Hyperbolic Procrustes Analysis Using Riemannian Geometry
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. In addition, we showcase its performance on three batch correction tasks involving gene expression and mass cytometry data. Specifically, we demonstrate high-quality unsupervised batch effect removal from data acquired at different sites and with different technologies that outperforms recent methods for label-free alignment in hyperbolic spaces.
Code (3)
Tasks
Computational EfficiencyTranslationMethods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
On Procrustes Analysis in Hyperbolic Space
Congruent Procrustes analysis aims to find the best matching between two point sets through rotation, reflection and translation. We formulate the Procrustes problem for hyperbolic spaces, review the canonical definition…
TranslationFast hyperboloid decision tree algorithms
Hyperbolic geometry is gaining traction in machine learning for its effectiveness at capturing hierarchical structures in real-world data. Hyperbolic spaces, where neighborhoods grow exponentially, offer substantial adva…
BenchmarkingRiemannian optimizationUltrahyperbolic Neural Networks
Riemannian space forms, such as the Euclidean space, sphere and hyperbolic space, are popular and powerful representation spaces in machine learning. For instance, hyperbolic geometry is appropriate to represent graphs w…
Node ClassificationUltrahyperbolic Neural Networks
Riemannian space forms, such as the Euclidean space, sphere and hyperbolic space, are popular and powerful representation spaces in machine learning. For instance, hyperbolic geometry is appropriate to represent graphs w…
Node ClassificationUltrahyperbolic Knowledge Graph Embeddings
Recent knowledge graph (KG) embeddings have been advanced by hyperbolic geometry due to its superior capability for representing hierarchies. The topological structures of real-world KGs, however, are rather heterogeneou…
Knowledge Graph Embeddings