Eikonal depth: an optimal control approach to statistical depths

Martin Molina-Fructuoso; Ryan Murray

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arXiv · 2201.05274

Eikonal depth: an optimal control approach to statistical depths

Martin Molina-Fructuoso·Ryan Murray

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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.

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Martin Molina-Fructuoso, Ryan Murray. 2022-01-14. Eikonal depth: an optimal control approach to statistical depths. https://arxiv.org/abs/2201.05274

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