A dual basis approach to multidimensional scaling
This is an incremental improvement for researchers in dimensionality reduction and distance geometry.
The paper tackles the problem of classical multidimensional scaling (CMDS) by proposing a dual basis approach, resulting in an explicit formula for dual basis vectors and a full characterization of the spectrum of an essential matrix.
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.