On FastMap and the convex hull of multivariate data: toward fast and robust dimension reduction

FastMap is a dimension reduction technique that operates on distances between objects. Although only distances are used, implicitly the technique assumes that the objects are points in a p-dimensional Euclidean space. It selects a sequence of k ≤ p orthogonal axes defined by distant pairs of points (called pivots) and computes the projection of the points onto the orthogonal axes. We show that FastMap uses only the outer envelope of a data set. Pivots are taken from the faces, usually vertices, of the convex hull of the data points in the original implicit Euclidean space. This provides a bridge to results in robust statistics, where the convex hull is used as a tool in multivariate outlier detection and in robust estimation methods. The connection sheds new light on the properties of FastMap, particularly its sensitivity to outliers, and provides an opportunity for a new class of dimension reduction algorithms, RobustMaps, that retain the speed of FastMap and exploit ideas in robust statistics.