Suppose we have a matrix $X\in\mathbb{R}^{m\times n}$ (with $n \le m$) with iid standard Gaussian entries, and suppose we have noise matrix $W\in\mathbb{R}^{m\times n}$ with iid Gaussian entries, but with some small variance $\sigma_W < 1$. We know that the columns of $X$ and $X+W$ are linearly independent almost surely, so they form a basis. I am interested in knowing how the noise $W$ changes the projection matrix $X(X^TX)^{-1}X^T$ of $X$. For instance, denoting $X_W = X+W$ and $x_j$ as the jth column of $X$, can we say anything about $X_W(X_W^TX_W)^{-1}X_W^Tx_j$? I am especially interested in references that discuss this kind of problem. I know standard matrix perturbation theory results could be applied on this, but I am in particular interested in the high-dimension regime, that is, when $n$ and/or $m$ are large, are there concentration results available?
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$\begingroup$ If you can reframe your question to be about the singular vectors of $X$ and $X_W$, there is some nice work by Vu that tells you how singluar vectors behave under random perturbation. In particular, the bounds are better than 'worse case' bounds from Wey's inequality for instance. The paper is here: arxiv.org/pdf/1004.2000.pdf $\endgroup$– Sandeep SilwalCommented Sep 10, 2020 at 13:28
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$\begingroup$ @SandeepSilwal This is a very interesting work, thank you for the reference! $\endgroup$– LongtiCommented Sep 10, 2020 at 15:41
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