Hierarchical Merging Strategy Overview

• Hierarchical merging strategy is a systematic approach that combines models, clusters, or predictions using a tree-like architecture to preserve local structure and enable efficient global consolidation.
• It employs techniques like pairwise merges, selective structural alignment, and adaptive clustering to improve scalability, reduce error propagation, and maintain accuracy.
• This strategy is widely applied in neural model fusion, data integration, graph merging, and robotics, providing measurable gains in efficiency and performance.

A hierarchical merging strategy refers to any systematic approach that combines multiple models, predictions, clusters, or trajectories by leveraging a multi-level, tree-like architecture. Such strategies are widely employed across machine learning, data integration, robotics, networking, and transportation systems where scalable, robust, or semantically meaningful aggregation is needed. Methods typically blend objectives such as preserving local structure while attaining efficient global consolidation, enabling them to circumvent limitations of naïve averaging or flat merging procedures. This article surveys foundational principles, leading algorithmic frameworks, and application domains for hierarchical merging strategy as substantiated in the arXiv literature.

1. Foundational Principles of Hierarchical Merging

Hierarchical merging organizes the fusion of entities (models, clusters, predictions, etc.) through a sequence of nested, typically pairwise, merges. This architecture is motivated by several recurring needs:

• Scalability: Reducing memory, compute, or communication costs by breaking global fusion into tractable units (e.g., (Wu et al., 9 Oct 2024, Franke et al., 10 Oct 2025, Ponomarenko, 21 May 2025)).
• Preservation of local structure: Maintaining information granularity in early hierarchy levels (e.g., domain-general features in shallow LLM layers (Timilsina et al., 17 Nov 2025); fine-grained clusters (Peterson et al., 2017)).
• Mitigation of error propagation and degeneration: Avoiding the loss basin evasion and accuracy drop-off associated with flat arithmetic averaging (e.g., Git Re-Basin (Franke et al., 10 Oct 2025)).
• Compatibility with system constraints: Enabling efficient distributed or privacy-preserving fusion, robust graph merging, or resource-aware deployment (e.g., (Timilsina et al., 17 Nov 2025, Ponomarenko, 21 May 2025, Wu et al., 9 Oct 2024)).

The pipeline is typically arranged as a merge-tree, where each node corresponds to a merged entity resulting from its immediate children. The strategy can be recursively applied at each level, e.g., layer merging within a model, cluster merging within partitions, or aggregation of intermediate predictions.

2. Algorithmic Frameworks and Methodologies

2.1 Pairwise and Multi-level Merging

Most hierarchical mergers proceed with binary or multi-way merges at each stage. In deep learning model fusions, the strategy can be formalized as follows:

• For $n=2^k$ models $\{M_1^{(0)},\dots,M_n^{(0)}\}$, at stage $s$, merge pairs $(M_{2i-1}^{(s-1)}, M_{2i}^{(s-1)})$ into new $M_{i}^{(s)}$ via an operation such as permutation alignment plus convex interpolation. This prevents the mean model from exiting the loss basin (Franke et al., 10 Oct 2025).

In clustering, agglomerative merges rely on a distance metric and linkage criterion. Clusters are successively fused to minimize an objective such as intra-cluster distance or maximize density (Peterson et al., 2017).

2.2 Selective Merge with Structural Alignment

Certain applications demand that only structurally compatible elements are merged:

• Attention-head permutation resolution: In transformer-based LLMs, hierarchical merging applies selective optimal transport (Hungarian assignment) to align attention heads before weighted parameter interpolation (Timilsina et al., 17 Nov 2025).
• Cluster correspondences: Co-clustering algorithms only merge those submatrix clusters with sufficient row/column overlap as measured by an overlap or similarity threshold (Wu et al., 9 Oct 2024).
• Layer or architecture path search: Reinforcement learning is used to discover optimal architecture merging paths by navigating design spaces across source models (Zhou et al., 27 Sep 2024).

