Multi-class Bayes error estimation with a global minimal spanning tree

Henze-Penrose (HP) divergence has been used in many information theory, statistics and machine learning contexts, including the estimation of two-class Bayes classification error. Previous work has shown HP divergence can be directly estimated using the Friedman-Rafsky (FR) multivariate run test statistic. For the multi-class classification problem, HP divergence can also be used to bound the Bayes error by estimating the sum of pairwise Bayes errors between classes. In situations in which the dataset and number of classes are large, this approach is infeasible. In this paper, we present a new generalized measure that allows us to estimate the Bayes error rate without the need to compute pairwise estimates. We compare our new approach with the pairwise HP bound and the bound proposed by Lin [1], and show that our upper bound on Bayes error is tighter, while also having lower computational complexity.