Nonnegative conorms, regular matroids, and the tropical Schottky problem

What the math gives to ML

The paper gives a constructive characterization of a class of positive-definite quadratic forms as nonnegative sums of integral rank-one forms whose covectors have unimodular, regular-matroid support. This suggests replacing an unconstrained dense Mahalanobis metric with a structured PSD metric whose positivity is automatic and whose support can be sparse and interpretable. The most direct neural-network test is to use such a metric for attention distances or metric-learning embeddings, with graph-incidence covectors providing a simple totally unimodular support. The likely benefits are reduced metric parameters, stable optimization, and useful inductive bias on small or graph-structured data.