Hilbert exclusion: improved metric search through finite isometric embeddings

Richard Connor, Franco Alberto Cardillo, Lucia Vadicamo, Fausto Rabitti

Research output: Contribution to journalArticlepeer-review

22 Citations (Scopus)


Most research into similarity search in metric spaces relies on the triangle inequality property. This property allows the space to be arranged according to relative distances to avoid searching some subspaces. We show that many common metric spaces, notably including those using Euclidean and Jensen-Shannon distances, also have a stronger property, sometimes called the four-point property: In essence, these spaces allow an isometric embedding of any four points in three-dimensional Euclidean space, as well as any three points in two-dimensional Euclidean space. In fact, we show that any space that is isometrically embeddable in Hilbert space has the stronger property. This property gives stronger geometric guarantees, and one in particular, which we name the Hilbert Exclusion property, allows any indexing mechanism which uses hyperplane partitioning to perform better. One outcome of this observation is that a number of state-of-the-art indexing mechanisms over high-dimensional spaces can be easily refined to give a significant increase in performance; furthermore, the improvement given is greater in higher dimensions. This therefore leads to a significant improvement in the cost of metric search in these spaces.

Original languageEnglish
Article number17
Number of pages27
JournalACM Transactions on Information Systems
Issue number3
Early online date15 Dec 2016
Publication statusPublished - Jun 2017


  • Four-point property
  • Hilbert embedding
  • Metric indexing
  • Metric Space
  • Similarity search


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