RAR203\.5 ± 1\.6331\.1 ± 0\.10\.836 ± 0\.0001012 ± 21\.82096 ± 52\.00\.850 ± 0\.007231\.6 ± 31\.4435\.1 ± 23\.10\.809 ± 0\.020w/o UM210\.3 ± 8\.7336\.7 ± 12\.70\.831 ± 0\.013950 ± 21\.42057 ± 19\.30\.856 ± 0\.003232\.1 ± 12\.5444\.6 ± 44\.90\.800 ± 0\.040w/o PID202\.9 ± 2\.6315\.5 ± 4\.10\.852 ± 0\.014975 ± 40\.01973 ± 450\.867 ± 0\.006237\.3 ± 4\.3422\.3 ± 18\.40\.820 ± 0\.016PRISM200\.7 ± 4\.8315\.1 ± 5\.70\.852 ± 0\.005819 ± 22\.01677 ± 460\.904 ± 0\.006204\.0 ± 10\.2364\.2 ± 15\.00\.866 ± 0\.013Crimew/o RAR56\.9 ± 3\.176\.5 ± 3\.10\.733 ± 0\.02278\.5 ± 3\.8105\.4 ± 3\.70\.642 ± 0\.025103\.2 ± 5\.9179\.3 ± 1\.30\.679 ± 0\.005w/o UM57\.6 ± 3\.277\.3 ± 5\.00\.726 ± 0\.03583\.3 ± 2\.0114\.3 ± 2\.40\.579 ± 0\.018102\.7 ± 1\.1176\.2 ± 6\.40\.690 ± 0\.023w/o PID56\.3 ± 1\.375\.0 ± 0\.70\.743 ± 0\.00484\.1 ± 6\.4113\.8 ± 2\.80\.583 ± 0\.021102\.0 ± 2\.1171\.9 ± 11\.40\.705 ± 0\.039PRISM53\.9 ± 0\.373\.2 ± 0\.50\.756 ± 0\.00474\.4 ± 4\.1101\.4 ± 5\.50\.668 ± 0\.03596\.5 ± 4\.4176\.0 ± 2\.80\.694 ± 0\.010Service callw/o RAR1311 ± 40\.61974 ± 15\.90\.482 ± 0\.009164\.8 ± 14\.5236\.0 ± 11\.70\.567 ± 0\.04281\.7 ± 2\.4139\.1 ± 5\.80\.628 ± 0\.031w/o UM1316 ± 15\.91967 ± 34\.10\.486 ± 0\.018169\.4 ± 4\.3233\.4 ± 16\.80\.576 ± 0\.06182\.1 ± 1\.1141\.0 ± 2\.10\.618 ± 0\.011w/o PID1296 ± 7\.81962 ± 31\.10\.489 ± 0\.016162\.4 ± 9\.0225\.8 ± 19\.70\.603 ± 0\.06980\.6 ± 3\.2137\.3 ± 9\.70\.637 ± 0\.051PRISM1268 ± 1\.01937 ± 160\.501 ± 0\.008149\.3 ± 3\.1212\.3 ± 4\.20\.650 ± 0\.01478\.9 ± 3\.1134\.6 ± 10\.20\.651 ± 0\.052
### 5\.2\.Overall Performance

Check\-in Prediction\.We begin by evaluating all methods on the check\-in prediction task, which measures a model’s ability to capture latent urban functionality and mobility dynamics\. As can be seen in Table[1](https://arxiv.org/html/2610.31850#S4.T1),PRISMachieves the best performance across all three cities, consistently outperforming the strongest baseline,HAF, on every evaluation metric\. In NYC,PRISMreduces the MAE from 202\.8 to 200\.7 and improves theR2R^\{2\}from 0\.844 to 0\.852\. Similar improvements are observed in both Chi and SF, wherePRISMachieves the lowest prediction errors and the highest explanatory power\. These results indicate thatPRISMmore effectively models cross\-region interactions and heterogeneous urban signals, which are essential for capturing complex check\-in behaviors\. Unlike existing baselines that focus on urban region representation learning, multi\-task objective optimization, or spatio\-temporal dependency modeling,PRISMjointly captures heterogeneous urban semantics, cross\-region relational structures, and task\-level interactions, thereby producing more discriminative and transferable region embeddings\.

