High-Dimensional Simplexes for Supermetric Search

Abstract

In a metric space, triangle inequality implies that, for any three objects, a triangle with edge lengths corresponding to their pairwise distances can be formed. The n-point property is a generalisation of this where, for any $(n+1)$ objects in the space, there exists an n-dimensional simplex whose edge lengths correspond to the distances among the objects. In general, metric spaces do not have this property; however in 1953, Blumenthal showed that any semi-metric space which is isometrically embeddable in a Hilbert space also has the n-point property.

We have previously called such spaces supermetric spaces, and have shown that many metric spaces are also supermetric, including Euclidean, Cosine, Jensen-Shannon and Triangular spaces of any dimension.

Here we show how such simplexes can be constructed from only their edge lengths, and we show how the geometry of the simplexes can be used to determine lower and upper bounds on unknown distances within the original space. By increasing the number of dimensions, these bounds converge to the true distance.

Finally we show that for any Hilbert-embeddable space, it is possible to construct Euclidean spaces of arbitrary dimensions, from which these lower and upper bounds of the original space can be determined. These spaces may be much cheaper to query than the original. For similarity search, the engineering tradeoffs are good: we show significant reductions in data size and metric cost with little loss of accuracy, leading to a significant overall improvement in exact search performance.