  Observables in random tensor theory are polynomials in the entries of a
tensor of rank $d$ which are invariant under $U(N)^d$. It is notoriously
difficult to evaluate the expectations of such polynomials, even in the
Gaussian distribution. In this article, we introduce singular value
decompositions to evaluate the expectations of polynomial observables of
Gaussian random tensors. Performing the matrix integrals over the unitary group
leads to a notion of effective observables which expand onto regular, matrix
trace invariants. Examples are given to illustrate that both asymptotic and
exact new calculations of expectations can be performed this way.
