A dual basis approach to multidimensional scaling
Abstract
Classical multidimensional scaling (CMDS) is a technique that embeds a set of objects in a Euclidean space given their pairwise Euclidean distances. The main part of CMDS involves double centering a squared distance matrix and using a truncated eigendecomposition to recover the point coordinates. In this paper, motivated by a study in Euclidean distance geometry, we explore a dual basis approach to CMDS. We give an explicit formula for the dual basis vectors and fully characterize the spectrum of an essential matrix in the dual basis framework. We make connections to a related problem in metric nearness.
Cite
@article{arxiv.2303.05682,
title = {A dual basis approach to multidimensional scaling},
author = {Samuel Lichtenberg and Abiy Tasissa},
journal= {arXiv preprint arXiv:2303.05682},
year = {2024}
}
Comments
7 pages. The proof of dual basis representation is now compact. It is not constructive compared to the previous version, but it uses bi-orthogonality relation to establish the result more directly. A minor error in the proof of the spectrum of the dual basis has been fixed. We also made few changes for better clarity and presentation