Eikonal depth: an optimal control approach to statistical depths
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Abstract (arXiv)
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.
Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Builds on 19 of its 65 references that are in the map. First posted on arXiv 14 Jan 2022.
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19 of 65 references.
References · 19
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7 more
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Citing papers per year: 2022: 1.
1 paper.