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A Geometric Approach to $k$-means

Jiazhen Hong, Wei Qian, Yudong Chen, Yuqian Zhang

TL;DR

This work tackles the nonconvexity of the $k$-means objective by exploiting the geometric structure of local minima. It introduces Fission-Fusion $k$-means (FFkm), a framework that detects mis-specified clusters and applies non-local fission and fusion steps to escape bad locals, interleaved with Lloyd updates. Theoretical results show recovery of ground-truth centers in $O(k)$ iterations under a stochastic ball model with sufficient separation, contrasting Lloyd’s exponential dependence on random initializations. Empirically, FFkm demonstrates robust performance across synthetic benchmarks and real-world tasks such as color quantization, often outperforming standard Lloyd’s algorithm and several modern alternatives, and even handling mis-specification of the number of clusters. The approach unifies and clarifies the efficacy of various heuristic improvements, offering a flexible toolkit for improving $k$-means in diverse data regimes.

Abstract

\kmeans clustering is a fundamental problem in many scientific and engineering domains. The optimization problem associated with \kmeans clustering is nonconvex, for which standard algorithms are only guaranteed to find a local optimum. Leveraging the hidden structure of local solutions, we propose a general algorithmic framework for escaping undesirable local solutions and recovering the global solution or the ground truth clustering. This framework consists of iteratively alternating between two steps: (i) detect mis-specified clusters in a local solution, and (ii) improve the local solution by non-local operations. We discuss specific implementation of these steps, and elucidate how the proposed framework unifies many existing variants of \kmeans algorithms through a geometric perspective. We also present two natural variants of the proposed framework, where the initial number of clusters may be over- or under-specified. We provide theoretical justifications and extensive experiments to demonstrate the efficacy of the proposed approach.

A Geometric Approach to $k$-means

TL;DR

This work tackles the nonconvexity of the -means objective by exploiting the geometric structure of local minima. It introduces Fission-Fusion -means (FFkm), a framework that detects mis-specified clusters and applies non-local fission and fusion steps to escape bad locals, interleaved with Lloyd updates. Theoretical results show recovery of ground-truth centers in iterations under a stochastic ball model with sufficient separation, contrasting Lloyd’s exponential dependence on random initializations. Empirically, FFkm demonstrates robust performance across synthetic benchmarks and real-world tasks such as color quantization, often outperforming standard Lloyd’s algorithm and several modern alternatives, and even handling mis-specification of the number of clusters. The approach unifies and clarifies the efficacy of various heuristic improvements, offering a flexible toolkit for improving -means in diverse data regimes.

Abstract

\kmeans clustering is a fundamental problem in many scientific and engineering domains. The optimization problem associated with \kmeans clustering is nonconvex, for which standard algorithms are only guaranteed to find a local optimum. Leveraging the hidden structure of local solutions, we propose a general algorithmic framework for escaping undesirable local solutions and recovering the global solution or the ground truth clustering. This framework consists of iteratively alternating between two steps: (i) detect mis-specified clusters in a local solution, and (ii) improve the local solution by non-local operations. We discuss specific implementation of these steps, and elucidate how the proposed framework unifies many existing variants of \kmeans algorithms through a geometric perspective. We also present two natural variants of the proposed framework, where the initial number of clusters may be over- or under-specified. We provide theoretical justifications and extensive experiments to demonstrate the efficacy of the proposed approach.
Paper Structure (50 sections, 8 theorems, 48 equations, 11 figures, 14 tables, 3 algorithms)

This paper contains 50 sections, 8 theorems, 48 equations, 11 figures, 14 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Structure of Local Solutions
  3. Properties of one-fit-many association
  4. Properties of many-fit-one association
  5. Properties of almost empty association
  6. From Structure to Algorithms
  7. Fission-Fusion $k$-means
  8. Theoretical Guarantees
  9. Detection Subroutines
  10. Detect one-fit-many: Standard Deviation (SD)
  11. Detect one-fit-many: $\epsilon$-Radius (RD)
  12. Detect one-fit-many: Total Deviation (TD)
  13. Detect many-fit-one: Pairwise Distance (PD)
  14. Detect many-fit-one: Objective Increment (OI)
  15. Other Detection Procedures
  16. ...and 35 more sections

Key Result

Theorem 3.1

Let $\left\{\boldsymbol\beta^\star_{i}\right\}_{i\in[k^{*}]}$ be $k^{*}$ unknown centers in $\mathbb{R}^{d}$, with maximum and minimum separations Suppose the data $\boldsymbol x_{1},\ldots,\boldsymbol x_{n}\in\mathbb{R}^{d}$ is generated independently from the stochastic ball model eq: SBM. Assume that $\frac{\Delta_{\min}}{r}\ge 30$. With probability at least $1-2k^{*}\exp\left(-\frac{n}{2k^{*2

Figures (11)

  • Figure 1: The one-fit-many and many-fit-one association relationships in a local minimizer of the $k$-means problem.
  • Figure 2: Illustration of the Fission-Fusion $k$-means algorithm.
  • Figure 3: Illustration of the Swap operation.
  • Figure 4: Comparison of the objective-based algorithm I-$k$-means$-+^\star$ and the geometry-based algorithm FFkm (SD+PD). Here $\ell$ is the iteration number, and $\beta^{(\ell)}$ is the cluster centers in iteration $\ell$. (a) shows the solution by Lloyd $k$-means, which also serves as the initial solution for both I-$k$-means$-+^\star$ and FFkm (SD+PD). (b--e) show each iteration of I-$k$-means$-+^\star$, while (f--i) show each iteration of FFkm (SD+PD).
  • Figure 5: Results of unsupervised color quantization using different numbers of clusters ($k$ values for colors). The images are organized in rows from top to bottom: Palace ($k=8$), Boat ($k=4$), Traffic ($k=8$). Each column shows (a) The original image ($k$ is provided in Table \ref{['tab:9']}). (b) The result of Lloyd $k$-means. (c) The result of FFkm (SD+PD). (d) The result of FFkm (TD+OI). (e) The result of I-$k$-means$-+^\star$.
  • ...and 6 more figures

Theorems & Definitions (9)

  • Theorem 3.1: Main Theorem
  • Theorem 3.2: Lloyd's Converges to Bad Locals
  • Lemma A.1: Upper Bound
  • Lemma A.2: Improvement
  • Lemma A.3: Monotonicity
  • Lemma A.4: Almost Equal Size
  • Definition B.1: Diffuse Stochastic Ball Model
  • Lemma B.2
  • Proposition B.3