# 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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