Lecture 7: Stability to Relative Perturbations (10/11 – 10/13)

In this lecture, we prove the stability of graph filters to relative perturbations. We show that integral Lipschitz filters are stable to relative perturbations, by showing that the operator norm modulo permutation of the difference between the graph filter for a graph and a relative perturbation of the graph is bounded. Following the proof will require using almost all the concepts learned so far plus some linear algebra. After this, we do the same but for the case of additive perturbations, this section however is part of the appendix of the lecture.

Video 7.1 – Definitions

We start softly, by defining the relative perturbation and stating some of its properties. We also recall the definition of integral Lipschitz filters and finally, we state the theorem of stability of graph filters to relative perturbations. Among others, we define the norm of the perturbation $\epsilon$ and the eigenvector misalignment constant $\delta$. These two constant will show up in the bound we are trying to prove.

• Covers Page 1 in the handout.

Video 7.2 – Eigenvector Perturbation Lemma

With the notion of additive perturbation at hand, we present the Eigenvector Perturbation Lemma. This auxiliary proof will be important to show the main result, as it combines the two constants, the norm of the perturbation $\epsilon$ and eigenvector misalignment $\delta$. The eigenvector perturbation lemma, bounds the norm of the part of the perturbation that does not have the same eigenvectors as the shift operator $S$.

• Covers Page 2 in the handout.

Video 7.3 – From Shift to Filter Perturbations

In this lecture, we unfold the filter on the polynomial on the shift operator and we take an approximation. Now the road of the proof is clear, we just need to prove each separate part. We will translate a shift operator perturbation into a filter output perturbation, there no conceptually challenging steps in this section. Consider this section as an algebraic manipulation of terms.

• Covers Page 3 in the handout.

Video 7.4 – Shifting to the GFT domain

In this part of the lecture, we switch to the GFT domain and we use the Eigenvector Perturbation Lemma. It was expected that at some point we would switch to the frequency domain, now is the time for us to do that. We will use concepts that we have learned in previous lectures. This section finished by stating two facts, that will be shown in the following two sections.

• Covers Page 4 in the handout.

Video 7.5 – Proof of Fact 1

In this lecture, we show Fact 1, namely a bound that will be used for the proof of the theorem. Here we will delve into integral Lipschitz properties of the filter to bound the error term. Fact 1 is not complicated to prove, we would just need to use some properties of the filter and more linear algebra.

• Covers Page 5 in the handout.

Video 7.6 – Proof of Fact 2

The theorem is just about to be proved, we just need to prove Fact 2. In this lecture we show that Fact 2 holds, this part of the proof involves the bound in which Pythagoras theorem does not hold, thus the $\sqrt{N}$ is introduced. This is the last part of the proof.

• Covers Page 6 in the handout.

Appendix: Proof of Stability to Additive Perturbations

We do similar proof as before but in this case for additive perturbations of the graph. Not many things change in this case, but it is worth taking a look at how the proof varies when a different perturbation is selected.