I have a matrix $X$ and I compute its Singular Value Decomposition: $$X = U \Sigma V^T$$ then, I take the lower rank approximization: $$X_k = U_k \Sigma_k V^T_k$$ where $k < rank(X)$, $U_k$ is made of the first $k$ columns of $U$, $\Sigma_k$ is the $k \times k$ diagonal matrix whose elements are the first $k$ singualar values of $X$ and $V_k$ is composed of the first $k$ columns of the $V$ matrix.
I would like to know how to estimante the information loss due to the lower rank approximization.
Is that possible to compute it?
Thanks