在所有最先进的聚类概要（sketching）与核心集（coreset）技术，以及已知最优的固定参数可处理近似算法中，随机性都起着关键作用。对于经典的$k$-中位数（$k$-median）和$k$-均值（$k$-means）问题，目前尚无已知的确定性降维流程或核心集构建方法，能够避免对输入维度$d$、精度参数$\u03b5^{-1}$或$k$的指数依赖。此外，也不存在成功概率为$1-1/n$且大小不依赖输入点数量$n$的核心集构建方法。这使得该领域的研究者提出疑问：随机性对于聚类概要的能力究竟有多大[Feldman, WIREs Data Mining Knowl. Discov'20]。类似地，在不引入维度指数级复杂度的前提下，确定性方法能达到的最优近似比对于$k$-中位数和$k$-均值均为$Ω(1)$，即便允许复杂度关于簇的数量$k$为固定参数可处理（FPT）也是如此。这与允许随机化时该场景下可达到的$(1+\u03b5)$-近似形成了鲜明对比。\n在本文中，我们提出了确定性的聚类概要构建方法，其大小边界接近已知最优的随机化方法的对应边界。我们还构建了一种确定性算法，用于在高维欧氏空间中计算$k$-中位数和$k$-均值的$(1+\u03b5)$-近似，时间复杂度为$2^{k^2/\u03b5^{O(1)}} poly(nd)$，接近最优的随机化复杂度。\n此外，我们在概要方面的新见解还催生了一种使用均匀采样的随机化核心集构建方法，其性能直接比[Braverman et al. FOCS '22]的最新结果提升了$k$倍。
英文摘要：
In all state-of-the-art sketching and coreset techniques for clustering, as well as in the best known fixed-parameter tractable approximation algorithms, randomness plays a key role. For the classic $k$-median and $k$-means problems, there are no known deterministic dimensionality reduction procedure or coreset construction that avoid an exponential dependency on the input dimension $d$, the precision parameter $\varepsilon^{-1}$ or $k$. Furthermore, there is no coreset construction that succeeds with probability $1-1/n$ and whose size does not depend on the number of input points, $n$. This has led researchers in the area to ask what is the power of randomness for clustering sketches [Feldman, WIREs Data Mining Knowl. Discov'20]. Similarly, the best approximation ratio achievable deterministically without a complexity exponential in the dimension are $Ω(1)$ for both $k$-median and $k$-means, even when allowing a complexity FPT in the number of clusters $k$. This stands in sharp contrast with the $(1+\varepsilon)$-approximation achievable in that case, when allowing randomization.
In this paper, we provide deterministic sketches constructions for clustering, whose size bounds are close to the best-known randomized ones. We also construct a deterministic algorithm for computing $(1+\varepsilon)$-approximation to $k$-median and $k$-means in high dimensional Euclidean spaces in time $2^{k^2/\varepsilon^{O(1)}} poly(nd)$, close to the best randomized complexity.
Furthermore, our new insights on sketches also yield a randomized coreset construction that uses uniform sampling, that immediately improves over the recent results of [Braverman et al. FOCS '22] by a factor $k$.
A Deterministic Streaming Sketch for Ridge Regression
University of Utah（犹他大学）
我们提供了一个确定性的空间有效算法估计岭回归。对于具有$d$特征和足够大的正则化参数的$n$数据点，我们仅使用$O(d/\varepsilon)$空间提供了$\varepsilon$ L $_2$误差内的解决方案。这是第一个$o(d^2)$空间确定性流算法，保证解决这个经典问题的错误和风险界。该算法通过频繁方向的变体来勾画协方差矩阵，这意味着它可以在仅插入流和各种分布式数据设置中操作。在合成和真实世界的数据集上的随机草图算法相比，我们的算法具有较少的经验误差，使用较少的空间和相似的时间。
