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, we show that 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 $N_i(x)$, assuming that $N(x)$ is $\mathrm{poly}(n)$-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.

翻译：对于 $\mathbb{R}^n$ 上的任意范数 $N_1,\ldots,N_m$ 以及 $N(x) := N_1(x)+\cdots+N_m(x)$，我们证明了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 $x \in \mathbb{R}^n$ 有 $|N(x) - \tilde{N}(x)| \leq \epsilon N(x)$，其中 $w_1,\ldots,w_m$ 为非负权重，且仅 $O(\epsilon^{-2} n \log(n/\epsilon) (\log n)^{2.5} )$ 个权重非零。此外，我们证明了在 $N(x)$ 与欧几里得范数 $\mathrm{poly}(n)$-等价的假设下，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 $T$ 为评估范数 $N_i(x)$ 所需时间。该结果直接蕴含对称子模函数之和稀疏化的类似结论。更一般地，我们展示了当范数 $p$ 次幂之和为 $p$-一致光滑时，如何稀疏化此类和式。