We introduce the first iterative algorithm for constructing a $\varepsilon$-coreset that guarantees deterministic $\ell_p$ subspace embedding for any $p \in [1,\infty)$ and any $\varepsilon > 0$. For a given full rank matrix $\mathbf{X} \in \mathbb{R}^{n \times d}$ where $n \gg d$, $\mathbf{X}' \in \mathbb{R}^{m \times d}$ is an $(\varepsilon,\ell_p)$-subspace embedding of $\mathbf{X}$, if for every $\mathbf{q} \in \mathbb{R}^d$, $(1-\varepsilon)\|\mathbf{Xq}\|_{p}^{p} \leq \|\mathbf{X'q}\|_{p}^{p} \leq (1+\varepsilon)\|\mathbf{Xq}\|_{p}^{p}$. Specifically, in this paper, $\mathbf{X}'$ is a weighted subset of rows of $\mathbf{X}$ which is commonly known in the literature as a coreset. In every iteration, the algorithm ensures that the loss on the maintained set is upper and lower bounded by the loss on the original dataset with appropriate scalings. So, unlike typical coreset guarantees, due to bounded loss, our coreset gives a deterministic guarantee for the $\ell_p$ subspace embedding. For an error parameter $\varepsilon$, our algorithm takes $O(\mathrm{poly}(n,d,\varepsilon^{-1}))$ time and returns a deterministic $\varepsilon$-coreset, for $\ell_p$ subspace embedding whose size is $O\left(\frac{d^{\max\{1,p/2\}}}{\varepsilon^{2}}\right)$. Here, we remove the $\log$ factors in the coreset size, which had been a long-standing open problem. Our coresets are optimal as they are tight with the lower bound. As an application, our coreset can also be used for approximately solving the $\ell_p$ regression problem in a deterministic manner.

翻译：我们提出了首个迭代算法，用于构建保证对任意$p \in [1,\infty)$和任意$\varepsilon > 0$均具有确定性$\ell_p$子空间嵌入的$\varepsilon$-核心集。对于给定满秩矩阵$\mathbf{X} \in \mathbb{R}^{n \times d}$（其中$n \gg d$），若对任意$\mathbf{q} \in \mathbb{R}^d$有$(1-\varepsilon)\|\mathbf{Xq}\|_{p}^{p} \leq \|\mathbf{X'q}\|_{p}^{p} \leq (1+\varepsilon)\|\mathbf{Xq}\|_{p}^{p}$，则$\mathbf{X}' \in \mathbb{R}^{m \times d}$称为$\mathbf{X}$的$(\varepsilon,\ell_p)$-子空间嵌入。具体而言，本文中$\mathbf{X}'$是$\mathbf{X}$行向量的加权子集（文献中通常称为核心集）。在每次迭代中，算法确保维护集合上的损失通过适当缩放被原始数据集上的损失上下界约束。因此，与典型的核心集保证不同，由于损失有界性，我们的核心集为$\ell_p$子空间嵌入提供了确定性保证。对于误差参数$\varepsilon$，算法耗时$O(\mathrm{poly}(n,d,\varepsilon^{-1}))$，并返回一个大小为$O\left(\frac{d^{\max\{1,p/2\}}}{\varepsilon^{2}}\right)$的确定性$\varepsilon$-核心集（用于$\ell_p$子空间嵌入）。此处我们去除了核心集大小中的对数因子，这曾是长期存在的开放问题。我们的核心集达到最优，因为其与下界严格匹配。作为应用，该核心集还可用于以确定性方式近似求解$\ell_p$回归问题。