I am trying to understand the effect of noise data on matrix. Here is the question,
I have $n \times d$ matrix A with rank r with SVD
A = USV^T
Now, by adding noise matric E of shape (nxd) I get a new matrix A1 = A + E, with rank r1.
$$A_1 = U_1S_1V_1^T$$
Suppose from $A_1$ matrix, only the top r (rank of A) entries are selected. So, that will be
$$\hat{A} = U_1\hat{S}V_1^T$$
I have to prove that,
||A(hat) - A||F (Frobenius norm) <= √8r * ||E||2 (2-norm)
I would really appreciate help. I am not getting the intuition on how to solve this.