No-Substitution k-means Clustering with Low Center Complexity and Memory
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Clustering is a fundamental task in machine learning. Given a dataset X = {x_1, … 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 Ω(n), meaning that it takes Ω(n) centers in expectation. Bhattacharjee and Moshkovitz (2020) refined this bound by defining a complexity measure called Lower_α, k(X), and proving that any α-approximation algorithm must have center complexity Ω(Lower_α, k(X)). They then complemented their lower bound by giving a O(k^3)-approximation algorithm with center complexity Õ(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 Õ(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.
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