Trans-Scale Variance Decoupling: An Information-Theoretic Impossibility Result by Osama Banat

This work by Osama Banat investigates fundamental limits of local learning from an information-theoretic perspective. We construct explicit continuous functions whose fine-scale components retain non-vanishing conditional variance when observed only through local neighborhoods. Our main contribution is a no-go theorem: any estimator restricted to a finite spatial radius (r-local) cannot achieve vanishing error, regardless of sample size or model complexity. This establishes locality as an inherent bottleneck and demonstrates the necessity of cross-scale interactions in consistent learning. The results provide rigorous theoretical insight for machine learning, nonparametric estimation, and the design of architectures such as convolutional neural networks, highlighting limits of purely local information. Numerical illustrations confirm the predicted nonzero risk for local estimators.