Consider the following statistical estimation problem: We observe a realization of a random matrix $Y$ of the form $Y = X° + W$, where $X°$ is an unknown $m\times n$ rank $k$ matrix and $W$ is a matrix with iid entries drawn from the standard Gaussian distribution $\mathcal N(0,1)$. The goal is to compute an estimate $X$ that is as close as possible to $X°$.
As we have seen, the maximum likelihood estimator $X_{\text{mle}}(Y)$ has the following form $X_{\text{mle}}(Y) = \text{argmin}_X\{||X - Y||^2_F,\;\text{rank}(X) \leq k\}$.
Prove the following approximation guarantee for this estimator $$E||X°-X(Y)||^2_F≤kO(n + m)$$
And we know that the spectral norm $||W||$ of $W$ satisfies $E(|| W||^2) \leq O(n+m)$.
Actually I know how to prove the rank-1 case, but how can I adapt the proof? Thank you very much