Cramér transform, half-space depth and threshold phenomena for convex bodies

What the math gives to ML

The paper supplies a computable bridge between a large-deviation energy, the Cramér transform \(\Lambda^*\), and geometric centrality measured by Tukey half-space depth. The key transferable asset is that \(\exp(\Lambda^*(x))\) estimates the number of independent samples needed for a point to appear in a random convex hull, while the depth comparison provides an interpretable geometric calibration up to a dimension-dependent factor. A practical neural use is to equip embedding models with a Cramér-energy confidence score and train or sample examples according to this score, rather than relying only on Euclidean distance or softmax confidence. The method is most suitable for bounded, approximately log-concave embedding distributions and should first be tested as an OOD detector and hard-example curriculum.