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Multiplying by E[xy'] where only some statistics of xy' are known

Computational Science Asked by Yaroslav Bulatov on October 7, 2020

(cross-posted on crossvalidated)

For random variable $(x,y)$ in $mathbb{R}^{d}times mathbb{R}^{d}$ and vector $v in mathbb{R}^d$, I need to perform the following matrix vector multiplication.

$$T(v)=E[xy’]v$$

The issue is that the expected matrix $E[xy’]$ is too large to represent in memory ($dapprox $1 million), so I can only afford to store $O(d)$ worth of statistics and perform $O(d^{1.5})$ worth of operations. Three such statistics are $E[x]$, $E[y]$ and $E[xodot y]$, where $odot$ refers to element-wise multiplication.

If I only had the first two, one could argue that the following modification of $T$ is appropriate, representing an unbiased guess subject to these constraints

$$T(v)approx E[x]E[y’]v$$

What’s an appropriate way to incorporate $E[xodot y]$?

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