Effective Resistance in Fixed-Rank External-Field Measures and Constant-Stretch Correlated Sampling on the Hypersimplex

What the math gives to ML

The paper proves a sharp covariance-pseudoinverse bound for fixed-cardinality external-field subset distributions, a structure that is directly useful for stochastic routers and fixed-budget selection layers. The indicator vector of a sampled subset forms an exponential family whose Fisher matrix is its covariance, but this matrix is singular because every sample has exactly m active coordinates. The effective-resistance inequality provides a principled stability bound for natural-gradient updates in the identifiable, sum-zero logit subspace, while the covariance lower bound prevents excessive degeneracy. The most promising transfer is a fixed-m MoE or token-selection router using exact external-field sampling and covariance-pseudoinverse preconditioning instead of independent Bernoulli or softmax routing.