Brownian Heads for Deep ReLU Representations: Activation Mass and the Cost of Same-Sample Selection

Abstract:Deep representation learning often selects hidden features and fits the final predictor on the same sample, so fixed-feature analysis performed after selection can omit selection cost. We study the conditional empirical Rademacher complexity of deep ReLU representations followed by bounded-norm predictors in additive or Lévy-Brownian RKHSs, termed Brownian heads. For a fixed representation, we derive an exact dual identity and sharp bounds in terms of activation mass, the average norm of the observed hidden vectors. Under same-sample selection, the representation supremum induces a quadratic Rademacher process. Brownian layer-cake and Gaussian-projection identities reduce it to coordinatewise or signed projected threshold traces, separating realized scale from selection complexity. For samples with pairwise-distinct inputs, explicit scalar ReLU families match the finite-trace and VC rates up to universal constants at the realized trace-and-envelope level. Induced-norm contraction also yields architecture-level bounds for rectangular, rank-deficient ReLU networks. Experiments verify the sharp bounds and rates, exhibit a selection gap at fixed activation mass, and assess the predictive feasibility of Brownian heads.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2609.21422 [stat.ML]

(or arXiv:2609.21422v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.21422 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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