Suppose I have a matrix $X in mathbb{R}^{a times b}$, with $a > b$ and $X$ full rank with singular value decomposition $UDV’=X$. Then, suppose I also have a “small” matrix $A$ (i.e., some perturbation matrix) which is not equal to $X$. I am interested if the following quantity could be lower bounded by a norm of $A$ and norm of $X$:

$$ sup_{|Q| leq 1} {rm tr}left[(Q – UV’)'(X – A) right],$$

where $|cdot|$ is the spectral norm. Using that the dual norm of the “nuclear norm” is the spectral norm, the above quantity could also be expressed as:

$$ |X – A|_* – |X|_* + {rm tr}(VU’A),$$

where $|cdot|_*$ is the nuclear norm which sums the singular values of its argument.

Any suggestions for directions or references would also be appreciated.