AI 中文总结
研究加权超图谱\(\varepsilon -\)稀疏化器，核心方法是全局字典链，通过该方法消除了秩依赖性，加强了相关工作，回答了开放性问题，其成果强化了许多后续继承采样界的保证。
AI 中文摘要
我们证明，每个具有\(n\)个顶点的加权超图都允许有一个具有\(O(n\log n/\varepsilon^2)\)条超边的谱\(\varepsilon -\)稀疏化器，这加强了2023年STOC会议上Lee以及Jambulapati - Liu - Sidford的独立工作，通过消除其秩依赖性并回答了Lee关于这种损失是否固有的开放性问题。关键思想是全局字典链：在选择具有平衡有效电阻的团边权重后，每个超边半范数相对于由归一化顶点对方向生成的相同全局字典范数是利普希茨连续的；局部秩复杂度由此被这个公共字典的高斯宽度所取代。由于这些2023年STOC会议上的工作已成为后续关于谱超图稀疏化及其变体的广泛文献中的标准分析原语，我们的与秩无关定理强化了许多后来继承其采样界的保证。
英文摘要
We show that every weighted hypergraph on $n$ vertices admits a spectral $\varepsilon$-sparsifier with $O(n\log n/\varepsilon^2)$ hyperedges, strengthening the independent STOC 2023 works of Lee and Jambulapati--Liu--Sidford by removing their rank dependence and answering Lee's open question on whether this loss is inherent. The key idea is global-dictionary chaining: after choosing clique edge weights with balanced effective resistances, every hyperedge seminorm is Lipschitz with respect to the same global-dictionary norm generated by normalized vertex-pair directions; the local rank complexity is thereby replaced by the Gaussian width of this common dictionary. Since these STOC 2023 works have become standard analytic primitives across a broad subsequent literature on spectral hypergraph sparsification and its variants, our rank-independent theorem sharpens many later guarantees that inherit their sampling bounds.
CommentsThe paper is withdrawn because the cited result does not support Theorem 3, invalidating the subsequent argument
