Let $\mathbf S \in \mathbb R^{n \times n}$ satisfy $\|\mathbf 1-\mathbf S\|_2\le\epsilon n$, where $\mathbf 1$ is the all ones matrix and $\|\cdot\|_2$ is the spectral norm. It is well-known that there exists such an $\mathbf S$ with just $O(n/\epsilon^2)$ non-zero entries: we can let $\mathbf S$ be the scaled adjacency matrix of a Ramanujan expander graph. We show that such an $\mathbf S$ yields a $universal$ $sparsifier$ for any positive semidefinite (PSD) matrix. In particular, for any PSD $\mathbf A \in \mathbb{R}^{n\times n}$ with entries bounded in magnitude by $1$, $\|\mathbf A - \mathbf A\circ\mathbf S\|_2 \le \epsilon n$, where $\circ$ denotes the entrywise (Hadamard) product. Our techniques also give universal sparsifiers for non-PSD matrices. In this case, letting $\mathbf S$ be the scaled adjacency matrix of a Ramanujan graph with $\tilde O(n/\epsilon^4)$ edges, we have $\|\mathbf A - \mathbf A \circ \mathbf S \|_2 \le \epsilon \cdot \max(n,\|\mathbf A\|_1)$, where $\|\mathbf A\|_1$ is the nuclear norm. We show that the above bounds for both PSD and non-PSD matrices are tight up to log factors. Since $\mathbf A \circ \mathbf S$ can be constructed deterministically, our result for PSD matrices derandomizes and improves upon known results for randomized matrix sparsification, which require randomly sampling ${O}(\frac{n \log n}{\epsilon^2})$ entries. We also leverage our results to give the first deterministic algorithms for several problems related to singular value approximation that run in faster than matrix multiplication time. Finally, if $\mathbf A \in \{-1,0,1\}^{n \times n}$ is PSD, we show that $\mathbf{\tilde A}$ with $\|\mathbf A - \mathbf{\tilde A}\|_2 \le \epsilon n$ can be obtained by deterministically reading $\tilde O(n/\epsilon)$ entries of $\mathbf A$. This improves the $1/\epsilon$ dependence on our result for general PSD matrices and is near-optimal.

翻译：设 $\mathbf S \in \mathbb R^{n \times n}$ 满足 $\|\mathbf 1-\mathbf S\|_2\le\epsilon n$，其中 $\mathbf 1$ 为全1矩阵，$\|\cdot\|_2$ 为谱范数。已知存在仅含 $O(n/\epsilon^2)$ 个非零元的此类 $\mathbf S$：可将 $\mathbf S$ 取为Ramanujan扩展图的缩放邻接矩阵。我们证明，这样的 $\mathbf S$ 可为任意半正定（PSD）矩阵提供$universal$ $sparsifier$（通用稀疏化器）。特别地，对任意元素幅值不超过1的PSD矩阵 $\mathbf A \in \mathbb{R}^{n\times n}$，有 $\|\mathbf A - \mathbf A\circ\mathbf S\|_2 \le \epsilon n$，其中 $\circ$ 表示逐元素（Hadamard）乘积。我们的技术亦适用于非PSD矩阵的通用稀疏化。此时，令 $\mathbf S$ 为具有 $\tilde O(n/\epsilon^4)$ 条边的Ramanujan图缩放邻接矩阵，可得 $\|\mathbf A - \mathbf A \circ \mathbf S \|_2 \le \epsilon \cdot \max(n,\|\mathbf A\|_1)$，其中 $\|\mathbf A\|_1$ 为核范数。我们证明上述PSD与非PSD矩阵的界在忽略对数因子下是紧的。由于 $\mathbf A \circ \mathbf S$ 可确定性构造，我们的PSD矩阵结果将需要随机采样 ${O}(\frac{n \log n}{\epsilon^2})$ 个元素的随机矩阵稀疏化方法去随机化并改进。我们进一步利用该结果给出了首个运行时间快于矩阵乘法的奇异值逼近相关问题确定性算法。最后，对PSD矩阵 $\mathbf A \in \{-1,0,1\}^{n \times n}$，我们证明通过确定性读取 $\tilde O(n/\epsilon)$ 个元素即可得到满足 $\|\mathbf A - \mathbf{\tilde A}\|_2 \le \epsilon n$ 的 $\mathbf{\tilde A}$，这改善了通用PSD矩阵结果中 $1/\epsilon$ 的依赖关系，且接近最优。