  We study the frequency and dimensions of inner and outer rings in the local
Universe as a function of disk parameters and the amplitude of
non-axisymmetries. We use the 1320 not-highly inclined disk galaxies
($i<65^{\circ}$) from the S$^4$G survey. The ring fraction increases with bar
Fourier density amplitude: this can be interpreted as evidence for the role of
bars in ring formation. The sizes of inner rings are positively correlated with
bar strength: this can be linked to the radial displacement of the 1/4
ultra-harmonic resonance while the bar grows and the pattern speed decreases.
The ring intrinsic ellipticity is weakly controlled by the non-axisymmetric
perturbation strength: this relation is not as strong as expected from
simulations, especially when we include the dark matter halo in the force
calculation. The ratio of outer-to-inner ring semi-major axes is uncorrelated
with bar strength: this questions the manifold origin of rings. In addition, we
confirm that i) $\sim 1/3$ ($\sim 1/4$) of the galaxies hosting inner (outer)
rings are not barred; ii) on average, the sizes and shapes of rings are roughly
the same for barred and non-barred galaxies; and iii) the fraction of inner
(outer) rings is a factor of $1.2-1.4$ ($1.65-1.9$) larger in barred galaxies
than in their non-barred counterparts. Finally, we apply unsupervised machine
learning (Self-Organizing Maps, SOMs) to show that, among early-type galaxies,
ringed or barred galaxies cannot be univocally distinguished based on 20
internal and external fundamental parameters. We confirm, with the aid of SOMs,
that rings are mainly hosted by red, massive, gas-deficient, dark-matter poor,
and centrally concentrated galaxies. We conclude that the present-day coupling
between rings and bars is not as robust as predicted by numerical models
(Abridged).
