  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.
