Suppose I need to compute the $L^2$ norm $| \cdot |$ of the sum of $n$ vectors $v_i \in \mathbb{R}^d$.
Denote $k = | \sum_{i=1}^n v_i |$, which can be achieved via adding all the vectors and then computing the $L^2$ norm. However, adding all the vectors first might be expensive if $n$ is big and I have many different subsets of the $n$ that needs to be computed.
I'm wondering what's a good way to approximate $k$ with $k^* = f(g(v_i))$, here $g: \mathbb{R}^d \to \mathbb{R}$ ? Namely, what kind of $f, g$ would ensure $k^*$ would be close to $k$ ?