Distance-Based Model Fusion

• Distance-based model fusion is a technique that integrates multiple models using pairwise or distributional distance measures, such as Euclidean and Wasserstein metrics, to adaptively weight and align outputs.
• It employs methodologies like regularization in joint losses, adaptive trust metrics, and gating functions in frameworks such as WR-EFM and InfiGFusion, yielding improved classification and sensor fusion performance.
• Empirical results across applications—from multimodal detection to sensor networks—demonstrate enhanced robustness, accuracy, and computational efficiency compared to naive averaging approaches.

Distance-based model fusion refers to a set of techniques where the integration of multiple models, estimators, or sensor outputs is governed directly or indirectly by pairwise or distributional distance measures between outputs, parameters, or learned feature spaces. The core principle is that fusion weights, transformations, or the fusion process itself are explicitly determined by metrics such as Euclidean distance, Wasserstein or other optimal transport distances, information-theoretic divergences, or specialized task-dependent similarity functions. This paradigm has been instantiated across diverse settings: expert neural-network ensembles, multimodal sensor fusion, structure learning, multi-source parameter integration, object detection, and more. The aim is usually improved robustness, fidelity, interpretability, or computational efficiency over naïve averaging or task-agnostic fusion.

1. Mathematical Foundations and Distance Metrics

Distance-based fusion methods leverage a variety of distances, either as regularizers in joint losses, as adaptive trust metrics, or as fusion criteria:

• Optimal Transport Distances: Wasserstein and Gromov–Wasserstein distances are used to compare distributions or structures of learned representations or output spaces. For example, the 1-Wasserstein distance between expert neural network embedding distributions guides adaptive weighting in WR-EFM, seeking to minimize representational mismatch (Ma et al., 21 Jul 2025). In the InfiGFusion framework, a sorting-based Gromov–Wasserstein distance over co-activation graphs between LLM logits captures and aligns higher-order semantic dependencies (Wang et al., 20 May 2025).
• Euclidean and Task-specific Distances: Euclidean distances between outputs or probability vectors quantify reliability or consensus among single-modality CNNs, as in the biologically inspired fusion for affective state recognition (Zhao et al., 2019). Similarly, the Gaussian Wasserstein distance between extended ellipse estimates yields the Minimum Mean Gaussian Wasserstein fusion estimator for multi-sensor elliptic target tracking (Thormann et al., 2019).
• Information-Theoretic Distances: Entropy-based metrics, notably the Normalised Information Distance (NID), are employed to evaluate and threshold the novelty of sensor frames or 3D surfaces, determining inclusion in the fusion process (Gallagher et al., 2021).
• Task-tailored Similarities: In multi-modal detection, the Distance-Decay IoU metric combines spatial displacement and scale-invariant overlap to robustly pair and fuse bounding-box results across sensors (Qingyun et al., 2021).

These distances can be embedded in objective functions, employed for gating, or interpreted as regularization terms to encourage compatible, structure-aware, or mutually consistent fused outputs.

2. Algorithmic Strategies for Fusion

Distance-based fusion architectures can be broadly categorized by mechanism and granularity:

• Expert and Representation Fusion: Models such as WR-EFM train specialized experts to capture different structural properties or class regimes and fuse their outputs using adaptive weights determined by distance regularization. The Wasserstein–Rubinstein distance between embedding distributions serves both as a fusion guide and as an alignment objective, with adaptive weights inversely proportional to this distance, yielding per-class or per-sample fusions (Ma et al., 21 Jul 2025).
• Decision-Level and Reliability-Weighted Fusion: In affective states recognition, fusion of probability vectors is performed at the decision level, where the contribution of each modality is weighted by modality-specific reliability defined via Gaussian kernels on Euclidean distances (Zhao et al., 2019).
• Sensor and Frame Selection: Surfel-based 3D mapping incorporates NID as a principled gating function to subsample frames prior to computationally expensive fusion. Only frames delivering sufficient nonredundant information, as measured by NID against the current model, are incorporated, optimizing the accuracy-cost trade-off (Gallagher et al., 2021).
• State-space and Extended Kalman Filtering: Distance-based fusion can take the form of filter-based estimation, as in UAV-person ranging, where an EKF fuses monocular depth predictions with sparse but accurate depth camera measurements, with trust assigned dynamically via covariance tuning (Šiktar et al., 24 Feb 2026).
• Pairwise Parameter Fusion: In integrative structure learning, model parameters from multiple studies or sources are fused using penalties on the sum of elementwise distances (e.g., fused lasso or MCP) across parameter vectors, inducing blockwise or clusterwise homogeneity where warranted by data (Hector, 2022).
• Domain-specific Pairing and Evidence Combination: In multi-sensor object detection, boxes are paired via a distance-decay similarity, and confidences are fused using the Dempster-Shafer framework, where weighting depends on relative evidence compatibility that is distance-informed (Qingyun et al., 2021).

3. Model Architectures and Implementation Schemes

The diversity of settings for distance-based fusion has fostered a range of architecture designs:

Framework/Domain	Core Distance Function	Fusion Mechanism
WR-EFM (node classification)	1-Wasserstein (dual form)	Adaptive weighted expert sum
InfiGFusion (LLM fusion)	Gromov-Wasserstein (approx.)	Graph-on-logits distillation
Multiscale CNN affective fusion	Euclidean + Gaussian kernel	Reliability-weighted sum
Surfel fusion (EF)	Normalised Information Distance	Frame gating
EKF-based sensor fusion	Implicit via process/measurement variance	Kalman update
Pairwise structure learning	L1 vector or group distance	Block/group-wise parameter fusion
Tracking fusion (ellipse)	2-Wasserstein (GW)	MMGW estimator in lifted space

WR-EFM runs two parallel GNN experts and fuses representations node-wise, with class-wise weights. The InfiGFusion approach maps model logit tensors to co-activation graphs, then applies an efficient Gromov–Wasserstein objective using node-degree feature matching and sorting. Higher-dimensional or non-Euclidean geometries are frequently handled via nonlinear lifting (e.g., shape space or T-space) for fusion in a metric-preserving domain, with back-projection.

