Do you have any idea about how we can find the principal eigenvectors of an unknown matrix ${H}$?
The elements of $H$ are unknown in general. If you are familiar with channel estimation procedure in wireless communications, $H$ is channel matrix, and every element $(i,j)$ represents the channel gain between antenna element $i$ of transmitter and antenna element $j$ of the receiver. The following are the information that we have:
- we do not know the channel matrix,
- we know second order statistics, covariance matrix of the channel,
- the channel matrix size tends to infinity (large number of antennas in Tx and Rx),
- the channel is sparse, that is, it has only two or three dominant eigenvectors regardless of the channel matrix size.
Note that the power method is not a good solution for me, since its complexity is huge, especially because it requires many iterations to find the dominant eigenvectors.
Some information about the interplay between channel and its covariance:
We consider Rayleigh correlated channel coefficients $h_k\sim CN (0,\mathbf{R}_k)$, mutually independent across the users $k$, and $CN$ is circularly symmetric complex Gaussian. We let $\mathbf{R}_k = \mathbf{U}_k \mathbf{\Lambda}_k \mathbf{U}_{k}^{H}$, where $\mathbf{U}_{k}$ is a tall matrix of the eigenvectors corresponding to the non-zero eigenvalues of $\mathbf{R}_k$, given as the diagonal elements of the diagonal matrix $\mathbf{\Lambda}_{k}$.
The Karhunen-Loeve representation of the user channel vector is given as $\mathbf{h}_{k} = \mathbf{U}_{k}\mathbf{\Lambda}_{k}^{\frac{1}{2}}\mathbf{w}_{k}$, where the entries of $\mathbf{w}_{k}$ are i.i.d. with zero mean Gaussian distributed with variance 1.