I am aware that if I have an expression $e^{\frac{(\sum_i v_i)^2}{2}}$ then the exponent can be 'linearized' by introducing a Gaussian distributed random variable: $$e^{\frac{(\sum_i v_i)^2}{2}} = \int_{-\infty}^{\infty} \frac{dx}{\sqrt{2\pi}}e^{-\frac{x^2}{2} + x\sum_i v_i}$$
My question is that is there a similar approximation that can be done if I have an expression like $e^{(\sum_i v_i)(\sum_j w_j)}$? I would like to linearize the product somehow if possible.
Thanks