Let $X$ be a tall $M\times N$ matrix with complex elements, i.e. $M >> N$, and $h$ an $N\times 1$ complex vector. Furthermore, $d$ is an $M\times 1$ vector, $\Sigma_h$ an $N\times N$ diagonal matrix with positive elements, $C_X$ an $MN\times MN$ positive semidefinite matrix and $vec(X)$ denotes the vectorization of $X$. I am wondering if there is any closed form expression for the following integral

$$\int_X \int_{h \in A_{\epsilon}(\tau)} e^{-(1/2)(c - Xh)^H (c-Xh) - (1/2)h^H\Sigma_h h - (1/2) vec(X)^H C_Xvec(X)} dX dh, $$
where $A_{\epsilon}(\tau) = \{ h | \sum_{j=\tau}^N |h_j|^2 < \epsilon\}$.

For example, I manage to evaluate the integral across the matrix $X$, but the final integral across $h$ then seems intractable. On the other hand, if I start by evaluating across $h$ first, then it seems that that integral has no closed form solution. Even if it can be expressed by some complicated functions, the final integral across $X$ then seems intractable.

Any comments or ideas are welcome.

Thanks!