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FlatLand: Personalized Graph Federated Learning via Tailored Lorentz Space

FlatLand is a novel personalized graph federated learning method that leverages tailored Lorentz space within hyperbolic geometry to effectively model client heterogeneity and graph structures through a parameter decoupling strategy, achieving superior performance without requiring explicit client similarity estimation.

Original authors: Jiahong Liu, Ram Samarth B B, Xinyu Fu, Menglin Yang, Weixi Zhang, Rex Ying, Irwin King

Published 2026-08-24

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CC BY 4.0

Original authors: Jiahong Liu, Ram Samarth B B, Xinyu Fu, Menglin Yang, Weixi Zhang, Rex Ying, Irwin King

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the digital age, machine learning often requires vast amounts of data to function well. Yet, much of this data is locked away in private silos, held by hospitals, banks, or individual devices, where sharing it would violate privacy laws or trust. To solve this, scientists developed a method called federated learning. Instead of gathering all the data in one place, this approach sends a shared computer model out to many different locations. Each location trains the model on its own private data and sends back only the learned improvements, not the data itself. This allows a global intelligence to emerge without ever exposing the raw information. However, a significant problem arises when the data at these different locations looks very different from one another. In the world of graphs—networks of connected points like social media friends or chemical molecules—these differences are not just about how many items exist, but about the very shape of the connections. Some networks are flat and uniform, while others are deeply curved and hierarchical. When a single, rigid model tries to learn from such diverse shapes, it often fails, performing worse than if each location had simply trained on its own.

Researchers have long tried to fix this by creating personalized versions of the model for each location, but these solutions usually force the data into a flat, uniform mathematical space, ignoring the natural curves of the network. A new study introduces a different approach called FlatLand, which embraces the curved nature of these networks rather than trying to flatten them. The team realized that real-world networks often possess a specific type of negative curvature, similar to the surface of a saddle, which standard flat mathematics cannot capture efficiently. To address this, they designed a system where each client, or data holder, is allowed to train its model within a uniquely curved mathematical space tailored to its specific data structure. This space is based on a geometry known as hyperbolic space, which naturally accommodates the way connections branch out in complex networks.

The core innovation of FlatLand lies in how it handles the exchange of information between these differently curved spaces. The researchers discovered that the differences between clients could be separated from the common knowledge they share. They split the model's internal settings into two distinct parts: one part that captures the unique, client-specific quirks of the local data, and another part that holds the universal patterns useful to everyone. The unique parts remain private and local, while the universal parts are sent to a central server to be combined. This separation is achieved by using a specific mathematical dimension that acts as a carrier for the differences, allowing the shared information to be aggregated cleanly without mixing in the noise of local variations. This process eliminates the need for complex calculations to guess how similar different clients are, a step that previous methods required and which often slowed things down.

To test their idea, the team applied FlatLand to a variety of real-world graph datasets, ranging from citation networks of scientific papers to social networks and chemical structures. They compared their method against existing techniques across different scenarios, including situations where only a small number of clients participated in each round of training. The results showed that FlatLand consistently outperformed other methods, particularly when the models were kept small and efficient. In many cases, the new approach achieved higher accuracy in predicting connections or classifying nodes than even the best local training methods, proving that collaboration could indeed be beneficial even when the data was highly diverse. The study also found that the method was especially robust when the data was sparse or when the network structures were very different from one another, a scenario where traditional flat models often struggle.

One of the most striking findings was how well the system worked with very low-dimensional representations. In machine learning, keeping the model small is crucial for saving bandwidth and energy, especially when communicating with many devices. While other methods lost significant accuracy when forced to use such compact models, FlatLand maintained its high performance. This suggests that by respecting the natural geometry of the data, the system can learn more effectively with fewer resources. The researchers also noted that the method did not require the clients to share any sensitive statistics about their data or their similarity to others, preserving privacy by design. Only the shared, universal components of the model were transmitted, while the personalized settings and the specific curvature of each client's space remained hidden.

The work does not claim that this curved approach is the perfect solution for every possible type of network, as some data might still fit better in a flat space. However, for the vast majority of complex, real-world networks that exhibit these natural curves, the study demonstrates a clear path forward. By moving away from the assumption that all data must fit into a single, flat framework, FlatLand offers a more natural and efficient way to learn together. It shows that when we allow our mathematical tools to bend and adapt to the shape of the information they are trying to understand, we can build smarter, more private, and more collaborative systems. The success of this approach suggests that the future of decentralized learning may lie not in forcing data to conform to a standard shape, but in finding the right curved space for each piece of the puzzle.

