Max-sliced Bures Distance for Interpreting Discrepancies
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. By scoring instances by an energy-based witness function, the proposed divergence can identify the specific subsets that are not well matched between the distributions. Localizing these discrepancies can be used to detect and correct for covariate shift and to evaluate generative adversarial networks. Similar to the Fréchet distance, the max-sliced Bures distance can be computed using the first and second-order moments and is able to capture changes in higher-order statistics through a non-linear mapping. 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 learned by 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.
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