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.

翻译：降维技术将高维空间中的数值映射到低维空间，从而降低物理内存需求并加速距离计算。这类方法在降维后空间能够以可接受的精度保留原始空间关键属性的场景中得到广泛应用。目前已存在多种此类变换方法，可分为两大类：线性方法与拓扑方法。线性方法如主成分分析（PCA）和随机投影（RP）通过矩阵变换实现欧几里得空间降维；拓扑方法如多维缩放（MDS）则致力于保留近邻关系等高阶特征，部分方法还可应用于非欧几里得空间。本文提出nSimplex Zen——一种新型拓扑降维方法。与MDS类似，该方法仅依赖原始空间中测量的成对距离。基于距离而非坐标的特性，使其可同时适用于欧几里得空间及其他希尔伯特空间，包括由余弦距离、詹森-香农散度与二次型距离所定义的空间。研究表明，得益于高维空间的几何特性，我们的新方法在几乎所有场景下均能提供优于其他方法的性能，尤其在降维至极低维度时优势显著。