Streaming algorithms for robust max-min diversification

Abstract

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 $n$.