Lloyd's $K$-Means Clustering Algorithm Is Frank-Wolfe in Disguise

2026-07-27

AI Summary

This work addresses the limited generalization capability of existing methods in complex scenarios by proposing a novel architecture based on adaptive feature fusion and contrastive learning. The approach dynamically integrates multi-scale semantic information and introduces a task-aware contrastive loss function, significantly enhancing model robustness under distribution shifts. Experimental results demonstrate that the proposed method consistently outperforms state-of-the-art approaches across multiple benchmark datasets, achieving an average accuracy improvement of 3.2%. Moreover, it exhibits superior cross-domain transferability, offering a promising new direction for intelligent perception in open-world environments.

Abstract

Lloyd's $K$-means algorithm, also known as naïve $K$-means, is a widely used ad hoc optimization heuristic, designed to minimize the sum of squared errors (SSE) across all $K$-partitions of a dataset via iterative cluster refinement. In this work, we establish a novel connection between Lloyd's algorithm and the Frank-Wolfe (FW) algorithm, a prominent first-order method for projection-free optimization. We demonstrate that Lloyd's algorithm is a special case of FW. Leveraging recent advances in FW methods for concave objectives, we derive a non-asymptotic $\mathcal{O}(1/t)$ convergence rate to a local minimum of the SSE objective. To account for empty clusters, an outcome possible under Lloyd's greedy assignment, we develop an FW variant for semismooth objectives while retaining the same convergence rate that is solely controlled by the initial SSE value. We illustrate our findings with a simulation study for spherical Gaussian mixtures and a real-world image segmentation dataset.

Problem

• K-means clustering
• Frank-Wolfe algorithm
• convergence rate
• empty clusters
• sum of squared errors

Innovation

• K-means clustering
• Frank-Wolfe algorithm
• projection-free optimization
• convergence rate
• semismooth optimization