Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks

תקציר מקורי באנגליתarXiv:2606.14954v4 Announce Type: replace-cross Abstract: We develop a general framework for analyzing representation costs induced by parameter-space regularizers in data-fitting methods. For an arbitrary parametric method, we define its representation cost and native function space, prove existence, and identify conditions under which parameter-space and function-space problems have equal infimal values and minimizers transfer between them. This framework yields representer theorems and recovers classical formulations---including kernel methods and RKHSs, wavelets and Besov spaces, and shallow neural networks and variation spaces---as special cases. Our main new results concern depth-$L$ feedforward ReLU networks with weight-decay regularization. For these networks, we prove that the rep