4. Empirical Results and Performance Gains

Distance-guided model fusion delivers notable empirical benefits across settings:

• Balanced and Stable Classification: On PubMed, WR-EFM achieves class-wise accuracies (C0=77.8%, C1=78.0%, C2=79.9%), exceeding GCN and baseline fusion especially for hard classes (Category 2 +5.5% vs. GCN), while reducing coefficient of variation (CV=0.013, 77.6% lower than GCN) (Ma et al., 21 Jul 2025).
• Efficient Model Fusion in LLMs: InfiGFusion with GLD loss outperforms state-of-the-art ensemble methods by an absolute mean gain (~+1.68%), with dramatic improvements for multi-hop and relational reasoning (+37.6 on BBH Multistep Arithmetic, +34.22 on BBH Causal Judgement), validating the importance of token co-activation structure (Wang et al., 20 May 2025).
• High-Fidelity, Cost-Effective 3D Mapping: Surfel selection via NID in ElasticFusion reduces the number of fused frames by 60–75% with <5% degradation in reconstruction error, and yields faster loop closures and sparser final surfel sets (Gallagher et al., 2021).
• Reliable, Low-Error Sensor Fusion: In UAV-person tracking, fusion via EKF reduces fused distance error RMSE and standard deviation by up to 15.3% compared to best single modality, maintaining closed-loop tracking accuracy in real-time applications (Šiktar et al., 24 Feb 2026).
• Optimal Fusion of Spatial Extents: For extended targets, the MMGW estimator demonstrably outperforms naïve or shape-mean averaging, with lowest average root mean Gaussian Wasserstein error (1.0470 or below) (Thormann et al., 2019).
• Parameter Efficiency and Structure Recovery: Fused structure learning via pairwise penalties consistently recovers correct partitions and shrinks parameter error/variance by up to 5x compared to heterogeneous models, with demonstrated gains in real-world neuroimaging (Hector, 2022).
• Robust Multi-modal Detection: In remote sensing object detection, DDIoU-guided pairing and weighted Dempster-Shafer fusion increase average precision by +20.13% vs. RGB-alone, with ~30Hz runtime (Qingyun et al., 2021).

5. Limitations, Extensions, and Open Challenges

While distance-based fusion provides principled regularization and task-adaptivity, several considerations arise:

• Computational Overhead: Optimal transport distances (Wasserstein/Gromov–Wasserstein) are costly for large domains, motivating approximations (sorting-based, sampling, or feature compressions as in InfiGFusion (Wang et al., 20 May 2025)).
• Choice and Calibration of Distance: Selection of the right metric (Euclidean, Wasserstein, NID, etc.) is crucial and may be task-dependent. Parametric choices (e.g., decay rate α, penalty parameters λ) often require empirical tuning or data-driven selection (e.g., GMM-BIC (Hector, 2022)).
• Non-Euclidean Geometries and Group Structures: In extended-object or shape-based fusion, nonlinear lifting is necessary; approximation errors can arise if underlying assumptions (e.g., commuting shape matrices) do not hold (Thormann et al., 2019).
• Data Imbalance and Distribution Shift: In class-imbalanced or heterogeneous data, adaptive distance-based fusion (WR-EFM) can exploit complementary learned features, but may require careful class-wise calibration to avoid overfitting or under-exploitation of hard classes (Ma et al., 21 Jul 2025).
• Interpretability: Fusion in transformed spaces (e.g., T-space, co-activation graphs) requires post-fusion back-mapping and may obscure direct interpretability relative to native model outputs.
• Scalability to Multi-source and Decentralized Settings: Pairwise or global fusion becomes rapidly more complex as the number of sources grows, necessitating scalable approximations and communication-efficient algorithms (e.g., decentralized T-space fusion (Thormann et al., 2019)).

Potential extensions comprise hierarchical spatial hashing in 3D, learned priors for scene fusion, decentralized multi-agent fusion, optimization of thresholds and weightings via Bayesian or meta-learning approaches, as well as the incorporation of structure-aware regularization into increasingly complex neural network architectures.

6. Applications and Generalization Scope

Distance-based model fusion spans multiple domains:

• Graph learning and node classification: WR-EFM achieves stable performance in class-imbalanced node classification by explicitly modeling embedding discrepancies and dynamically weighting specialized experts.
• LLMs: InfiGFusion fuses LLMs by aligning higher-order structure in output logits, critical for domains with strong multi-step or relational dependencies.
• Sensor fusion and robotics: Kalman filter variants and NID-based frame selection address sensor heterogeneity, missing data, and real-time performance requirements.
• Data integration and structure learning: Pairwise distance penalties allow data-driven clustering of study/model parameters, supporting both full and partial parameter sharing.
• Remote sensing and multimodal object detection: Fusion at the detection and confidence level, informed by geometric and reliability distances, supports robust cross-modality performance.
• Ecological modeling: Fusion of different observational modalities (distance sampling and capture-recapture) enables unbiased, high-efficiency estimation under missingness and partial information (Mohankumar et al., 2022).

These examples confirm that distance-based fusion strategies, regardless of specific model class or application, offer a flexible and theoretically grounded approach to the principled integration of heterogeneous information sources.