  The $\beta$-functions describe how couplings run under the renormalization
group flow in field theories. In general, all couplings that respect the
symmetry and locality are generated under the renormalization group flow, and
the exact renormalization group flow is characterized by the $\beta$-functions
defined in the infinite dimensional space of couplings. In this paper, we show
that the renormalization group flow is highly constrained so that the
$\beta$-functions defined in a measure zero subspace of couplings completely
determine the $\beta$-functions in the entire space of couplings. We provide a
quantum renormalization group-based algorithm for reconstructing the full
$\beta$-functions from the $\beta$-functions defined in the subspace. As
examples, we derive the full $\beta$-functions for the $O(N)$ vector model and
the $O_L(N) \times O_R(N)$ matrix model entirely from the $\beta$-functions
defined in the subspace of single-trace couplings.
