In this paper, we revisit spectral sparsification for sums of arbitrary positive semidefinite (PSD) matrices. Concretely, for any collection of PSD matrices $\mathcal{A} = \{A_1, A_2, \ldots, A_r\} \subset \mathbb{R}^{n \times n}$, given any subset $T \subseteq [r]$, our goal is to find sparse weights $μ\in \mathbb{R}_{\geq 0}^r$ such that $(1 - ε) \sum_{i \in T} A_i \preceq \sum_{i \in T} μ_i A_i \preceq (1 + ε) \sum_{i \in T} A_i.$ This generalizes spectral sparsification of graphs which corresponds to $\mathcal{A}$ being the set of Laplacians of edges. It also captures sparsifying Cayley graphs by choosing a subset of generators. The former has been extensively studied with optimal sparsifiers known. The latter has received attention recently and was solved for a few special groups (e.g., $\mathbb{F}_2^n$). Prior work shows any sum of PSD matrices can be sparsified down to $O(n)$ elements. This bound however turns out to be too coarse and in particular yields no non-trivial bound for building Cayley sparsifiers for Cayley graphs. In this work, we develop a new, instance-specific (i.e., specific to a given collection $\mathcal{A}$) theory of PSD matrix sparsification based on a new parameter $N^*(\mathcal{A})$ which we call connectivity threshold that generalizes the threshold of the number of edges required to make a graph connected. Our main result gives a sparsifier that uses at most $O(ε^{-2}N^*(\mathcal{A}) (\log n)(\log r))$ matrices and is constructible in randomized polynomial time. We also show that we need $N^*(\mathcal{A})$ elements to sparsify for any $ε< 0.99$. As the main application of our framework, we prove that any Cayley graph can be sparsified to $O(ε^{-2}\log^4 N)$ generators. Previously, a non-trivial bound on Cayley sparsifiers was known only in the case when the group is $\mathbb{F}_2^n$.

翻译：本文重新审视任意正定半定矩阵和的谱稀疏化问题。具体而言，对于任意正定半定矩阵集合 $\mathcal{A} = \{A_1, A_2, \ldots, A_r\} \subset \mathbb{R}^{n \times n}$ 及任意子集 $T \subseteq [r]$，我们的目标是找到稀疏权重 $μ\in \mathbb{R}_{\geq 0}^r$，使得 $(1 - ε) \sum_{i \in T} A_i \preceq \sum_{i \in T} μ_i A_i \preceq (1 + ε) \sum_{i \in T} A_i$。该问题推广了图的谱稀疏化（对应 $\mathcal{A}$ 为边拉普拉斯矩阵集合的情形），并通过选择生成元子集实现了凯莱图的稀疏化。前者已有深入研究并获得了最优稀疏化方案，后者近期受到关注且仅在少数特殊群（如 $\mathbb{F}_2^n$）中得以解决。已有研究表明任意正定半定矩阵和可稀疏化为 $O(n)$ 个元素，但该界过于宽松，尤其无法为凯莱图稀疏化提供非平凡边界。本文基于新参数 $N^*(\mathcal{A})$（称为连通性阈值，其推广了使图连通所需边数的阈值）建立了针对特定矩阵集合 $\mathcal{A}$ 的正定半定矩阵稀疏化理论。主要成果是构造了一个最多使用 $O(ε^{-2}N^*(\mathcal{A}) (\log n)(\log r))$ 个矩阵的稀疏化方案，且可在随机多项式时间内构建。同时证明对于任意 $ε< 0.99$，稀疏化至少需要 $N^*(\mathcal{A})$ 个元素。作为框架的主要应用，我们证明任意凯莱图可稀疏化为 $O(ε^{-2}\log^4 N)$ 个生成元。此前仅当群为 $\mathbb{F}_2^n$ 时存在凯莱稀疏化的非平凡边界。