Consider the problem of sparse principal component analysis: $$\max_{||{\bf x}||_0=k,||{\bf x}||_2=1} {\bf x}^T{\bf A}{\bf x}$$ where a $k$-sparse $n$-dim. unit vector that "maximizes variance" is to be found. I was wondering if there were any practical (i.e. not artificial :)) examples where the correlation matrix ${\bf A}$ has fixed and low rank, or it is a low rank update, i.e. ${\bf A} = c{\bf I}+{\bf C}{\bf C}^T$, where $c$ is a constant and ${\bf C}$ has low and fixed rank independent of $n$.

thank you