Explicit recovery of a probability measure from its geometric depth

We prove that in any Euclidean space, an arbitrary probability measure can be reconstructed explicitly by its geometric rank. The reconstruction takes the form of a (potentially fractional) linear PDE given in closed form. While this relation holds in the sense of distributions for an arbitrary probability measure, when it admits a density we provide sufficient conditions to ensure that the density can be recovered pointwise through the PDE. Surprisingly, the reconstruction procedure is of a local nature when the dimension is odd, and of a non-local nature in even dimensions. We give examples of the reconstruction in dimension 2 and 3. We use our results to characterise the regularity of depth contours. We conclude the paper with a partial counterpart to the non-localisability in even dimensions.