I have a matrix $P$ with size $(d,d)$ and two vectors $x$ and $y$ of size $(d,1)$. I want to reduce the size of the quantity $(Px-Py)$ by setting the size of $P$ to be $(p,d)$ where $p \leq d$ to ensure that the scalar product $\|P^T (Px-Py)\|$ is as close as possible to the one when using the matrix with size $(d,d)$.
If this was about reducing the number of columns, I know that SVD can be used but the question here is about reducing the number of rows in $P$. Would it make sense to apply SVD decomposition to $P^T$ instead? In this sense, we will reduce the rows of $P$. Otherwise, is there a method that can be used to achieve this goal?