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Let $X$ be a given $d \times T$ matrix, and let $M$ be an $n \times d$ random matrix (say i.i.d. centered coefficients). Define $Y=MX$ in $\mathbb{R}^n$ and $H=Y'(YY')^{-1}Y$, where $'$ denotes the transpose operator. We assume $n<T$. The question is: is it possible to estimate the scalar quantity $$E=\mathbb{E}[\|HX - X\|^2]$$ as a function of $n, T, d$ and probably some quantities related to $X$?

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It depends on the matrix norm that you are referring inside the expectation.

There is a huge literature on the related problem of approximating solving Least Squares by random projections. Hope this helps.

Paper

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