RegD: Hierarchical Embeddings via Distances over Geometric Regions

Hierarchical data are common in many domains like life sciences and e-commerce, and their embeddings often play a critical role. Although hyperbolic embeddings offer a grounded approach to representing hierarchical structures in low-dimensional spaces, their utility is hindered by optimization difficulties in hyperbolic space and dependence on handcrafted structural constraints. We propose RegD, a novel Euclidean framework that addresses these limitations by representing hierarchical data as geometric regions with two new metrics: (1) depth distance, which preserves the representational power of hyperbolic spaces for hierarchical data, and (2) boundary distance, which explicitly encodes set-inclusion relationships between regions in a general way. Our empirical evaluation on diverse real-world datasets shows consistent performance gains over state-of-the-art methods and demonstrates RegD's potential for broader applications beyond hierarchy alone tasks.