Istituto di Scienza e Tecnologie dell'Informazione     
Connor R., Cardillo F. A., Vadicamo L., Rabitti F. Hilbert exclusion: improved metric search through finite isometric embeddings. In: ACM Transactions on Information Systems, vol. 35 (3) article n. 17. ACM, 2017.
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.
URL: http://doi.acm.org/10.1145/3001583
DOI: 10.1145/3001583
Subject Similarity search
Metric space
Metric indexing
Four-point property
Hilbert embedding
H. Information systems. Data structures
H. Information systems. Multidimensional range search
H. Information systems. Proximity search
H. Information systems. Database query processing
H. Information systems. Retrieval models and ranking
H. Information systems. Retrieval efficiency
H. Information systems. Multimedia information systems
F. Theory of computation. Random projections and metric embeddings

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