Suppose $X \in \mathbb{R}^{m \times n}$ is a rank $r$ matrix. Let $\Omega$ be a generic subset of $\in \{1, \ldots, m\} \times \{1, \ldots, n\}$ of cardinality $r(m + n - r)$. Denote by $X_{\Omega}$ the matrix that is equal to $X$ on $\Omega$, and 0 on $\Omega^\complement$. Putting it mathematically, $$X_{\Omega}(i, j) = \begin{cases} X(i, j), \quad & (i, j) \in \Omega\\ 0, \quad & (i, j) \notin \Omega \end{cases}$$

Then, I was wondering that could we prove that $$\|X - Y\|_{\mathrm{F}} \lesssim \|X_{\Omega} - Y_{\Omega}\|_{\mathrm{F}}$$ for any rank $r$ matrices $X$ and $Y$. Any comments related and literature are very welcome.