Given $\mathbf A \in \mathbb{R}^{n \times n}$ with entries bounded in magnitude by $1$, it is well-known that if $S \subset [n] \times [n]$ is a uniformly random subset of $\tilde{O} (n/\epsilon^2)$ entries, and if ${\mathbf A}_S$ equals $\mathbf A$ on the entries in $S$ and is zero elsewhere, then $\|\mathbf A - \frac{n^2}{s} \cdot {\mathbf A}_S\|_2 \le \epsilon n$ with high probability, where $\|\cdot\|_2$ is the spectral norm. We show that for positive semidefinite (PSD) matrices, no randomness is needed at all in this statement. Namely, there exists a fixed subset $S$ of $\tilde{O} (n/\epsilon^2)$ entries that acts as a universal sparsifier: the above error bound holds simultaneously for every bounded entry PSD matrix $\mathbf A \in \mathbb{R}^{n \times n}$. One can view this result as a significant extension of a Ramanujan expander graph, which sparsifies any bounded entry PSD matrix, not just the all ones matrix. We leverage the existence of such universal sparsifiers to give the first deterministic algorithms for several central problems related to singular value computation that run in faster than matrix multiplication time. We also prove universal sparsification bounds for non-PSD matrices, showing that $\tilde{O} (n/\epsilon^4)$ entries suffices to achieve error $\epsilon \cdot \max(n,\|\mathbf A\|_1)$, where $\|\mathbf A\|_1$ is the trace norm. We prove that this is optimal up to an $\tilde{O} (1/\epsilon^2)$ factor. Finally, we give an improved deterministic spectral approximation algorithm for PSD $\mathbf A$ with entries lying in $\{-1,0,1\}$, which we show is nearly information-theoretically optimal.

翻译：给定 $\mathbf A \in \mathbb{R}^{n \times n}$，其元素绝对值不超过 $1$，众所周知，若 $S \subset [n] \times [n]$ 是大小为 $\tilde{O} (n/\epsilon^2)$ 的均匀随机子集，且 ${\mathbf A}_S$ 在 $S$ 中的元素上等于 $\mathbf A$，其余位置为零，则高概率地有 $\|\mathbf A - \frac{n^2}{s} \cdot {\mathbf A}_S\|_2 \le \epsilon n$，其中 $\|\cdot\|_2$ 为谱范数。我们证明，对于半正定矩阵，该结论完全无需随机性：存在一个固定的子集 $S$ 大小为 $\tilde{O} (n/\epsilon^2)$，可作为通用稀疏化器——上述误差界同时适用于任意有界元素的半正定矩阵 $\mathbf A \in \mathbb{R}^{n \times n}$。这一结果可视为对 Ramanujan 扩展图的重要推广，后者仅稀疏化全一矩阵，而我们的方法适用于任意有界半正定矩阵。利用此类通用稀疏化器的存在性，我们首次给出了若干奇异值计算核心问题的确定性算法，其运行速度快于矩阵乘法时间。我们还证明了非半正定矩阵的通用稀疏化界：$\tilde{O} (n/\epsilon^4)$ 个元素足以实现误差 $\epsilon \cdot \max(n,\|\mathbf A\|_1)$，其中 $\|\mathbf A\|_1$ 为迹范数。我们证明此结果在 $\tilde{O} (1/\epsilon^2)$ 因子内最优。最后，我们改进了针对元素取值为 $\{-1,0,1\}$ 的半正定矩阵 $\mathbf A$ 的确定性谱逼近算法，并证明该算法在信息论意义上近乎最优。