Distance Profile Embedding for Independence and Conditional Independence Testing of Random Objects

What the math gives to ML

The paper introduces a distribution-faithful representation for arbitrary metric-space objects: each object is represented by its entire distance profile to the reference space, viewed as an element of an L2 function space. The transferable asset is that this representation does not require the original metric space to admit an isometric Hilbert embedding, while the paper claims injectivity and preservation of full distributional information under mild conditions. A practical neural-network adaptation is a landmark-approximated DPE layer that converts graphs, trees, point clouds, distributions, or other structured objects into fixed-width distance-profile vectors before a standard encoder, with the landmark measure controlling approximation quality.