Proposing a Dimensionality Reduction Technique With an Inequality for Unsupervised Learning from High-Dimensional Big Data

Data-clustering task can be considered as the most important unsupervised learning algorithms. For about all clustering algorithms, finding the Nearest Neighbors of a point within a certain radius <inline-formula> <tex-math notation="LaTeX">${r}$ </tex-math></inline-formula> (NN-<inline-formula> <tex-math notation="LaTeX">${r}$ </tex-math></inline-formula>), is a critical task. For a high-dimensional dataset, this task becomes too time consuming. This article proposes a simple dimensionality reduction (DR) technique. For point <inline-formula> <tex-math notation="LaTeX">${p}$ </tex-math></inline-formula> in <inline-formula> <tex-math notation="LaTeX">${d}$ </tex-math></inline-formula>-dimensional space, it produces point <inline-formula> <tex-math notation="LaTeX">$p^{\prime }$ </tex-math></inline-formula> in <inline-formula> <tex-math notation="LaTeX">$d^{\prime }$ </tex-math></inline-formula>-dimensional space, where <inline-formula> <tex-math notation="LaTeX">$d^{\prime } < < {d}$ </tex-math></inline-formula>. In addition, for any pair of points <inline-formula> <tex-math notation="LaTeX">${p}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>, and their maps <inline-formula> <tex-math notation="LaTeX">$p^{\prime }$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$q^{\prime }$ </tex-math></inline-formula> in the target space, it is proved that <inline-formula> <tex-math notation="LaTeX">$|p, q| > |p^{\prime }, q^{\prime }|$ </tex-math></inline-formula> is preserved, where <inline-formula> <tex-math notation="LaTeX">$|,|$ </tex-math></inline-formula> used to denote the Euclidean distance between a pair of points. This property can speed up finding NN-<inline-formula> <tex-math notation="LaTeX">${r}$ </tex-math></inline-formula>. For a certain radius <inline-formula> <tex-math notation="LaTeX">${r}$ </tex-math></inline-formula>, and a pair of points <inline-formula> <tex-math notation="LaTeX">${p}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>, whenever <inline-formula> <tex-math notation="LaTeX">$|p^{\prime }, q^{\prime }| > r$ </tex-math></inline-formula>, then <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula> can not be in NN-<inline-formula> <tex-math notation="LaTeX">${r}$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">${p}$ </tex-math></inline-formula>. Using this trick, the task of finding the NN-<inline-formula> <tex-math notation="LaTeX">${r}$ </tex-math></inline-formula> is speeded up. Then, as a case study, it is applied to accelerate the <inline-formula> <tex-math notation="LaTeX">${k}$ </tex-math></inline-formula>-means, one of the most famous unsupervised learning algorithms, where it can automatically determine the <inline-formula> <tex-math notation="LaTeX">$d^{\prime }$ </tex-math></inline-formula>. The proposed NN-<inline-formula> <tex-math notation="LaTeX">${r}$ </tex-math></inline-formula> method and the accelerated <inline-formula> <tex-math notation="LaTeX">${k}$ </tex-math></inline-formula>-means are compared with recent state-of-the-arts, and both yield favorable results.