Finding spectrum of a Kronecker factored + block-partitioned matrix

I have dense $$d\times d$$ matrices $$A$$, $$B$$, $$C$$ with $$d\approx 1000$$ and want to find the top $$10^5$$ eigenvalues of the following positive definite matrix:

$$\Sigma= \left(\begin{matrix} A\otimes A & C^T\otimes C\\ C\otimes C^T & B\otimes B \end{matrix}\right)$$

• Cheap: assume $$C=0$$ and compute $$\lambda(A)$$, $$\lambda(B)$$, since $$\lambda$$ commutes with $$\otimes$$.

• Expensive: power method. Storing $$10^5$$ eigenvectors is 40 GB.

Is there something in between the two?

Motivation

$$\Sigma$$ is a

• covariance of $$x\otimes x$$, where $$x$$ is a partitioned Gaussian vector with mean 0 and $$E[xx']= \left(\begin{matrix} A&C^T\\ C&B \end{matrix}\right)$$
• Gauss-Newton block for neural network linear layer if we assume forward/backward values are jointly Gaussian