Neuc-MDS: Non-Euclidean Multidimensional Scaling Through Bilinear Forms

Abstract

We introduce \textbf{N}on-\textbf{Euc}lidean-\textbf{MDS} (Neuc-MDS), which extends Multidimensional Scaling (MDS) to generate outputs that can be non-Euclidean and non-metric. The main idea is to generalize the inner product to other symmetric bilinear forms to utilize the negative eigenvalues of dissimiliarity Gram matrices. Neuc-MDS efficiently optimizes the choice of (both positive and negative) eigenvalues of the dissimilarity Gram matrix to reduce STRESS, the sum of squared pairwise error. We provide an in-depth error analysis and proofs of the optimality in minimizing lower bounds of STRESS. We demonstrate Neuc-MDS's ability to address limitations of classical MDS raised by prior research, and test it on various synthetic and real-world datasets in comparison with both linear and non-linear dimension reduction methods.