Hilbert Exclusion: Improved Metric Search through Finite Isometric Embeddings
This work provides a foundational improvement for metric search in high-dimensional spaces, benefiting applications in data retrieval and machine learning.
The paper tackled the problem of similarity search in metric spaces by identifying that many common metric spaces have a stronger geometric property (the four-point property) due to isometric embeddability in Hilbert space, which allows for improved indexing mechanisms. The result is a significant increase in performance, especially in higher dimensions, leading to reduced search costs.
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 extended 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.