The Endpoint Fractional Riesz Estimate on the Hamming Cube

What the math gives to ML

The paper proves a dimension-free endpoint estimate on the Boolean cube: for 1 < p < 2, aggregate coordinate sensitivity is controlled by a fractional Laplacian of order 1/p, even though the usual square-root Riesz estimate fails in this range. The transferable asset is a principled way to penalize sensitivity to many binary feature flips using fractional spectral energy rather than enumerating every perturbation. A practical neural-network adaptation is a robustness regularizer for binary inputs, discrete features, or learned binary gates, with the fractional Laplacian computed through a Walsh transform on small cubes or a truncated spectral estimator at larger dimension.