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It is common in many applications to quantify similarities between objects of a domain by embedding the domain into a Euclidean space. Any such embedding induces a metric on the domain, which can be used for measuring similarity. Suppose that new knowledge about the domain has emerged. Are this knowledge and the metric induced by the embedding consistent? If not, the embedding must be adjusted to attain consistency. In this paper, we consider the case in which the new knowledge emerges in the form of a taxonomic hierarchy on a subset of the domain. We give a polynomial-time algorithm that adjusts the embedding of this subset so that the corresponding adjusted metric is consistent with the taxonomic hierarchy. The algorithm is geometric in nature, namely, it uses an isometric embedding of a finite ultrametric space into a Euclidean space.

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Quantification of Similarities: Embeddings Into Euclidean Spaces and Taxonomies

  • 会议论文

  • 发表日期：

    2026年

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  • Evgeny Dantsin,
  • Alexander Wolpert

摘要

It is common in many applications to quantify similarities between objects of a domain by embedding the domain into a Euclidean space. Any such embedding induces a metric on the domain, which can be used for measuring similarity. Suppose that new knowledge about the domain has emerged. Are this knowledge and the metric induced by the embedding consistent? If not, the embedding must be adjusted to attain consistency. In this paper, we consider the case in which the new knowledge emerges in the form of a taxonomic hierarchy on a subset of the domain. We give a polynomial-time algorithm that adjusts the embedding of this subset so that the corresponding adjusted metric is consistent with the taxonomic hierarchy. The algorithm is geometric in nature, namely, it uses an isometric embedding of a finite ultrametric space into a Euclidean space.