Many-point tropical relaxation and the Monge--Ampère equation

What the math gives to ML

The paper constructs concave piecewise-affine tropical potentials whose Monge–Ampere measure encodes a finite source measure, with an explicit weak discrepancy bound of order N^{-1/2}. This suggests a geometric decoder or regularizer for point clouds in which mass is represented by curvature or supergradient-cell area rather than by independent coordinate predictions. The most practical adaptation is a soft minimum of affine functions, trained against empirical source points through random test functions and then annealed toward a hard tropical representation. The mathematical guarantee should be treated as a design principle and evaluation metric, not as a direct neural-network convergence theorem.