# Understanding stability in frame theory. Wavelets

I'm trying to grasp some understanding of the concept of stability in the context of the frame theory. The definition of frame in ref1 states that a sequence of functions $$\{ \phi_n \}_{n \in \Gamma}$$ belonging to a dictionary $$D$$ is a frame if there exist two constants $$B \geq A > 0$$ so that:

$$$$\tag{1} A \|f\|^2 \leq \sum_{n\in\Gamma} |\langle f, \phi_n \rangle|^2 \leq B \|f\|^2$$$$

where $$f$$ belongs to a Hilbert space $$H$$.

From my understanding, the definition of frame entails stability since the variation of the norm of the representation $$\|\Phi(f)\|^2 = \sum_{\Gamma} |\langle f, \phi_n \rangle |^2$$ is tied to the variation of $$\|f\|^2$$ and bounded by the constants $$A$$ and $$B$$. In ref2, the concept of stability is explained with reference to the time-warped deformations $$f_\tau(t) = f(t - \tau(t))$$ saying that, to be stable, there should be a constant $$C>0$$ so that:

$$$$\tag{2} \| \Phi(f) - \Phi(f_\tau) \| \leq C \sup_t |\tau'(t)| \|f\|$$$$

It is also shown that, in the case of the Fourier transform, $$\| \Phi(f) - \Phi(f_\tau) \|$$ do not to decrease proportionally to $$\epsilon$$ ($$\tau(t) = \epsilon t$$) and equation (2) cannot be satisfied.

The first question is: to what extent conditions (1) and (2) are equivalent?

The second question relates to the STFT. In 1, section 4.2.1 says that the STFT is stable. Referring to the results in 2 mentioned above, my understanding is that the stability comes from the fact that the window operates some kind of averaging over a finite interval of frequencies. However, for windows very long in time, I was expecting to find again the same kind of instability seen for the Fourier transform. What am I missing here?

They aren't equivalent; "stability" is used differently in each context.

$$(1)$$ guarantees a stable inverse. If $$A=0$$, we lose information. Existence of such $$A$$ and $$B$$ ensure the representation's norm is bound by input's norm alone, and not its other characteristics (e.g. regularity, frequency).

$$(2)$$ guarantees a stable representation, in that the distance between representations of deformed and undeformed inputs grows by, and only by, the size of deformation. STFT is stable per $$(1)$$ but not per $$(2)$$.

In context of scattering, $$(1)$$ is necessary to guarantee $$(2)$$, and we must also have $$B \leq 1$$, else cascaded operators explode the input norm. Explained further here and here.

• You beat me to it. Very nice answer :-).
– Royi
Dec 12, 2021 at 10:53
• Thanks! Clear explanation :)
– dac
Dec 14, 2021 at 2:10