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更低维的空间，从而得到所需物理内存更少且距离计算更快的空间。这类技术广泛应用于降维后空间在保持原始空间可接受精度的条件下满足所需特性的场景。目前已有多类此类变换方法被提出，主要分为线性方法与拓扑方法两类。线性方法如主成分分析（PCA）与随机投影（RP）通过矩阵变换将数据映射到欧几里得空间的低维子空间。拓扑方法如多维尺度分析（MDS）则致力于保留更高级别的结构特征（如最近邻关系），且部分方法可适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。由于使用距离而非坐标，该技术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、Jensen-Shannon距离和二次型距离主导的空间。研究表明，基于高维空间的几何特性，本方法在绝大多数场景下（尤其是降维至极低维度时）相比其他方法具有更优的性质。