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Claims about the benefit of depth depend on the complexity assigned to a representation. We introduce the \emph{Variation Brownian Kernel Ladder} (VBKL), a path-atomic function-space framework that separates nonlinear recursive dictionary construction from linear variation superposition. Starting from linear projection...
One finite-dimensional covariance object is identified linking unrestricted kernel adaptation, Gaussian geometry, common subspaces, and certifiable computation and is identified as the minimum-trace common covariance that dominates the unrestricted empirical union generated by Brownian kernel ladders.
The sample-size-weighted trace norm of the end-to-end map from task coefficients to input-space predictors is studied instead the sample-size-weighted trace norm of the end-to-end map from task coefficients to input-space predictors.
This work studies the conditional empirical Rademacher complexity of deep ReLU representations followed by bounded-norm predictors in additive or L\'evy-Brownian RKHSs, termed Brownian heads, and derives an exact dual identity and sharp bounds in terms of activation mass, the average norm of the observed hidden vectors...
Float64 tests over five grids numerically verify the finite identities, mapped one-layer and recursive trajectories, conditioning predictions, and theorem-matched Adam separation of a controlled finite Brownian RKHS with nodal, increment, and spectral coordinates.
Operator-theoretic generalization bounds for deep multi-output function classes are developed by representing network layers as Koopman composition operators on vector-valued reproducing kernel Hilbert spaces and derive Rademacher complexity bounds for invertible and width-expanding injective architectures.
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