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# Singular Value Decomposition of Noisy Matrices

I am an engineer who makes measurements of a variable over a grid of, say, $m\times n$. Since these are actual measurements, the true values are always corrupted by noise, and what I measure is a noisy version of the true set of values. Let $\mathbf{U}$ be the true (unknown) matrix of values and $\widetilde{\mathbf{U}}=\mathbf{U}+\mathbf{E}$ be the (measured) noisy version, where $\mathbf{E}$ is a matrix of error values. It is commonly assumed that elements of $\mathbf{E}$ are obtained from a zero-mean Gaussian distribution with some variance $\sigma^{2}$.

Let $\lambda_{i}$ be the singular values of $\mathbf{U}$ arranged in descending order and $\mathbf{l}_{i}$ and $\mathbf{r}_{i}$ the corresponding left and right eigenvectors respectively, i.e. $\mathbf{U}_{(m\times n)}=\mathbf{L}_{(m\times m)}\mathbf{S}_{(m\times n)}\mathbf{R}_{(n\times n)}^{T}$, where $\mathbf{l}_{i}$ and $\mathbf{r}_{i}$ are the columns respectively of $\mathbf{L}$ and $\mathbf{R}$. Let $\widetilde{\lambda_{i}}$, $\widetilde{\mathbf{l}_{i}}$ and $\widetilde{\mathbf{r}_{i}}$ be the correpsonding quantities for $\widetilde{\mathbf{U}}$.

I would like to estimate the singular values and eigenvectors of $\mathbf{U}$ knowing only the corresponding quantities of $\widetilde{\mathbf{U}}$ and the variance $\sigma^{2}$ of the error. Therefore, I would like to know the answer to the following two questions:

1. what is the relationship between $\widetilde{\lambda_{i}}$ and $\lambda_{i}$?
2. what is the relationship between the eigenvectors of $\widetilde{\mathbf{U}}$ and those of $\mathbf{U}$?