No-Substitution $k$-means Clustering with Low Center Complexity and Memory.

Clustering is a fundamental task in machine learning. Given a dataset $X = \{x_1, \ldots x_n\}$, the goal of $k$-means clustering is to pick $k$ "centers" from $X$ in a way that minimizes the sum of squared distances from each point to its nearest center. We consider $k$-means clustering in the online, no substitution setting, where one must decide whether to take $x_t$ as a center immediately upon streaming it and cannot remove centers once taken. The online, no substitution setting is challenging for clustering--one can show that there exist datasets $X$ for which any $O(1)$-approximation $k$-means algorithm must have center complexity $\Omega(n)$, meaning that it takes $\Omega(n)$ centers in expectation. Bhattacharjee and Moshkovitz (2020) refined this bound by defining a complexity measure called $Lower_{\alpha, k}(X)$, and proving that any $\alpha$-approximation algorithm must have center complexity $\Omega(Lower_{\alpha, k}(X))$. They then complemented their lower bound by giving a $O(k^3)$-approximation algorithm with center complexity $\tilde{O}(k^2Lower_{k^3, k}(X))$, thus showing that their parameter is a tight measure of required center complexity. However, a major drawback of their algorithm is its memory requirement, which is $O(n)$. This makes the algorithm impractical for very large datasets. In this work, we strictly improve upon their algorithm on all three fronts; we develop a $36$-approximation algorithm with center complexity $\tilde{O}(kLower_{36, k}(X))$ that uses only $O(k)$ additional memory. In addition to having nearly optimal memory, this algorithm is the first known algorithm with center complexity bounded by $Lower_{36, k}(X)$ that is a true $O(1)$-approximation with its approximation factor being independent of $k$ or $n$.