Max-sliced Bures Distance for Interpreting Discrepancies
Authors: Austin J. Brockmeier, Claudio Cesar Claros, Carlos H. Mendoza-Cardenas, Yüksel Karahan, Matthew S. Emigh, Luis Gonzalo Sanchez Giraldo
Keywords: covariance, covariate shift, distance metrics, divergence, generative adversarial networks, interpretable approaches, kernel methods, probability metric, RKHS
Abstract: We propose the max-sliced Bures distance, a lower bound on the max-sliced Wasserstein-2 distance, to identify the instances associated with the maximum discrepancy between two samples. The max-slicing can be decomposed into two asymmetric divergences each expressed in terms of an optimal slice or equivalently a witness function that has large magnitude evaluations on a localized subset of instances in one distribution versus the other. We show how witness functions can be used to detect and correct for covariate shift through reweighting and to evaluate generative adversarial networks.  Unlike heuristic algorithms for the max-sliced Wasserstein-2 distance that may fail to find the optimal slice, we detail a tractable algorithm that finds the global optimal slice and scales to large sample sizes.  As the Bures distance quantifies differences in covariance, we generalize the max-sliced Bures distance by using non-linear mappings, enabling it to capture changes in higher-order statistics. We explore two types of non-linear mappings: positive semidefinite kernels where the witness functions belong to a reproducing kernel Hilbert space, and task-relevant mappings corresponding to a neural network. In the context of samples of natural images, our approach provides an interpretation of the Fréchet Inception distance by identifying the synthetic and natural instances that are either over-represented or under-represented with respect to the other sample. We apply the proposed measure to detect imbalances in class distributions in various data sets and to critique generative models.