Information-Theoretic Foundations of the AaroNet Multi-Scale Detector

We study the information-theoretic structure of multi-scale binary detection, a problem in which a noisy observation is to be classified at each of $L$ disjoint scale strata, with a known prior $p_S$ over the strata and a total channel budget $\Cl_{\rm tot}$ to be apportioned across the strata. The problem is the classification analogue of the parallel-Gaussian-channel water-filling problem of classical rate-distortion theory, and we develop a parallel set of analytical results. Three theorems are proved. First, under a localised-generator model in which each stratum's contribution is band-limited in Fourier space, the scale-matched sufficient statistics for two distant strata are asymptotically conditionally independent given the labels, with conditional mutual information $O((s/s')^2)$. The decoupling is a property of the sufficient statistics; the full feature representations need not, and in general do not, decouple. Second, the rate-distortion problem of maximising prior-weighted Bayes accuracy subject to a budget on total channels admits a unique Karush--Kuhn--Tucker solution, which in the high-resolution asymptotic regime reduces to a closed-form water-filling expression $\Cl_\Lev^\star \propto (p_\Lev \, \Ent(Y_{s_\Lev})\,\log(1+\SNR_\Lev))^{2/3}\,\tau_\Lev^{-2/3}$. We show that this allocation strictly dominates prior-proportional allocation whenever the per-stratum signal-to-noise profiles differ in shape, and coincides with it whenever they do not. Third, the binary detection error rate at each stratum admits a Fano-type lower bound $h_2(P_e^{(s)}) \ge \Ent(Y_s) - \I(\Feat_s;Y_s)$, providing a direct information-theoretic objective for representation learning at each stratum. Synthetic experiments on a 1-D scale-stratified detection problem with computable ground-truth information quantities validate each theorem: the sufficient-statistic mutual-information matrix is strongly diagonal (peak ratio $>90{:}1$ between matched and mismatched strata), and the closed-form rate-distortion allocation matches brute-force grid search exactly across nine channel budgets, outperforming prior-proportional allocation by up to $1.02$ percentage points of weighted accuracy in the differentiated-SNR regime. An information-driven detector architecture, AaroNet, is derived as a direct application of the three theorems and discussed as a case study.