A Matrix Computation View of FastMap and RobustMap Dimension Reduction Algorithms

Abstract

Given a set of pairwise object distances and a dimension k, FastMap and RobustMap algorithms compute a set of k-dimensional coordinates for the objects. These metric space embedding methods implicitly assume a higher-dimensional coordinate representation and are a sequence of translations and orthogonal projections based on a sequence of object pair selections (called pivot pairs). We develop a matrix computation viewpoint of these algorithms that operates on the coordinate representation explicitly using Householder reflections. The resulting coordinate mapping algorithm (CMA) is a fast approximate alternative to truncated principal component analysis (PCA), and it brings the FastMap and RobustMap algorithms into the mainstream of numerical computation where standard BLAS building blocks are used. Motivated by the geometric nature of the embedding methods, we further show that truncated PCA can be computed with CMA by specific pivot pair selections. Describing FastMap, RobustMap, and PCA as CMA computations with different pivot pair choices unifies the methods along a pivot pair selection spectrum. We also sketch connections to the semidiscrete decomposition and the QLP decomposition.