发表机构
University of Padova（帕多瓦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对鲁棒最大最小多样化问题，提出一种确定性基于核心集的流式算法，在内部点-离群点分离假设下返回恰好 $k$ 个内部点，达到 $(2+\varepsilon)$ 近似，内存和更新代价与 $n$ 无关。
AI 中文摘要
给定度量空间中的一组 $n$ 个点 $X$ 和一个整数 $k$，最大最小多样化旨在选择 $X$ 中的 $k$ 个点，最大化它们之间的最小成对距离。然而，该目标函数极易受到噪声点的影响。在 [Amagata, AAAI23] 中，提出了一种鲁棒公式，通过排除包含 $z$ 个离群点（定义为 $X$ 中具有最大最近邻距离的 $z$ 个点）的任何解来解决此脆弱性。该论文还提出了一种基于核心集的流式算法，该算法基于合适的内部点-离群点分离假设。然而，我们发现了 [Amagata, AAAI23] 算法的三个缺点：其核心集构建需要对 $X$ 进行离线计算，这需要线性于 $n$ 的内存，与流处理的典型目标形成鲜明对比；用于从核心集提取解的单遍过程可能返回少于 $k$ 个点（因此产生不可行解），因为它永久丢弃了距离当前解太远的点；其离群点排除保证仅是概率性的，并且随着核心集大小的缩小而减弱。相比之下，我们提出了一种确定性的基于核心集的算法，在自然的内部点-离群点分离假设（类似于 [Amagata, AAAI23] 中使用的假设）下，该算法返回恰好 $k$ 个内部点，它们是 $(2+\varepsilon)$ 近似解，对于任何 $\varepsilon>0$，因此仅比最佳多项式时间顺序近似高出 $\varepsilon$，即使没有离群点也是如此。其单遍流式实现自适应地适应数据集的双倍维度 $D$，并且对于广泛的 $k$、$z$、$\varepsilon$ 和 $D$ 值，它使用的内存与 $n$ 无关。对于足够长的流，其摊还更新时间与核心集大小成正比，因此也与 $n$ 无关。
英文摘要
Given a set of $n$ points $X$ in a metric space and an integer $k$, max-min diversification aims to select $k$ points of $X$ maximizing their minimum pairwise distance. This objective function is however highly vulnerable to noisy points. In[Amagata, AAAI23], a robust formulation is proposed which addresses this vulnerability by excluding solutions containing any of $z$ outliers, defined as the $z$ points in $X$ with the largest nearest-neighbor distances. That paper also presents a coreset-based streaming algorithm for the new formulation, based on a suitable inlier-outlier separation assumption. However, we identify three shortcomings in the algorithm by [Amagata, AAAI23]: its coreset construction requires an offline computation over $X$, which needs memory linear in $n$, in stark contrast with the typical goals of stream processing; the one-pass procedure used to extract the solution from the coreset may return fewer than $k$ points (hence, an unfeasible solution) because it permanently discards points too far from the current solution; and its outlier-exclusion guarantee is only probabilistic and weakens as the coreset size shrinks. In contrast, we present a deterministic coreset-based algorithm that, under a natural inlier-outlier separation assumption (similar to the one used in [Amagata, AAAI23]), returns exactly $k$ inliers which are a $(2+\varepsilon)$-approximate solution, for any $\varepsilon>0$, thus only $\varepsilon$ above the best polynomial-time sequential approximation, even without outliers. Its one-pass streaming implementation adapts obliviously to the dataset's doubling dimension $D$ and, for wide ranges of $k$, $z$, $\varepsilon$, and $D$, it uses memory independent of $n$. For sufficiently long streams, its amortized update time is proportional to the coreset size, thus also independent of $n$.
