A Two-regime Khintchine Inequality and an Improved Bound on the Degree-1 Fourier Weight for Linear Threshold Functions

What the math gives to ML

The paper provides a quantitative certificate that a Rademacher linear form cannot remain close to its minimum expected absolute magnitude unless its weight vector is close to a two-coordinate, equal-magnitude extremizer. This can be transferred into a neuron regularizer that discourages degenerate filters or feature projections whose energy is concentrated in only two coordinates, while remaining invariant to signs and permutations. The practical implementation is to estimate expected absolute responses under random sign perturbations and combine them with the paper's explicit distance-sensitive lower bound, yielding a falsifiable feature-diversity and robustness experiment.