Median-Based Classifiers for High-Dimensional Data

Abstract

Conventional distance-based classifiers use standard Euclidean distance, and so can suffer from excessive volatility if vector components have heavy-tailed distributions. This difficulty can be alleviated by replacing the L2 distance by its L1 counterpart. For example, the L1 version of the popular centroid classifier would allocate a new data value to the population to whose centroid it was closest in L1 terms. However, this approach can lead to inconsistency, because the centroid is defined using L2, rather than L1 distance. In particular, by mixing L1 and L2 approaches, we produce a classifier that can seriously misidentify data in cases where the means and medians of marginal distributions take different values. These difficulties motivate replacing centroids by medians. However, in the very-high-dimensional settings commonly encountered today, this can be problematic if we attempt to work with a conventional spatial median. Therefore, we suggest using componentwise medians to construct a robust classifier that is relatively insensitive to the difficulties caused by heavy-tailed data and entails straightforward computation. We also consider generalizations and extensions of this approach based on, for example, using data truncation to achieve additional robustness. Using both empirical and theoretical arguments, we explore the properties of these methods, and show that the resulting classifiers can be particularly effective. Supplementary materials are available online.