Suppose I have a $n \times p$ data matrix $X$, $p>>n$. To reduce the dimension of the data, I use principal component analysis as follows: I perform SVD and find matrices U ($n \times r$) and V ($r \times p$) such that $X=UDV$, where $D$ is a diagonal matrix. Now I reduce the dimension of $X$ using the matrix $V$, i.e., use the PC scores $Z=XV^{\prime}$. My question is in that case does the property like 'restricted isometry' hold for the projected data point. In particular, if I consider the rows of $X$ are independently generated from some distribution, then what are the most sharp bounds ($m$, $M$) for which the following hold
$$ m \| x \|^2 \leq \| Vx \|^2 \leq M \| x \|^2 ?$$