Eikonal depth: an optimal control approach to statistical depths
This work addresses the need for interpretable and robust statistical depths in higher dimensions, particularly for multi-modal and non-Euclidean data, representing a novel method rather than an incremental improvement.
The paper tackles the problem of generalizing quantiles and medians to high-dimensional data by proposing a new statistical depth based on control theory and eikonal equations, which measures the minimal probability density traversed to reach points outside the distribution's support, and demonstrates its robustness under adversarial models and applicability to non-Euclidean data with examples in mixture models and MNIST.
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.