Crime Forecasting\.Next, crime forecasting requires modeling subtle spatial dependencies and socio\-environmental factors\. As shown in Table[1](https://arxiv.org/html/2610.31850#S4.T1),PRISMagain delivers the strongest results across all datasets, achieving the lowest MAE and RMSE values while obtaining the highestR2R^\{2\}values\. Relative toHAF,PRISMimprovesR2R^\{2\}from 0\.734 to 0\.756 in NYC and from 0\.631 to 0\.668 in Chi, indicating enhanced predictive capability in explaining crime variability\. The moderate performance gains are expected, given the inherently stochastic nature of crime data\. The results suggest thatPRISMlearns region embeddings that preserve fine\-grained spatial semantics well, thereby supporting more accurate modeling of localized crime patterns\.Service Call Prediction\.Finally, service call prediction is challenging due to high demand volatility and weak spatial regularity\. Here,PRISMachieves the best performance across all cities\. For example, in Chi,PRISMimprovesR2R^\{2\}from 0\.613 to 0\.650, representing a meaningful relative gain\. Comparable improvements are observed in SF, wherePRISMsurpasses the second\-best method across all metrics\. The larger performance gap observed for this task suggests thatPRISMis particularly effective in complex prediction settings, where robust representation learning is critical\.

### 5\.3\.Ablation Studies

Table[2](https://arxiv.org/html/2610.31850#S5.T2)reports the ablation results ofPRISM\. Overall, the full model achieves strong performance across most settings, but the contribution of each component varies by city and task\.

Effect of Region\-Aware Routing \(RAR\)\.Replacing region\-aware routing with simple feature concatenation \(w/o RAR\) results in the most consistent performance drop across nearly all settings\. The degradation is particularly evident for the check\-in task, whereR2R^\{2\}decreases from0\.8520\.852to0\.8360\.836in NYC and from0\.9040\.904to0\.8500\.850in Chi\. This confirms that static concatenation tends to entangle heterogeneous signals and cannot adapt to region\-specific informativeness\. In contrast, region\-aware routing prioritizes the most informative views for each region dynamically, enabling the learned embeddings to capture functional heterogeneity better\. Similar trends observed for crime and service call prediction further suggest that adaptive routing offers a general advantage rather than being task\-specific\.

Effect of Uncertainty Weighting \(UM\)\.Removing uncertainty\-based weighting and assigning equal importance to all objectives \(w/o UM\) yields both lower accuracy and higher variance\. For example, for check\-in prediction in NYC, the MAE increases to210\.3210\.3with substantially larger variance compared to the full model\. This behavior indicates less stable optimization dynamics\. Since urban representation learning relies on multiple supervisory signals derived from different modalities, naïvely combining them with fixed weights can introduce gradient conflicts and bias the model toward particular objectives\. Uncertainty weighting mitigates this issue by automatically balancing loss contributions, thereby improving convergence stability and cross\-city generalization\.

Effect of PID\-MSE \(Feedback\-Regulated Error Modeling\)\.Replacing PID\-MSE with standard MSE \(w/o PID\) also leads to performance degradation in most cases, showing the benefit of feedback\-inspired discrepancy modeling\. By decomposing prediction errors into proportional, accumulated, and differential components, PID\-MSE simultaneously suppresses systematic bias and local oscillations during training\. This is particularly beneficial for noisy urban signals, as reflected in the consistent improvements observed for crime and service call prediction\. The results suggest that regulating optimization dynamics is as important as improving the fusion strategy itself\.

