  In this paper, we revisit scalar field theories in $d$ space-time dimensions
possessing $U(N)$ global symmetry. Following our recent work arXiv:1402.1430v2,
we consider the generating function of correlation functions of all
$U(N)$-invariant, single-trace operators at the free fixed point. The exact
renormalization group equations are cast as Hamilton equations of radial
evolution in a model space-time of one higher dimension, in this case
$AdS_{d+1}$. The geometry associated with the RG equations is seen to emerge
naturally out of the infinite jet bundle corresponding to the field theory, and
suggests their interpretation as higher-spin equations of motion. While the
higher-spin equations we obtain are remarkably simple, they are non-local in an
essential way. Nevertheless, solving these bulk equations of motion in terms of
a boundary source, we derive the on-shell action and demonstrate that it
correctly encodes all of the correlation functions of the field theory, written
as `Witten diagrams'. Since the model space-time has the isometries of the
fixed point, it is possible to construct new higher spin theories defined in
terms of geometric structures over other model space-times. We illustrate this
by explicitly constructing the higher spin RG equations corresponding to the
$z=2$ non-relativistic free field theory in $D$ spatial dimensions. In this
case, the model space-time is the Schr\"odinger space-time, $Schr_{D+3}$.
