Sparsifying Sums of Norms

Abstract-For any norms $N_{1}, \ldots, N_{m}$ on $\mathbb{R}^{n}$ and $N(x):= N_{1}(x)+\cdots+N_{m}(x)$, we show there is a sparsified norm $\tilde{N}(x)= w_{1} N_{1}(x)+\cdots+w_{m} N_{m}(x)$ such that $|N(x)-\tilde{N}(x)| \leqslant \varepsilon N(x)$ for all $x \in \mathbb{R}^{n}$, where $w_{1}, \ldots, w_{m}$ are non-negative weights, of which only $O\left(\varepsilon^{-2} n \log (n / \varepsilon)(\log n)^{2.5}\right)$ are non-zero. Additionally, we show that such weights can be found with high probability in time $O\left(m(\log n)^{O(1)}+\right.$ poly $\left.(n)\right) T$, where T is the time required to evaluate a norm $N_{i}(x)$, assuming that $N(x)$ is poly $(n)$ equivalent to the Euclidean norm. This immediately yields analogous statements for sparsifying sums of symmetric submodular functions. More generally, we show how to sparsify sums of p th powers of norms when the sum is p-uniformly smooth.¹