Technical Summary: FlatLand

Problem Statement
Federated Learning (FL) enables privacy-preserving collaborative training but struggles with data heterogeneity, a challenge that is particularly acute in Graph Federated Learning (GFL). In GFL, clients possess structurally diverse graphs, leading to pronounced structural heterogeneity that standard FL models cannot satisfy. Existing Personalized Federated Learning (PFL) methods for graphs typically operate within Euclidean space, implicitly assuming a uniform, flat geometry across all client data distributions. This assumption forces existing methods to rely on intricate mechanisms—such as client similarity estimation, parameter disentanglement, or auxiliary modules—to handle heterogeneity, often incurring significant computational overhead. Furthermore, empirical analysis reveals that real-world graph data frequently exhibits negative Ricci curvature (hyperbolic structure) and that curvature values vary substantially across clients, suggesting that a unified Euclidean geometry is fundamentally misaligned with the intrinsic nature of graph data.

Methodology: FlatLand
The authors propose FlatLand, a personalized GFL framework that embeds client data in tailored Lorentz spaces of hyperbolic geometry to faithfully capture intrinsic geometric properties. The method consists of three core components:

Tailored Lorentz Spaces: Instead of a single global geometry, FlatLand assigns each client a specific Lorentz scale parameter ( $K_c$ ), defining a unique hyperbolic surface ( $L^{d}_{K_c}$ ) with sectional curvature $-1/K_c$ . This allows the model to adapt to the varying Ricci curvature observed across different client graphs. The curvature parameter is initialized using Forman-Ricci curvature but is treated as a learnable quantity during local training to account for feature and label heterogeneity.
Parameter Decoupling Strategy: Leveraging the structure of the Lorentz model, the framework decomposes the transformation parameters of the neural network into two distinct sets:
- Space-like parameters ( $\theta_s$ ): These correspond to the spatial dimensions ( $x_s$ ) and carry common, transferable knowledge.
- Time-like parameters ( $\theta_c$ ): These correspond to the time-like dimension ( $x_t$ ) and encode client-specific heterogeneity.
  The server aggregates only the space-like parameters ( $\theta_s$ ) using a standard averaging approach (similar to FedAvg), while the time-like parameters and the curvature scale $K_c$ remain local to each client. This design theoretically ensures that client-specific heterogeneity is preserved without requiring explicit client similarity estimation or auxiliary modules.
Fully Lorentz Neural Networks: The framework utilizes Fully Lorentz networks, which operate entirely within the hyperbolic manifold, avoiding the need for repeated projections between Euclidean and hyperbolic spaces and enhancing computational efficiency.

Key Contributions

Geometric Perspective on PFL: The paper is the first to bridge hyperbolic geometry and PFL to address client heterogeneity in a principled manner. It theoretically establishes that tailored curvature is necessary for minimizing embedding distortion when client graphs possess varying Ricci curvatures (Theorem 4.1).
Intrinsic Heterogeneity Encoding: The authors demonstrate that the time-like dimension in Lorentz space naturally encodes client-specific geometric heterogeneity, while space-like dimensions preserve shared information (Theorem 4.3).
Succinct Framework: FlatLand achieves personalization without the computational overhead of client clustering or similarity estimation. It is built directly upon the FedAvg paradigm, requiring only the aggregation of space-like parameters.
Theoretical Guarantees: The paper provides proofs for the correctness of the method (ensuring representations remain valid in Lorentz space after aggregation), convergence rates comparable to FedAvg ( $O(1/T)$ ), and minimal computational overhead.

Experimental Results
The authors evaluated FlatLand on diverse node classification and graph classification tasks across multiple real-world datasets (e.g., Cora, CiteSeer, ogbn-arxiv, Amazon-Photo, and heterophilic datasets like Roman-Empire and Minesweeper).

Performance: FlatLand consistently outperformed state-of-the-art baselines, including Euclidean PFL methods (FedPer, GCFL), standard FL methods (FedAvg, FedProx), and hyperbolic FL baselines (FedHGCN). It showed particularly strong gains in low-dimensional settings and on highly heterogeneous datasets.
Low-Dimensional Efficiency: In experiments reducing embedding dimensions from 64 to 4, FlatLand maintained superior performance compared to Euclidean counterparts, highlighting the representational power of hyperbolic geometry in compact settings.
Robustness: The method demonstrated robustness under partial client participation (e.g., 10% participation rate), significantly outperforming FedAvg, which suffered from sensitivity to sampled clients.
Ablation Studies: Experiments confirmed that both the adaptive curvature initialization and the time-like parameter decoupling strategy are critical. Removing the decoupling strategy led to significant performance degradation, and using a fixed curvature parameter yielded inferior results compared to learnable, client-specific curvature.

Significance and Claims
The paper claims that FlatLand offers a new, succinct, and effective perspective for personalized federated learning by leveraging geometric properties rather than statistical heuristics. By mapping diverse data distributions onto different Lorentz surfaces, the method naturally accommodates the intrinsic negative curvature of real-world graphs. The authors assert that this approach effectively mitigates heterogeneity without requiring extra calculation modules or client similarity estimation, making it particularly suitable for communication-efficient federated learning scenarios where low-dimensional representations are crucial. The work positions non-Euclidean geometry as a promising direction for addressing heterogeneity in federated settings, moving beyond the limitations of uniform Euclidean assumptions.

Technical Summary: FlatLand

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