Hilbert Exclusion: improved metric search through finite isometric embeddings

TY - JOUR

T1 - Hilbert Exclusion

T2 - ACM Transactions on Information Systems

AU - Connor, Richard

AU - Cardillo, Franco Alberto

AU - Vadicamo, Lucia

AU - Rabitti, Fausto

N1 - © ACM, 2016. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in ACM Transactions on Information Systems, Vol. 35, No. 3, 17, 15.12.2016. http://doi.acm.org/10.1145/3001583

PY - 2016/12/15

Y1 - 2016/12/15

N2 - Most research into similarity search in metric spaces relies upon 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 which 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.

AB - Most research into similarity search in metric spaces relies upon 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 which 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.

KW - Similarity search

KW - Metric Space

KW - Metric Indexing

KW - Four-point property

KW - Hilbert Embedding

UR - http://tois.acm.org/

U2 - 10.1145/3001583

DO - 10.1145/3001583

M3 - Article

VL - 35

JO - ACM Transactions on Information Systems

JF - ACM Transactions on Information Systems

SN - 1046-8188

IS - 3

M1 - 17

ER -