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 looking for more systematic and refined analysis.

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?

Suppose we have a matrix $X\in\mathbb{R}^{m\times n}$ (with $m \le n$) 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 looking for more systematic and refined analysis.

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 looking for more systematic and refined analysis.