  We present an ab initio study of the correlation between nuclear matrix
elements of neutrinoless double-beta ($0\nu\beta\beta$) decay and
nucleon-nucleon scattering phase shifts in the $^1S_0$ channel. Starting from
thirty-four statistically weighted interactions derived from chiral effective
field theory, we apply the valence-space in-medium similarity renormalization
group to calculate nuclear matrix elements in four key experimental isotopes.
Comparing with the $^1S_0$-channel phase shifts given from each interaction, in
all cases we observe a strong correlation for scattering energies above 75 MeV.
Furthermore, a global sensitivity analysis, enabled by newly developed
machine-learning emulators, confirms that the nuclear matrix elements of the
decay depend primarily on the $C_{1S0}$ low-energy constant, which is
associated with the phase shifts in that partial wave. These results provide
the first clear correlation between $0\nu\beta\beta$ decay nuclear matrix
elements and a measured observable and will therefore serve as a crucial
component in ongoing and future refinements of ab initio uncertainty estimates.
