Intrinsic Geometry of Hard Disk Clusters

What the math gives to ML

The paper's transferable contribution is an active-set geometry for systems defined by pairwise non-overlap inequalities: realised contacts determine both the local feasible cone and the equality subspace of contact-preserving motions. This is more informative than imposing all pairwise distance constraints uniformly, because only currently active constraints should shape the local update. A practical neural-network adaptation is a cone-aware optimizer for learned prototypes, codebooks, or embedding anchors that enforces minimum separation while allowing inactive pairs to move freely. The resulting method is an explicit quadratic-program projection and can be tested against ordinary repulsion losses and penalty-based constrained training.