Generalizing Graph Signal Processing: High Dimensional Spaces, Models and Structures

Abstract

This correspondence points out a technical error in Proposition 4 of the paper [1]. Because of this error, the proofs of Lemma 3, Theorem 1, Theorem 3, Proposition 2, and Theorem 4 in that paper are no longer valid. We provide counterexamples to Proposition 4 and discuss where the flaw in its proof lies. We also provide numerical evidence indicating that Lemma 3, Theorem 1, and Proposition 2 are likely to be false. Since the proof of Theorem 4 depends on the validity of Proposition 4, we propose an amendment to the statement of Theorem 4 of the paper using convergence in operator norm and prove this rigorously. In addition, we also provide a construction that guarantees convergence in the sense of Proposition 4.

Graphon Signal Processing

“…We simply provided the wrong proof for this result. We also point out that the results in [5] require graph and graphon filters of finite length. This is weaker than the results we provide in [1], which only require filters to have a Lipschitz continuous spectral representation.…”

“…We thank Dr. Feng Ji for pointing out the mistake in the proof of [1, Proposition 4]. Along with collaborators, Dr. Ji published an extensive correction of our results [5]. These corrections are unnecessary since [1, Proposition 4] holds.…”