Sharp Metric $X_p$ Inequalities via Martingales

What the math gives to ML

The paper proves a sharp higher-order inequality on the Hamming cube: the average L^p magnitude of a conditional projection onto a random k-coordinate subset is controlled by coordinate-wise discrete derivatives and the global centered L^p norm, with the sharp factor p/log p. This suggests a robustness regularizer for neural networks whose inputs contain binary nuisance or augmentation variables, suppressing predictable responses to randomly revealed nuisance coordinates while explicitly controlling coordinate sensitivity. The random-reveal martingale construction gives an efficient unbiased estimator based on random permutations and partial masking rather than enumerating all subsets. The most direct test is on nuisance-invariance benchmarks, measuring robustness at matched clean accuracy and training cost.