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.

Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth

“…Shortest path distances on graphs have found applications in many areas of data science and machine learning, including dimensionality reduction (e.g., the ISOMAP algorithm [61]), semisupervised learning on graphs [5,25,51,56,64], graph classification [6], and data depth [19,49,50]. In many applications, the shortest paths are density weighted, to make path lengths shorter in high density regions of the graph, and longer in sparse regions [5].…”

“…We were recently made aware of another paper [50] that was developed in parallel with ours, and proposes to use the eikonal equation for data depth. The method in [50] requires identifying boundary points first, and then the depth is defined as the length of a shortest density-weighted path back to the boundary. This approach, without density weighting, was also used in [19], in combination with a method for detecting boundary points.…”