Volume and Projection Inequalities I: Zonoids and Courtade's Conjecture

What the math gives to ML

The paper develops a determinant-based volume polynomial for zonotopes, where each coefficient measures the geometric diversity of a subset of generating vectors. Its central structural result identifies log-submodularity with the Rayleigh inequality for this multiaffine polynomial, with guarantees in low degree or codegree and counterexamples in higher dimensions. This machinery can be transferred into neural networks as a diversity regularizer for attention heads, MoE experts, or embedding prototypes whose directions should span complementary subspaces. Because the paper exhibits failures in dimensions at least four, an implementation should optimize the volume objective directly and measure Rayleigh defects empirically rather than assume universal submodularity.