Say we have several rectangular matrices (J,K) of different shapes, and we want to evaluate the Expected value, or Trace, of their Product
$$ \mathbb{E}[J_{1}^{T}J_{2}^{T}J_{3}^{T}K_{3}K_{2}K_{1}] $$
where by the expected value here I mean the gaussian integral $$ \mathbb{E}[\mathbf{M}]=\int d^{n}x e^{\frac{-1}{2}\mathbf{x}^{T}\mathbf{Mx}} $$
Where the matrices are real but rectangular, and of different shapes, and non-random
i.e say the dimensions of the matrices are
$$J_{3}, K_{3}\;are\;(1000\times300), $$ $$J_{2}, K_{2}\;are\;(300\times100), $$ $$J_{1}, K_{1}\;are\;(100\times10)$$
Is there a way to express this using a Generalization of Wick's theorem (or Isserlis' theorem), to give something like
$$ \mathbb{E}[J_{1}^{T}K_{1}]\times\mathbb{E}[J_{2}^{T}K_{2}]\times\mathbb{E}[J_{3}^{T}K_{3}]+\cdots $$