  We present the shift-dimension of multipersistence modules and investigate
its algebraic properties. This gives rise to a new invariant of multigraded
modules over the multivariate polynomial ring arising from the hierarchical
stabilization of the zeroth total multigraded Betti number. We give a fast
algorithm for the computation of the shift-dimension of interval modules in the
bivariate case. We construct multipersistence contours that are parameterized
by multivariate functions and hence provide a large class of feature maps for
machine learning tasks.
