Submodular Hamming Metrics
This work addresses the problem of defining and optimizing discrete metrics for binary vectors, which is incremental in advancing submodular optimization techniques.
The paper introduces positive polymatroids as a class of functions that define proper discrete metrics for binary vectors, showing they are tractable to optimize over using submodularity, with empirical results on clustering and diverse k-best list generation tasks.
We show that there is a largely unexplored class of functions (positive polymatroids) that can define proper discrete metrics over pairs of binary vectors and that are fairly tractable to optimize over. By exploiting submodularity, we are able to give hardness results and approximation algorithms for optimizing over such metrics. Additionally, we demonstrate empirically the effectiveness of these metrics and associated algorithms on both a metric minimization task (a form of clustering) and also a metric maximization task (generating diverse k-best lists).