Eikonal depth: an optimal control approach to statistical depths
Abstract
Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which measures the smallest amount of probability density that has to be passed through in a path to points outside the support of the distribution: for example spatial infinity. This depth is easy to interpret and compute, expressively captures multi-modal behavior, and extends naturally to data that is non-Euclidean. We prove various properties of this depth, and provide discussion of computational considerations. In particular, we demonstrate that this notion of depth is robust under an aproximate isometrically constrained adversarial model, a property which is not enjoyed by the Tukey depth. Finally we give some illustrative examples in the context of two-dimensional mixture models and MNIST.
Cite
@article{arxiv.2201.05274,
title = {Eikonal depth: an optimal control approach to statistical depths},
author = {Martin Molina-Fructuoso and Ryan Murray},
journal= {arXiv preprint arXiv:2201.05274},
year = {2022}
}