Embedding Methods and Robust Statistics for Dimension Reduction

Recently, several non-deterministic distance embedding methods that can be used for fast dimension reduction have been proposed in the machine learning literature. These include FastMap, MetricMap, and SparseMap. Among them, FastMap, implicitly 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 picks all of its pivots from the vertices of the convex hull of the data points in the original implicit Euclidean space. This provides a connection 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 a new light on some properties of FastMap and provides an opportunity for a robust class of dimension reduction algorithms that we call RobustMaps, which retain the speed of FastMap and exploit ideas in robust statistics. One simple RobustMap algorithm is shown to outperform principal components on contaminated data both in terms of clean variance captured and in terms of time complexity.