nSimplex Zen: A Novel Dimensionality Reduction for Euclidean and Hilbert Spaces

TL;DR

nSimplex Zen is a new topological dimensionality reduction method that preserves pairwise distances and works in both Euclidean and Hilbert spaces, outperforming existing techniques especially in very low dimensions.

Contribution

It introduces nSimplex Zen, a novel distance-based topological reduction method applicable to Euclidean and Hilbert spaces, improving low-dimensional embeddings.

Findings

Outperforms existing methods in low-dimensional reductions

Applicable to various distance measures including Cosine and Jensen-Shannon

Provides better geometric properties in high-dimensional spaces

Abstract

Dimensionality reduction techniques map values from a high dimensional space to one with a lower dimension. The result is a space which requires less physical memory and has a faster distance calculation. These techniques are widely used where required properties of the reduced-dimension space give an acceptable accuracy with respect to the original space. Many such transforms have been described. They have been classified in two main groups: linear and topological. Linear methods such as Principal Component Analysis (PCA) and Random Projection (RP) define matrix-based transforms into a lower dimension of Euclidean space. Topological methods such as Multidimensional Scaling (MDS) attempt to preserve higher-level aspects such as the nearest-neighbour relation, and some may be applied to non-Euclidean spaces. Here, we introduce nSimplex Zen, a novel topological method of reducing dimensionality. Like MDS, it relies only upon pairwise distances measured in the original space. The use of distances, rather than coordinates, allows the technique to be applied to both Euclidean and other Hilbert spaces, including those governed by Cosine, Jensen-Shannon and Quadratic Form distances. We show that in almost all cases, due to geometric properties of high-dimensional spaces, our new technique gives better properties than others, especially with reduction to very low dimensions.