Minkowski和与半范数和的优化稀疏化器
Optimal Sparsifiers for Minkowski Sums and Sums of Seminorms
- Princeton University（普林斯顿大学）
- University of California, Berkeley（加州大学伯克利分校）
- Department of EECS, University of Michigan, Ann Arbor（密歇根大学安娜堡分校电气工程和计算机科学系）
- School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, USA（哈佛大学工程与应用科学学院）
- Microsoft Research, Redmond（微软研究院雷德蒙德分部）
- Department of Computer Science and Engineering, The Ohio State University（俄亥俄州立大学计算机科学与工程系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种稀疏化中心对称凸集Minkowski和的方法，仅用O(n/ε²)个非零权重即可近似，并推广至半范数和、超图割及对称子模函数和，达到最优规模。
AI中文摘要：
我们将Reis和Rothvoss近期关于稀疏化ℓ₁范数之和的工作扩展到更一般的任务：稀疏化中心对称凸集的（Minkowski）和。作为我们的主要结果，我们证明对于任意ε>0和中心对称凸集C₁,…,Cₘ⊆ℝⁿ，存在权重λ₁,…,λₘ∈ℝ≥₀的选择，使得至多O(n/ε²)个权重非零，并且(1−ε)·C⊆∑ᵢ₌₁ᵐλᵢ·Cᵢ⊆(1+ε)·C，其中C:=C₁+⋯+Cₘ指集合C₁,…,Cₘ的Minkowski和，λ·C指集合C的膨胀。作为该结果的直接应用，我们获得了大小为O(n/ε²)的稀疏化器，用于稀疏化n维空间中的半范数和，改进了Jambulapati、Lee、Liu和Sidford（FOCS 2023）工作中O(n log(n/ε)·log²·⁵(n)/ε²)大小的稀疏化器。这进一步产生了具有O(n/ε²)条超边的最优大小超图割稀疏化器，改进了Chen、Khanna和Nagda（FOCS 2020）工作中O(n log(n)/ε²)大小的稀疏化器。更一般地，这也为对称子模函数之和提供了最优大小的稀疏化器。
英文摘要：
We extend the recent work of Reis and Rothvoss on sparsifying sums of $\ell_1$ norms to the more general task of sparsifying (Minkowski) sums of centrally symmetric, convex sets. As our main result, we prove that for any $\varepsilon > 0$ and centrally symmetric, convex sets $C_1, \ldots, C_m\subseteq\mathbb{R}^n$ there is a choice of weights $λ_1, \dots , λ_m \in \mathbb{R}_{\geq 0}$ such that at most $O(n / \varepsilon^2)$ of the weights are non-zero, and \[(1 - \varepsilon)\cdot C\subseteq\sum_{i = 1}^mλ_i\cdot C_i\subseteq(1 + \varepsilon)\cdot C,\] where $C:= C_1 + \cdots + C_m$ refers to the Minkowski sums of the sets $C_1, \ldots, C_m$, and $λ\cdot C$ refers to the dilation of the set $C$. As immediate applications of this result, we obtain sparsifiers of size $O(n / \varepsilon^2)$ for sparsifying sums of seminorms in $n$-dimensional space, improving on the $O\left ( \frac{n \log(n/\varepsilon) \cdot \log^{2.5}(n)}{\varepsilon^2} \right )$ size sparsifiers from the work of Jambulapati, Lee, Liu, and Sidford (FOCS 2023). This further yields optimal size hypergraph cut sparsifiers with $O(n / \varepsilon^2)$ hyperedges, improving on the $O(n \log(n) / \varepsilon^2)$ size sparsifiers from the work of Chen, Khanna, and Nagda (FOCS 2020). More generally, this also gives optimal size sparsifiers for sums of symmetric submodular functions.
