  We complete the reformulation of the holographic correspondence as a
\emph{highly efficient RG flow} that can also determine the UV data in the
field theory in the strong coupling and large $N$ limit. We introduce a special
way to define operators at any given scale in terms of appropriate
coarse-grained collective variables, without requiring the use of the
elementary fields. The Wilsonian construction is generalised by promoting the
cut-off to a functional of these collective variables. We impose three criteria
to determine the coarse-graining. The first criterion is that the effective
Ward identities for local conservation of energy, momentum, etc. should
preserve their standard forms, but in new scale-dependent background metric and
sources which are functionals of the effective single trace operators. The
second criterion is that the scale-evolution equations of the operators in the
actual background metric should be state-independent, implying that the
collective variables should not explicitly appear in them. The final criterion
is that the endpoint of the scale-evolution of the RG flow can be transformed
to a fixed point corresponding to familiar non-relativistic equations with a
finite number of parameters, such as incompressible non-relativistic
Navier-Stokes, under a certain universal rescaling of the scale and of the time
coordinate. Using previous work, we explicitly show that in the hydrodynamic
limit each such highly efficient RG flow reproduces a unique classical gravity
theory with precise UV data that satisfy our IR criterion. We obtain the
explicit coarse-graining which reproduces Einstein's equations. In a simple
example, we are also able to compute the beta function. Finally, we show how
our construction can be interpolated with the traditional Wilsonian RG flow at
a suitable scale, and can be used to develop new non-perturbative frameworks
for QCD-like theories.
