Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding
This work addresses the need for efficient and accurate hyperbolic embeddings to extract hierarchical information from data, with incremental improvements in speed and performance over existing algorithms.
The paper tackles the problem of learning low-dimensional hyperbolic embeddings for data by introducing a fast algorithm, TreeRep, that learns a tree structure from a δ-hyperbolic metric, which can then be used for embedding or hierarchical extraction. The result shows that TreeRep is orders of magnitude faster than previous methods and achieves lower average distortion and higher mean average precision in empirical evaluations.
Given data, finding a faithful low-dimensional hyperbolic embedding of the data is a key method by which we can extract hierarchical information or learn representative geometric features of the data. In this paper, we explore a new method for learning hyperbolic representations by taking a metric-first approach. Rather than determining the low-dimensional hyperbolic embedding directly, we learn a tree structure on the data. This tree structure can then be used directly to extract hierarchical information, embedded into a hyperbolic manifold using Sarkar's construction \cite{sarkar}, or used as a tree approximation of the original metric. To this end, we present a novel fast algorithm \textsc{TreeRep} such that, given a $δ$-hyperbolic metric (for any $δ\geq 0$), the algorithm learns a tree structure that approximates the original metric. In the case when $δ= 0$, we show analytically that \textsc{TreeRep} exactly recovers the original tree structure. We show empirically that \textsc{TreeRep} is not only many orders of magnitude faster than previously known algorithms, but also produces metrics with lower average distortion and higher mean average precision than most previous algorithms for learning hyperbolic embeddings, extracting hierarchical information, and approximating metrics via tree metrics.