Say a reconstruction of matrix $A$ is $A'$ and it's defined as

$ A' = PDP^TA $

where $P$ is an orthonormal matrix, $D$ is a diagonal binary (1 or 0) matrix. In a trivial case, when all diagonal elements are 1, we have a perfect reconstruction ($A'=A$).

Now we constrain the number of 1's in the diagonal entries of $D$ to, say, $n$. How do I find the best $D$ s.t. $Tr(D)=n$ that would minimize $||A-A'||$? $P$ and $A$ are fixed.

I think I need to inspect the singular vectors of $A$, but I am not sure what to do exactly. I think the 1's in $D$ should indicate columns of $P$ that best align with the column space of $A$, but what how do I determine the "best" alignment?

Say a reconstruction of matrix $A$ is $A'$ and it's defined as $$ A' = PDP^TA $$ where $P$ is an orthogonal matrix, $D$ is a diagonal binary (1 or 0) matrix. In a trivial case, when all diagonal elements are 1, we have a perfect reconstruction ($A'=A$).

Now we constrain the number of 1's in the diagonal entries of $D$ to, say, $n$. How do I find the best $D$ s.t. $\operatorname{Tr}(D)=n$ that would minimize $\lVert A-A'\rVert$? $P$ and $A$ are fixed.

I think I need to inspect the singular vectors of $A$, but I am not sure what to do exactly. I think the 1's in $D$ should indicate columns of $P$ that best align with the column space of $A$, but what how do I determine the "best" alignment?