![Refer to caption](https://arxiv.org/html/2610.31850v2/regionsize1.png)Figure 3\.Region Embedding Size Study
### 5\.4\.Parameter Sensitivity Analysis

Effects of Region Embedding Size\.We investigate the impact of region embedding dimensionality on model performance by varying the embedding size from 32 to 288 dimensions\. Figure[3](https://arxiv.org/html/2610.31850#S5.F3)summarizes the MAE and RMSE results\. Overall, the performance exhibits a non\-monotonic trend as the embedding size increases\. Very small embeddings \(e\.g\., 32\) tend to underperform, suggesting insufficient representational capacity to capture complex spatial semantics and cross\-region interactions\. As the dimensionality increases, the performance improves consistently, indicating that richer embeddings encode heterogeneous urban signals better\. However, further enlarging the embeddings does not always yield gains\. When the dimensionality becomes very large \(e\.g\., 288\), the performance either saturates or decreases slightly in several settings\. This phenomenon likely stems from over\-parameterization, which can introduce redundant features, amplify noise, and increase optimization difficulty\. Interestingly, the optimal embedding size remains relatively stable across cities, typically falling within the mid\-range \(96–144 dimensions\)\. This suggests that urban region representations require sufficient capacity to model functional diversity but also benefit from controlled compactness to preserve generalization\. From a task perspective, service call prediction appears more sensitive to embedding dimensionality than crime forecasting, possibly because service demand is influenced by a broader set of socio\-economic factors that require richer representations\.

These findings reveal an important trade\-off between representational capacity and model robustness\. While larger embeddings enhance expressiveness, overly high dimensionality may harm generalization\.

![Refer to caption](https://arxiv.org/html/2610.31850v2/tem1.png)Figure 4\.Effect of Router Temperature![Refer to caption](https://arxiv.org/html/2610.31850v2/top-k-scalability1.png)Figure 5\.Model Scalability vs\. Top\-kkEffect of Top\-kkRouter Temperature\.We evaluate the impact of the Top\-kkrouter temperature, varying the temperature over\{0\.1,0\.3,0\.5,1\.0,2\.0,5\.0\}\\\{0\.1,0\.3,0\.5,1\.0,2\.0,5\.0\\\}\. We report the results using theR2R^\{2\}metric\. As shown in Figure[4](https://arxiv.org/html/2610.31850#S5.F4), prediction performance varies across different temperature settings\. Lower temperatures lead to sharp routing, higher temperatures provide smooth routing distributions\. We can see that the best results for all three tasks across the three cities occur when the temperature is set to1\.01\.0\.

### 5\.5\.Model Scalability Analysis

Figure[5](https://arxiv.org/html/2610.31850#S5.F5)reports on the scalability ofPRISM’s Router model under different Top\-kksettings\. We compare the Router model with the Concat baseline \(i\.e\.,PRISM\-Concat\) in terms of prediction performance \(R2R^\{2\}\) and computational cost \(GFLOPs\)\.

Performance vs\. Top\-kk\.We see that, the Router model is sensitive to the choice ofkk\. Thus, the prediction performance improves when increasingkkfrom 1 to 2, and then it either stabilizes or decreases slightly whenk=3k=3\. This trend is consistent across multiple settings, suggesting that a moderate number of selected experts provides a good balance between representation capacity and noise control\. Specifically, the Router model often achieves its bestR2R^\{2\}atk=2k=2, outperforming bothk=1k=1andk=3k=3\. Whenk=1k=1, the model may suffer from limited expert diversity, while largerkk\(e\.g\.,k=3k=3\) can introduce redundant or less relevant experts, leading to marginal performance degradation\. In contrast, the Concat baseline remains relatively stable across differentkkvalues, as it does not explicitly perform dynamic expert selection\. However, its performance is generally lower than or comparable to the Router model at the optimalkksetting, highlighting the effectiveness of the adaptive routing mechanism\.

Computational Cost\.The GFLOPs comparison shows that the Router model maintains competitive computational efficiency compared to the Concat baseline\. Although increasingkkslightly increases computation due to additional expert activation, the overall cost remains comparable to that of Concat\. Notably, on the NYC and SF datasets, the Router model even exhibits slightly lower GFLOPs than Concat, indicating that the dynamic routing mechanism can achieve competitive—or even improved—computational efficiency while offering strong predictive performance\.

