The need to understand the structure of hierarchical or high-dimensional data
is present in a variety of fields. Hyperbolic spaces have proven to be an
important tool for embedding computations and analysis tasks as their
non-linear nature lends itself well to tree or graph data. Subsequently, they
have also been used in the visualization of high-dimensional data, where they
exhibit increased embedding performance. However, none of the existing
dimensionality reduction methods for embedding into hyperbolic spaces scale
well with the size of the input data. That is because the embeddings are
computed via iterative optimization schemes and the computation cost of every
iteration is quadratic in the size of the input. Furthermore, due to the
non-linear nature of hyperbolic spaces, Euclidean acceleration structures
cannot directly be translated to the hyperbolic setting. This paper introduces
the first acceleration structure for hyperbolic embeddings, building upon a
polar quadtree. We compare our approach with existing methods and demonstrate
that it computes embeddings of similar quality in significantly less time.
Implementation and scripts for the experiments can be found at
this https URL

本研究使用极坐标四叉树构建超几何嵌入的加速结构，相比现有方法在更短的时间内计算出类似质量的嵌入结果。