Approximate computation of leading eigenvector of average of PSD matrices

Question

Some ongoing research has led me to consider the following two problems.

Question 1

Let $v$ be a random column vector. Is there an efficient way to compute leading eigenvector of the psd matrix $\mathbb E_{v}[vv^T]$ ?

Question 2

Let $V$ be a fixed $m\times n$ real matrix with $i$th row $v_i \in \mathbb R^{m}$, and let $\pi$ be a probabilitty distribution on $\{1,\ldots,n\}$.

Is there an efficient approximate way to compute the leading eigenvector of the $m \times m$ psd matrix $M^\pi$ defined by $M^\pi := \mathbb E_{i \sim \pi}[V_iV_i^T]$ ?

Answer 1

Here are some thoughts, but I'm not sure how useful this will be. The basic idea for both is just to compute the expectation and then apply the power method. You can use the explicit form of the expectations to guide the choice of initial guess for the power method.

Question 1

Note that, $$ \mathbb{E}[vv^T] = \mathbb{E}[v]\mathbb{E}[v]^T + \mathbb{C}\textrm{oV}[v,v] $$

If you know the covariance matrix, then you can construct $\mathbb{E}[vv^T]$ explicitly and apply standard methods (power method) to find the top eigenvector.

Question 2

Note that, $$ \mathbb{E}[V_iV_i^T] = \sum_i\pi_iV_iV_i^T = V\Pi V^T $$ where $\Pi$ is the diagonal matrix with entries $\pi_i$.

Again, this gives you an explicit matrix which you can apply classic techniques to to find the top eigenvector.