The Zero Pattern of a Design Matrix Drives Multiple Descent in Over-parameterized Regression

What the math gives to ML

The paper identifies a discrete structural cause of multiple-descent spikes that is invisible to ordinary covariance summaries: the zero pattern of the variance profile. A bipartite graph connecting sample coordinates to feature coordinates exposes whether the design has enough combinatorial support, while maximum matchings and the Dulmage–Mendelsohn decomposition locate configurations where variance becomes singular. This suggests designing sparse or grouped neural layers by matching analysis before training, rather than selecting widths using parameter counts alone. The most direct transfer is a matching-safe sparse layer or routing mask that prevents unmatched feature groups and adds the smallest number of skip connections needed to restore combinatorial coverage.