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Shift of Pairwise Similarities for Data Clustering

Morteza Haghir Chehreghani

TL;DR

This work introduces Shifted Min Cut, an approach that regularizes the classic Min Cut by adaptively shifting pairwise similarities to avoid degenerate, unbalanced partitions. It establishes a principled link to Correlation Clustering under shifted similarities and develops an efficient local-search optimization with fast convergence. The method supports adaptive, edge-level shifts and yields improved clustering performance across numerous datasets, outperforming several baselines on multiple evaluation criteria. The approach also analyzes how shifting affects other clustering paradigms, showing invariant or near-invariant behavior for many common methods, which underscores the method's applicability and robustness in graph-based clustering tasks.

Abstract

Several clustering methods (e.g., Normalized Cut and Ratio Cut) divide the Min Cut cost function by a cluster dependent factor (e.g., the size or the degree of the clusters), in order to yield a more balanced partitioning. We, instead, investigate adding such regularizations to the original cost function. We first consider the case where the regularization term is the sum of the squared size of the clusters, and then generalize it to adaptive regularization of the pairwise similarities. This leads to shifting (adaptively) the pairwise similarities which might make some of them negative. We then study the connection of this method to Correlation Clustering and then propose an efficient local search optimization algorithm with fast theoretical convergence rate to solve the new clustering problem. In the following, we investigate the shift of pairwise similarities on some common clustering methods, and finally, we demonstrate the superior performance of the method by extensive experiments on different datasets.

Shift of Pairwise Similarities for Data Clustering

TL;DR

This work introduces Shifted Min Cut, an approach that regularizes the classic Min Cut by adaptively shifting pairwise similarities to avoid degenerate, unbalanced partitions. It establishes a principled link to Correlation Clustering under shifted similarities and develops an efficient local-search optimization with fast convergence. The method supports adaptive, edge-level shifts and yields improved clustering performance across numerous datasets, outperforming several baselines on multiple evaluation criteria. The approach also analyzes how shifting affects other clustering paradigms, showing invariant or near-invariant behavior for many common methods, which underscores the method's applicability and robustness in graph-based clustering tasks.

Abstract

Several clustering methods (e.g., Normalized Cut and Ratio Cut) divide the Min Cut cost function by a cluster dependent factor (e.g., the size or the degree of the clusters), in order to yield a more balanced partitioning. We, instead, investigate adding such regularizations to the original cost function. We first consider the case where the regularization term is the sum of the squared size of the clusters, and then generalize it to adaptive regularization of the pairwise similarities. This leads to shifting (adaptively) the pairwise similarities which might make some of them negative. We then study the connection of this method to Correlation Clustering and then propose an efficient local search optimization algorithm with fast theoretical convergence rate to solve the new clustering problem. In the following, we investigate the shift of pairwise similarities on some common clustering methods, and finally, we demonstrate the superior performance of the method by extensive experiments on different datasets.

Paper Structure

This paper contains 18 sections, 2 theorems, 32 equations, 2 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Notations and Definitions
  3. Shift of Pairwise Similarities for Clustering
  4. Adaptive shift of pairwise similarities.
  5. Relation to Correlation Clustering
  6. Optimization of the Shifted Min Cut Cost Function
  7. On the convergence rate of the local search optimization.
  8. Shift Analysis of Other Clustering Methods
  9. Shift of pairwise similarities for flat clustering.
  10. Shifted Dominant Set Clustering.
  11. Shift of pairwise similarities for hierarchical clustering.
  12. Experiments
  13. Data.
  14. Methods.
  15. Evaluation criteria.
  16. ...and 3 more sections

Key Result

Proposition 1

Single linkage, complete linkage and average linkage methods are invariant with respect to the shift of the pairwise dissimilarities $\mathbf D$ by constant $\alpha$.

Figures (2)

  • Figure 1: The Min Cut cost function has a bias to split small (singleton) sets of objects. Any cut that splits one of the objects on the right half will have smaller cost than the cut that splits the objects into the left and right halves. The figure has been adapted from JS:JM:PAMI:2000.
  • Figure 2: The impact of the shift parameter $\alpha$ on the results of the Shifted Min Cut cost function. A very large $\alpha$ might yield splitting large clusters, instead of separating true small clusters.

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • Proposition 2