I've come across the following claim in a paper of Mallat:

"High frequency instabilities [of a signal representation] to deformations can be avoided by grouping frequencies into dyadic packets in $\mathbb{R}^d$, with a wavelet transform."

I know what the wavelet packet decomposition is and I know what stability to deformations is (in the case of $C^2$ diffeomorphisms), but I don't understand why the quoted statement is true.

Perhaps I missed the explanation in the paper, but could someone provide a reference, an explanation or both? I assume it has something to do with the fact that a wavelet is localized.

¹ Stéphane Mallat: Group Invariant Scattering, Communications on Pure and Applied Mathematics, Vol. LXV, 1331–1398 (2012), DOI: 10.1002/cpa.21413