On the Expected Exclusion Power of Binary Partitions for Metric Search

Abstract

The entire history and, we dare say, future of similarity search is governed by the underlying notion of partition. A partition is an equivalence relation defined over the space, therefore each element of the space is contained within precisely one of the equivalence classes of the partition. All attempts to search a finite space efficiently, whether exactly or approximately, rely on some set of principles which imply that if the query is within one equivalence class, then one or more other classes either cannot, or probably do not, contain any of its solutions.

In most early research, partitions relied only on the metric postulates, and logarithmic search time could be obtained on low dimensional spaces. In these cases, it was straightforward to identify multiple partitions, each of which gave a relatively high probability of identifying subsets of the space which could not contain solutions. Over time the datasets being searched have become more complex, leading to higher dimensional spaces. It is now understood that even an approximate search in a very high-dimensional space is destined to require $\mathcal {O}(n)$ time and space.

Almost entirely missing from the research literature however is any analysis of exactly when this effect takes over. In this paper, we make a start on tackling this important issue. Using a quantitative approach, we aim to shed some light on the notion of the exclusion power of partitions, in an attempt to better understand their nature with respect to increasing dimensionality.