### 5\.6\.Model Efficiency Analysis

![Refer to caption](https://arxiv.org/html/2610.31850v2/Prism_Eff1.png)Figure 6\.Pareto Analysis of Accuracy — Efficiency Trade\-offBeyond the prediction capabilities, we evaluate the computational efficiency and resource footprint of different fusion strategies\. Figure[6](https://arxiv.org/html/2610.31850#S5.F6)presents the accuracy–efficiency trade\-off across the3×33\\times 3city–task settings\. We useR2R^\{2\}to measure prediction performance, report the average training time per epoch \(in seconds\) to quantify computational cost, and report the peak GPU memory consumption \(in GB\) to report the footprint\. we use marker size to represent the latter\. These metrics provide a three\-dimensional characterization of computational and memory efficiency\. To quantitatively assess the effectiveness–efficiency trade\-off, we define𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦=R2𝑇𝑖𝑚𝑒\\mathit\{Efficiency\}=\\frac\{R^\{2\}\}\{\\mathit\{Time\}\}, where𝑇𝑖𝑚𝑒\\mathit\{Time\}denotes the average training time per epoch\.

Pareto Dominance\.As shown in Figure[6](https://arxiv.org/html/2610.31850#S5.F6),PRISMconsistently lies on or near the Pareto frontier across all nine city–task combinations\. In particular, no competing method simultaneously achieves higher prediction accuracy and lower training time\. AlthoughHAFusionoffers shorter training times, this is accompanied by lower prediction performance\.PRISM\-Concat, while marginally faster in some scenarios, offers lower accuracy than the routing\-based variant\.

Importantly, the improvements obtained byPRISMare achieved without noticeable increases in memory consumption, as evidenced by comparable GPU memory usage across methods\. This indicates that the adaptive routing mechanism enhances representation quality without introducing significant additional resource overhead\.

Quantitative Efficiency Comparison\.Across the nine settings, the average efficiency score ofPRISMis 6\.76, compared to 6\.20 forPRISM\-Concatand 10\.38 forHAFusion\.HAFusionachieves a higher numerical efficiency score due to its substantially shorter training time, and the higher score primarily reflects reduced model complexity rather than improved prediction capabilities\. In contrast,PRISMconsistently achieves betterR2R^\{2\}values than doesPRISM\-Concatwhile incurring only marginal additional training cost \(typically 0\.01–0\.02 seconds per epoch\)\. From a practical perspective, such minor computational overheads are negligible relative to the observed gains in prediction performance\.

### 5\.7\.Model Behavior Analysis

To understand the mechanisms underlying the performance improvements offered byPRISM, we analyze its behavior from two perspectives: modality heterogeneity and optimization stability\.

Model Heterogeneity\.Figure[7\(a\)](https://arxiv.org/html/2610.31850#S5.F7.sf1)reports the Top\-1 routed view ratio across different city–task combinations, thus indicating how frequently each modality is selected as the dominant expert\. The results show clear task\- and city\-specific modality preferences\. In NYC, the Land modality consistently dominates across all three tasks, suggesting that land\-use information plays a primary role in regional prediction\. InChi, routing decisions appear more balanced, particularly for Crime and Service Call, where POI and Land exhibit comparable selection frequencies, reflecting stronger cross\-modal complementarity\. In SF, the Crime task is dominated by the Mobility modality, indicating a strong dependence on human mobility patterns\.

These observations indicate substantial modality heterogeneity across city–task settings\. The relative importance of modalities varies markedly depending on both the urban context and the prediction objective\. Therefore, a fixed fusion strategy is unlikely to perform optimally across all scenarios\. In contrast, the adaptive routing mechanism inPRISMadjusts its modality selection dynamically according to task\-specific signals, enabling more context\-aware representation learning\.

![Refer to caption](https://arxiv.org/html/2610.31850v2/Top1View1.png)\(a\)Modality heterogeneity\.
![Refer to caption](https://arxiv.org/html/2610.31850v2/TrainingEfficiency1.png)\(b\)Optimization instability\.

