Label-free alignment between datasets collected at different times, locations, or by different instruments is a fundamental scientific task. Hyperbolic spaces have recently provided a fruitful foundation for the development of informative representations of hierarchical data. Here, we take a purely geometric approach for label-free alignment of hierarchical datasets and introduce hyperbolic Procrustes analysis (HPA). HPA consists of new implementations of the three prototypical Procrustes analysis components: translation, scaling, and rotation, based on the Riemannian geometry of the Lorentz model of hyperbolic space. We analyze the proposed components, highlighting their useful properties for alignment. The efficacy of HPA, its theoretical properties, stability and computational efficiency are demonstrated in simulations.

Neural Information Processing Systems http://nips.cc/

In our experiments, we show that hyperbolic GNNs can lead to substantial improvements on various benchmark datasets.

Hyperbolic geometry has emerged as a powerful tool for modeling complex, structured data, particularly where hierarchical or tree-like relationships are present. By enabling embeddings with lower distortion, hyperbolic neural networks offer promising alternatives to Euclidean-based models for capturing intricate data structures. Despite these advantages, they often face performance challenges, particularly in computational efficiency and tasks requiring high precision. In this work, we address these limitations by simplifying key operations within hyperbolic neural networks, achieving notable improvements in both runtime and performance. Our findings demonstrate that streamlined hyperbolic operations can lead to substantial gains in computational speed and predictive accuracy, making hyperbolic neural networks a more viable choice for a broader range of applications. Recent datasets with complex geometric structures have highlighted the limitations of traditional Euclidean spaces in capturing hierarchical or tree-like data found in domains like complex networks (Krioukov et al., 2010), natural language processing (L opez et al., 2019; Zhu et al., 2020; L opez & Strube, 2020), and protein interactions (Zitnik et al., 2019). Hyperbolic neural networks (HNNs) (Ganea et al., 2018) and hyperbolic graph convolutional networks (HGCNs) (Chami et al., 2019) have demonstrated superior performance over Euclidean models in learning from hierarchical data.