Suppose I have a large matrix ${\bf X} \in {\Bbb R}^{m \times n}$. By independently sampling $r$ rows and $c$ columns of $\bf X$, we obtain a random submatrix $\bf S$. From $\bf S$, how to obtain an unbiased estimate of the eigenvalues of $\frac1c {\bf X} {\bf X}^\top$?
I am not assuming the entries of $\bf X$ are sampled independently from some distribution, e.g., the normal distribution. I am assuming that there is a correlation between the entries in a given row and in a given column. That is, $X_{ij}$ and $X_{kl}$ are correlated iff $i=k$ or $j=l$. I am not sure we need the following assumption, but I think it might make the problem easier/tractable if we also assume the entries of $\bf X$ have finite moments.