Figure 7\.Model Behavior Analysis\.Optimization Instability\.Figure[7\(b\)](https://arxiv.org/html/2610.31850#S5.F7.sf2)examines the optimization dynamics by comparing the training stability ofPRISM\(PID\-MSE\) andHAFusion\(MSE\)\. Across all city–task combinations,PRISMconsistently achieves lower instability scores throughout the training\.HAFusionexhibits higher and more fluctuating stability values, indicating stronger gradient imbalance and inter\-modal conflicts\. The performance gap remains stable across epochs, suggesting that the improvement is due to systematic optimization advantages rather than transient effects\. Lower instability scores imply better alignment among modality\-specific gradients and reduced scale disparity during joint optimization\. By incorporating routing and PID\-based regulation,PRISMmitigates gradient competition across modalities and promotes more coordinated multi\-view learning\. Consequently, the adaptive routing mechanism not only captures heterogeneous modality relevance but also improves optimization stability, thereby explaining the consistent performance gains observed across city–task settings\.

### 5\.8\.Robustness to View Corruption and Incompleteness

Real\-world urban data is inherently prone to view incompleteness and corruption caused by data acquisition limitations and sensor noise\. To systematically evaluate the robustness ofPRISMandHAFusionunder these challenging conditions, we perform experiments in missing\-view and noisy\-view settings\.

![Refer to caption](https://arxiv.org/html/2610.31850v2/ViewMissing1.png)Figure 8\.Performance Across Different Missing View Ratio#### 5\.8\.1\.Missing Views\.

Motivated by the modality heterogeneity analysis in Figure[7\(a\)](https://arxiv.org/html/2610.31850#S5.F7.sf1), we investigate the robustness ofPRISMin different missing\-view scenarios\. Specifically, we simulate missing views for the dominant modality in each city: we mask the Land view for NYC, the POI view for Chi, and the Mobility view for SF\. We vary the missing\-view ratio in\{0,0\.1,0\.3,0\.5,0\.7,0\.9,1\.0\}\\\{0,0\.1,0\.3,0\.5,0\.7,0\.9,1\.0\\\}, where a ratio of 0 indicates no masking and 1\.0 corresponds to complete removal of the selected view\. Figure 8 reports the prediction performance findings\. As the missing ratio increases, the performance of both methods degrades consistently, confirming the importance of the masked modality\. However,PRISMmaintains the best performance across nearly all missing ratios and city–task combinations\. The performance gap becomes more pronounced under moderate to high missing ratios, indicating that adaptive routing enables the model to better compensate for incomplete information by dynamically reallocating attention to the remaining informative views\. Figure[8](https://arxiv.org/html/2610.31850#S5.F8)also shows thatHAFusionslightly outperformsPRISMon SF Crime when the mobility missing ratio is 0\.7\. This case reveals a limitation of sparse routing under severe dominant\-view incompleteness\. Since SF Crime relies strongly on mobility\-related signals, heavily masking the mobility view can make top\-kkrouting decisions less reliable\. By contrast, the dense fusion strategy inHAFusionaggregates all available views and may provide more stable behavior at this specific corruption level\. This result suggests that sparse view selection is generally beneficial, but may require missingness\-aware or uncertainty\-aware routing when the dominant modality is severely corrupted\. Even when the dominant modality is fully removed \(ratio = 1\.0\),PRISMcontinues to outperform the baseline in most cases\.

#### 5\.8\.2\.Noisy Views\.

We evaluate the robustness ofPRISMby injecting three types of perturbations—additive Gaussian noise, mixture\-of\-Gaussians noise, and column\-wise shuffling—with corruption ratios varied in\{0\.1,0\.5,0\.9\}\\\{0\.1,0\.5,0\.9\\\}to simulate progressively severe view degradation\. These perturbations collectively capture realistic degradation patterns, including random feature corruption, heterogeneous distribution shifts, and structural information distortion\. Following the missing\-view setting, we inject noise into the dominant view of each city: the Land view for NYC, the POI view for Chi, and the Mobility \(Mob\) view for SF, as these views contribute most significantly to the corresponding downstream tasks in each city\.