2.3 Hierarchical Clustering

Agglomerative hierarchical clustering, a core technique in both classical and modern settings, provides a general blueprint:

• Entities (clusters, experts, or model fragments) are represented via feature vectors (e.g., expert-average outputs for sparse MoE (Chen et al., 11 Oct 2024)) with a similarity metric (e.g., cosine or task performance).
• At each iteration, the closest pair is merged, employing linkage methods (average, single, or complete).
• Merging may entail averaging parameters (for models/experts), union of sets (for clusters), or optimization-guided fusion.

Stopping criteria include achieving a desired granularity, surpassing a similarity threshold, or optimizing a validation measure.

Table: Selected Merging Operations

Domain	Merge Operation	Alignment/Linkage
Model fusing	Weighted averaging with OT permutation (LLM)	Selective OT, cosine similarity
Clustering	Set union, within/between density maximization	Overlap, linkage, density gain
Adapter merging	Concatenation, pruning, scalar weighting	Cosine similarity on delta
MoE pruning	Parameter averaging of clustered experts	Cosine distance on output means

3. Applications Across Domains

3.1 Neural Model Fusion

In distributed and continual learning, hierarchical merging addresses key issues such as catastrophic forgetting, privacy, and compute constraints:

• Distributed medical LLM merging: Selective OT aligns attention, layerwise cosine similarity weights interpolate parameters, yielding efficient knowledge consolidation without catastrophic forgetting (Timilsina et al., 17 Nov 2025).
• Continual learning (HAM): Low-rank adapters for sequential tasks are grouped and hierarchically merged based on similarity, using importance scalars and pruning to avoid interference and scale to long task sequences (Coleman et al., 16 Sep 2025).
• Multi-objective model fusion: HM3 generalizes to both parameter- and architecture-space merging, constructing a Pareto front of trade-offs between multiple tasks (Zhou et al., 27 Sep 2024).
• Sparse mixture-of-experts: Hierarchical clustering using expert output representations achieves retraining-free parameter reduction with controlled degradation (Chen et al., 11 Oct 2024).

3.2 Data Combination and Label Integration

• Hierarchical prediction consolidation: Label hierarchies induce both constraints and similarity penalties in the fusion of source predictions, solved via a Laplacian-regularized quadratic program with iterative consensus (Zhang et al., 2016).
• Co-clustering large data matrices: After parallel submatrix clustering, overlapping clusters are greedily merged by overlap similarity, preserving granularity and improving clustering quality (Wu et al., 9 Oct 2024).

3.3 Graph and Trajectory Fusion

• Hierarchical navigation-structure merging (HNSW): Multi-stage algorithms (IGTM, CGTM) balance candidate propagation, local search efficiency, and neighbor selection to reduce computational cost while retaining search quality (Ponomarenko, 21 May 2025).
• Multi-robot coverage planning: Morse-theory-based cycle partitioning, cyclic merging search, and hierarchical partition concatenation yields optimally balanced, conflict-free robot routes (Zheng et al., 7 Aug 2025).

3.4 Information Summarization and Contextualization

• Hierarchical document summarization: Typed merges (replace, support) and extractive/retrieval-augmented context enrichment at each summary level improve factuality and reduce hallucination, especially for ultra-long document inputs (Ou et al., 3 Feb 2025).
• Hierarchical context merging for LLMs: Divide-and-conquer token reduction and chunk merging at successive transformer layers achieves O(log L) memory scaling and promotes efficient long-context understanding (Song et al., 16 Apr 2024).

