Let $X$ represent an $N \times K$ matrix, $\Gamma$ a $K\times K$ matrix, and $E$ a $N\times K$ matrix. Let $Z$ be a matrix satisfying $ZZ^T = (I_N+X\Gamma X^T)$ My goal is to quickly compute the quantity $ZE$. Assume that: $K << N$.
In this case, I am primarily concerned with the speed of the actual matrix multiplication $ZE$ as $N$ is very large and naive solutions ends up having $Z$ involving the matrix multiplication of an $N\times N$ with an $N\times K$ matrix. I am not actually concerned with the time it takes to compute any matrix decompositions of $I_N+X\Gamma X^T$.
My feeling is that there must be some matrix identity that would allow me to reduce the dimension of these operations by exploiting the fact that $X\Gamma X^T$ is low rank and that multiplication by the identity matrix leaves a quantity unchanged (essentially allowing me to ignore that multiplication).
If need be I am ok with taking a low-rank approximation of the quantity $ZE$.
Any ideas?
