We develop a general framework for analyzing representation costs of parametric data-fitting methods through their parameter-space regularizers. From this abstract perspective, we define representation costs for arbitrary parametric models and reveal their induced (native) function spaces. This unifies recent function-space views of data-fitting methods. We also prove that many natural results hold in this abstract setting, including representer theorems for parametric methods on their native spaces. The framework also rigorously connects parametric methods with their equivalent nonparametric descriptions under sufficient overparameterization. Classical methods and their native spaces, such as kernel methods / reproducing kernel Hilbert spaces, wavelets / Besov spaces, and shallow neural networks / variation spaces emerge as special cases of our abstract framework. A byproduct of "axiomatizing" the study of representation costs is that we also immediately obtain new results for deep neural networks: For depth-$L$ feedforward ReLU networks, their induced native spaces are $p$-normable quasi-Banach spaces with $p = 2/L$. This reveals that the inductive bias of deep neural networks (as given by the representation cost) cannot be captured by norms for depths $L > 2$.

In practice, when training denoisers, we sample multiple noisy samples for each clean image and minimize the Mean Squared Error (MSE) loss for recovering the clean image.

In practice, when training denoisers, we sample multiple noisy samples for each clean image and minimize the Mean Squared Error (MSE) loss for recovering the clean image.

This formalizes a balance between learning low-dimensional representations and minimizing complexity/irregularity in the feature maps, allowing the network to learn the'right' inner dimension.

For different parameterizations (mappings from parameters to predictors), we study the regularization cost in predictor space induced by $l_2$ regularization on the parameters (weights). We focus on linear neural networks as parameterizations of linear predictors. We identify the representation cost of certain sparse linear ConvNets and residual networks. In order to get a better understanding of how the architecture and parameterization affect the representation cost, we also study the reverse problem, identifying which regularizers on linear predictors (e.g., $l_p$ norms, group norms, the $k$-support-norm, elastic net) can be the representation cost induced by simple $l_2$ regularization, and designing the parameterizations that do so.