Let $A \in \mathbb{R}^{n \times n}$ a positive definite matrix and $x \in \mathbb{R}^n$.
I am intetested in $x^TAx$.
However, I do not have $A$ explicitly and I need to approximate it by $\widetilde{A}$, the best rank-$k$ of $A$ in some norm (for some given $k$). In what norm is it best to approximate $A$ (i.e., to minimize $\lVert A - \widetilde{A}\rVert_?$) in order to minimize $|x^TAx - x^T\widetilde{A}x|$?