Geometric Deep Learning HT26, Scale separation
Flashcards
Suppose:
- We have a multiscale coarsening of a data domain $\Omega$ into a hierarchy $\Omega _ 1, \ldots, \Omega _ J$
- $\mathcal X _ j(\Omega _ j, \mathcal C _ j) = \{ x _ j \mid \Omega _ j \to \mathcal C _ j \}$ are the signals over each coarsened domain
@Define what it means for a function $f : \mathcal X(\Omega) \to \mathcal Y$ to be locally stable. @Visualise what this looks like in the context of an image classification task.
It admits a factorisation
\[f \approx f _ k \circ P _ j\]where $P _ j : \mathcal X(\Omega) \to \mathcal X _ j(\Omega _ j)$ is a non-linear coarse graining and $f _ j : \mathcal X _ j (\Omega _ j) \to \mathcal Y$.
