A Bayesian Approach to Non-Metric Hyperbolic Multi-Dimensional Scaling

Abstract

This paper explores the intersection of hyperbolic geometry, non-metric techniques, and Bayesian frameworks to extend the capabilities of Bayesian Hyperbolic Multi-Dimensional Scaling (HMDS). While hyperbolic geometry is gaining attention for its ability to represent hierarchical relationships, traditional metrics impose constraints on distances. Non-metric techniques offer flexibility in capturing complex structures, making them suitable for scenarios where metric distances are less meaningful. The paper introduces a novel extension of Bayesian HMDS, incorporating non-metric techniques, enabling the embedding of Euclidean data within a hyperbolic space. The approach simultaneously fits for curvature and coordinates, leveraging the scaling properties of hyperbolic space. The non-metric Bayesian Hyperbolic MDS is expected to unveil new insights into hierarchical structures within complex datasets, providing a versa-tile tool for analyzing high-dimensional data flexibly and accurately. The efficacy of the proposed method is demonstrated through synthetic data experiments, showcasing its ability to capture non-linear transformations and accurately predict underlying curvature, with an emphasis on its ro-bustness to hyperparameter choices.