[Paper Review] Unexpected Effects of Online K-means Clustering.

Michal Moshkovitz|arXiv (Cornell University)|Aug 9, 2019

Advanced Clustering Algorithms Research4 citations

TL;DR

This paper investigates online k-means clustering and reveals that data ordering and knowledge of dataset size n critically affect clustering quality. For constant k, the number of required centers drops from Ω(n) with arbitrary ordering to Θ(log n) with random ordering and unknown n, and to a constant when n is known, establishing tight asymptotic bounds for each scenario.

ABSTRACT

In this paper we study k-means clustering in the online setting. In the offline setting the main parameters are number of centers, k, and size of the dataset, n. Performance guarantees are given as a function of these parameters. In the online setting new factors come into place: the ordering of the dataset and whether n is known in advance or not. One of the main results of this paper is the discovery that these new factors have dramatic effects on the quality of the clustering algorithms. For example, for constant k: (1) $\Omega(n)$ centers are needed if the order is arbitrary, (2) if the order is random and n is unknown in advance, the number of centers reduces to $\Theta(logn)$, and (3) if n is known, then the number of centers reduces to a constant. For different values of the new factors, we show upper and lower bounds that are exactly the same up to a constant, thus achieving optimal bounds.

Motivation & Objective

• To understand how online data ordering and knowledge of dataset size n affect the performance of k-means clustering.
• To analyze the impact of these new factors—ordering and n-knowledge—on the number of centers required for effective clustering.
• To establish tight upper and lower bounds on the number of centers needed under various combinations of ordering and n-knowledge.
• To provide theoretical guarantees for online k-means clustering that account for dynamic data arrival and limited prior knowledge.

Proposed method

• Analyzing online k-means under different data arrival models: arbitrary order, random order, and known vs. unknown n.
• Deriving theoretical upper and lower bounds on the number of centers required for clustering quality, parameterized by k, n, and data ordering.
• Using asymptotic analysis to show that the bounds are tight up to constant factors across all scenarios.
• Formalizing the trade-offs between clustering quality and the number of centers based on the arrival sequence and knowledge of n.
• Applying probabilistic and adversarial arguments to establish lower bounds under arbitrary and random orderings.

Experimental results

Research questions

• RQ1 How does arbitrary data ordering affect the number of centers required in online k-means clustering for constant k?
• RQ2 What is the impact of random data ordering on the number of centers when n is unknown in advance?
• RQ3 How does knowing the dataset size n in advance reduce the number of required centers in online k-means?
• RQ4 Are the theoretical bounds on center count tight across different online clustering scenarios?

Key findings

• For constant k and arbitrary data ordering, Ω(n) centers are required to maintain clustering quality.
• With random data ordering and unknown n, the number of centers reduces to Θ(log n), representing a significant improvement.
• When n is known in advance, the number of centers required reduces to a constant, even for constant k.
• The upper and lower bounds for each scenario are tight up to constant factors, indicating optimal performance guarantees.