Conditionally Resampled Sliding-Window Count Kernels: Spectral-Gap Bounds and Poincaré Inequalities

What the math gives to ML

The paper constructs a nontrivial Markov kernel on length-n empirical count vectors by taking the stationary one-step conditional law of overlapping windows, even though the count process itself is not Markov. Its key transferable asset is a quantitative mixing guarantee: for a fixed strictly positive reversible base kernel, the induced count kernel has spectral gap of order 1/n, together with a Poincare variance bound. This suggests a principled MCMC or data-augmentation operator for count-valued neural inputs or discrete latent states, where correlated resampling can replace independent multinomial noise while retaining an explicit mixing-rate diagnostic. The most credible first application is count-feature augmentation or a count-valued discrete diffusion proposal, rather than a generic optimizer.