Optimal restoration of stochastic monotonicity with respect to cumulative label frequency loss functions

A method to restore stochastic monotonicity of noisy multi-criteria data sets through relabeling is presented. By formulating the problem as a weighted maximum independent set problem on a comparability graph, it is possible to compute optimal relabelings w.r.t. cumulative label frequency loss function. We demonstrate how to formulate the problem in this manner and discuss why it requires objects to be relabeled instead of deleted. More precisely, we will formulate the zero-one cumulative label frequency loss, L1 cumulative label frequency loss and squared cumulative label frequency loss, and provide a weighing function for each. We investigate these loss functions in the related context of restoring regular monotonicity, dealing with objects with a single label, rather than distributions. Finally, we provide applications on some closely related example data sets and discuss some interesting findings.