4. Performance Characteristics and Empirical Findings

Hierarchical merging's strengths and limitations are evidenced across multiple axes:

• Robustness to information loss or drift: Tree-based merge procedures retain intermediate entity integrity, yielding higher clean and adversarial accuracy for model fusion compared to flat averaging (e.g., +6 points on CIFAR-10 over MergeMany in (Franke et al., 10 Oct 2025)).
• Parameter- and memory-efficiency: Pruning- and OT-based schemes keep overhead within a few percent of simple averaging (Timilsina et al., 17 Nov 2025), while expert clustering can halve MoE parameters for negligible accuracy drop (Chen et al., 11 Oct 2024).
• Accuracy–efficiency trade-off: For graph indexing, IGTM achieves 68% reduction in merge cost for <1% recall loss (Ponomarenko, 21 May 2025). For co-clustering, hierarchical merge adds <5% to computation time but improves NMI/ARI by 3–5% over non-hierarchical merging (Wu et al., 9 Oct 2024).
• Versatility: Hierarchical frameworks are adapted to distributed AI, privacy-critical data integration, high-dimensional graph or matrix structuring, large-scale continual learning, multimodal fusion, and complex scheduling/trajectory domains.

5. Selected Theoretical Analyses and Complexity

Hierarchical strategies frequently admit tractable analysis due to their recursive structure. Examples include:

• Convexity and closed-form solution: For hierarchical label consolidation, the quadratic Laplacian objective ensures convexity, with closed-form updates alternating with similarity matrix inference (Zhang et al., 2016).
• Complexity bounds: Hierarchical merge for co-clustering is $O(K d \log K)$ under adjacency pruning heuristics (Wu et al., 9 Oct 2024), and O(L·H³ + P) for selective-OT-based model merging (Timilsina et al., 17 Nov 2025).
• Optimality Guarantees: Algorithms such as HCMR in robotics show that all Morse-bounded tours are exhaustively enumerated, ensuring optimality and completeness for sweep coverage (Zheng et al., 7 Aug 2025).
• Hyperparameter and stopping criteria: Distance or overlap thresholds, or objective function plateaus, guide the depth of merges to prevent either under- or over-merging (Peterson et al., 2017, Wu et al., 9 Oct 2024).

6. Practical Considerations and Implementation Guidelines

Implementation best practices are highly domain-specific but share core themes:

• Structural compatibility: All merging steps require architectural or feature compatibility. For transformers, head dimensions and model shapes must be matched (Timilsina et al., 17 Nov 2025).
• Alignment and normalization: Proper alignment (OT, permutation) must be performed in layers or clusters where permutation variance is fatal to merger stability.
• Resource constraints: Merges are designed so that expensive operations (e.g., OT, large-scale clustering) are performed offline, with the merged entity (model, index, cluster set) deployed for fast online use (Timilsina et al., 17 Nov 2025, Ponomarenko, 21 May 2025).
• Granularity tuning: Merge thresholds, cluster counts, or density factors must be tuned relative to application needs, often by cross-validation or target accuracy/memory levels (Chen et al., 11 Oct 2024, Meimaris et al., 2018).
• Interpretability: Hierarchical merges (particularly labelling, co-clustering, and cluster-metric-based methods) offer interpretable merging paths and hierarchies, improving traceability in critical applications.

7. Future Directions and Limitations

Emergent patterns in hierarchical merging research suggest further areas of paper:

• End-to-end learnable fusion paths: Reinforcement learning approaches (e.g., RL-based architecture merges) are a leading candidate for learning optimal merge structures adaptively (Zhou et al., 27 Sep 2024).
• Dynamic, context- or data-driven merge depth: Adaptive scheduling of merge thresholds or layers depending on input complexity or system state offers the prospect of variable resource use and finer trade-offs (Song et al., 16 Apr 2024).
• Theoretical limits of permutation/structure mismatch: While current methods resolve architectural permutations heuristically or via explicit alignment (OT, Hungarian), fundamental understanding may unlock more efficient or generalizable algorithms.
• Cross-modal hierarchical merging: Integrating semantic hierarchies from language, vision, or multimodal sources is largely unexplored, as are methods that can reconcile heterogeneous feature spaces robustly.

In sum, hierarchical merging strategy forms a versatile, theoretically rich, and practically impactful family of techniques driving advances in deep learning, data integration, clustering, robotics, and autonomous systems, with future work poised to expand its applicability and performance envelope across new domains and tasks.