  In this article, we prove that for a finite quiver $Q$ the equivalence class
of a potential up to formal change of variables of the complete path algebra
$\widehat{\mathbb{C} Q}$, is determined by its Jacobi algebra together with the
class in its 0-th Hochschild homology represented by the potential assuming the
Jacobi algebra is finite dimensional. This is an noncommutative analogue of the
famous theorem of Mather and Yau on isolated hypersurface singularities. We
also prove that the right equivalence class of a potential is determined by its
sufficiently high jet assuming the Jacobi algebra is finite dimensional. These
two theorems can be viewed as a first step towards the singularity theory of
noncommutative power series. As an application, we show that if the Jacobi
algebra is finite dimensional then the corresponding complete Ginzburg
dg-algebra, and the (topological) generalized cluster category thereof, are
determined by the isomorphic type of the Jacobi algebra together with the class
in its 0-th Hochschild homology represented by the potential.
