Identical fixed points in state evolutions of AMP and VAMP

AMP and VAMP are two different algorithms that solve high-dimensional standard linear regression problems effectively and efficiently. In previous studies, they were long observed to have identical fixed points in their state evolutions (SEs), however, a formal justification is still missing. This issue is addressed in the paper. Instead of looking directly into the problem, we consider a scope more ambitious which puts aside all assumptions underlying the proven SEs. We then show that: 1) Having the spectrum of a Marchenko-Pastur law is a necessary and sufficient condition for the AMP-style and VAMP-style SEs to share an identical fixed point; 2) As the zero-mean i.i.d. Gaussian ensemble have a Marchenko-Pastur spectrum and satisfies all conditions required by the proven AMP SE and VAMP SE, this particular ensemble is the first example of having an identical fixed point. After that, we conjecture the proven VAMP SE might be extended to cover ensembles that break the right rotational invariance, among which the i.i.d. zero-mean sub-Gaussian ensemble could be a second example to yield identical fixed points. This conjecture is supported by experiments from wireless communications and compressive sampling.