Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks
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 regularizati...
| Authors | Greg Ongie, Rahul Parhi |
|---|---|
| Primary category | math.FA |
| Published | 12 June 2026 |
| Last updated at source | 5 September 2026 |
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