![Refer to caption](https://arxiv.org/html/2610.31850v2/Gaussian1.png)Figure 9\.Robustness Analysis — Gaussian PerturbationGaussian Noise\.Figure[9](https://arxiv.org/html/2610.31850#S5.F9)presents the performance under additive Gaussian perturbations across three cities and three tasks\. In NYC, performance declines steadily as the corruption ratio increases, with the most pronounced degradation observed in the Crime task; nevertheless,PRISMconsistently maintains higherR2R^\{2\}scores thanHAFusion, and the performance gap widens under severe noise \(0\.90\.9\)\. In Chicago, both methods exhibit sharper drops compared to NYC, particularly for Crime and Service Call prediction; however,PRISMdemonstrates slower performance decay, indicating improved robustness under moderate\-to\-high corruption levels\. For San Francisco, the degradation trend is comparatively smoother across all three tasks, andPRISMpreserves a stable margin over the baseline even at the highest noise ratio, suggesting stronger resistance to stochastic perturbations across diverse urban prediction scenarios\.

![Refer to caption](https://arxiv.org/html/2610.31850v2/MultiGaussian1.png)Figure 10\.Robustness Analysis — Multi\-Gaussian Perturbation![Refer to caption](https://arxiv.org/html/2610.31850v2/Col1.png)Figure 11\.Robustness Analysis — Column\-Wise Shuffle\.Multi\-Gaussian Noise\.Figure[10](https://arxiv.org/html/2610.31850#S5.F10)presents the results for mixture\-of\-Gaussians perturbations\. In NYC, performance degrades monotonically with increasing corruption, andPRISMachieves more competitive performance than the strongest baselineHAFusionacross all three tasks\. In Chi, both methods decline as noise intensifies; notably, for the Check\-in task at ratio0\.10\.1,HAFusionslightly surpassesPRISM, whereasPRISMis the best at higher corruption levels\. For Crime and Service Call prediction in Chi,PRISMremains consistently better\. In SF, degradation is relatively smooth, andPRISMpreserves higherR2R^\{2\}scores across all tasks and noise ratios\.

Column\-Wise Shuffle\.Figure[11](https://arxiv.org/html/2610.31850#S5.F11)presents the results for column\-wise shuffling, which permutes feature dimensions to disrupt intra\-view structural dependencies\. In NYC, both methods exhibit nearly identical degradation patterns across Check\-in, Crime, and Service Call prediction\. Performance decreases consistently as the shuffle ratio increases, while the gap betweenPRISMandHAFusionremains marginal across all corruption levels\. In contrast, for Chi and SF,PRISMconsistently achieves higherR2R^\{2\}scores across all three tasks and shuffle ratios\. The performance gap becomes more pronounced under moderate\-to\-high shuffle ratios, particularly for Crime and Service Call prediction, indicating improved stability under severe structural perturbation\.

## 6\.Conclusion

We presentPRISM, a self\-adjusting framework for multi\-view urban region representation learning\. By integrating region\-wise sparse routing with PID\-inspired discrepancy regulation,PRISMenables adaptive modality allocation and stabilizes multi\-objective optimization\. Extensive experiments on three large\-scale urban datasets across three tasks exhibit consistent performance improvements over state\-of\-the\-art proposals while maintaining comparable model complexity\. Our study highlights the importance of adaptive fusion and feedback\-aware optimization in heterogeneous urban representation learning, particularly for achieving stable training and preserving model robustness\. Overall, the results show that adaptive mechanisms are essential for task\-specific urban region representation learning\. Future work will focus on cross\-city generalization and adaptive fusion in dynamic urban environments\.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China \(Nos\. 62472174 and 62402082\), the ECNU Multifunctional Platform for Innovation \(No\. 001\), and the Chongqing Municipal Education Commission \(No\. KJQN202400637\)\.

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