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 $|N(x) - \tilde{N}(x)| \leq \epsilon N(x)$ for all $x \in \mathbb{R}^n$, where $w_1,\ldots,w_m$ are non-negative weights, of which only $O(\epsilon^{-2} n \log(n/\epsilon) (\log n)^{2.5} )$ are non-zero. Additionally, if $N$ is $\mathrm{poly}(n)$-equivalent to the Euclidean norm on $\mathbb{R}^n$, then such weights can be found with high probability in time $O(m (\log n)^{O(1)} + \mathrm{poly}(n)) T$, where $T$ is the time required to evaluate a norm $N_i$. 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.

翻译：对于 $\mathbb{R}^n$ 上的任意范数 $N_1,\ldots,N_m$ 以及 $N(x) := N_1(x)+\cdots+N_m(x)$，我们证明存在一个稀疏化范数 $\tilde{N}(x) = w_1 N_1(x) + \cdots + w_m N_m(x)$，使得对所有 $x \in \mathbb{R}^n$ 满足 $|N(x) - \tilde{N}(x)| \leq \epsilon N(x)$，其中 $w_1,\ldots,w_m$ 为非负权重，且仅有 $O(\epsilon^{-2} n \log(n/\epsilon) (\log n)^{2.5} )$ 个权重非零。此外，若 $N$ 与 $\mathbb{R}^n$ 上的欧几里得范数 $\mathrm{poly}(n)$-等价，则可在时间 $O(m (\log n)^{O(1)} + \mathrm{poly}(n)) T$ 内以高概率找到此类权重，其中 $T$ 为计算单个范数 $N_i$ 所需时间。这一结果直接导出对对称子模函数之和进行稀疏化的类似结论。更一般地，我们展示了当范数之和为 $p$ 阶一致光滑时，如何对范数的 $p$ 次幂之和进行稀疏化。