Pego theorem for Hilbert space-valued functions on compact groups

What the math gives to ML

The paper supplies a compactness template for families of Hilbert-valued functions on a compact group: control small translations, suppress high Fourier modes, and, in infinite-dimensional feature spaces, enforce uniform tightness in the value-space. This is transferable to group-structured neural features, where spectral smoothness alone can still allow energy to escape into new feature directions as width, tasks, or training time increase. The practical adaptation is a three-part representation regularizer evaluated over a finite transformation group, with a low-rank feature-space projection enforcing the missing tightness condition.