Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Alternatives to the Laplacian for Scalable Spectral Clustering with Group Fairness Constraints

Iván Ojeda-Ruiz, Young Ju-Lee, Malcolm Dickens, Leonardo Cambisaca

TL;DR

The paper tackles scalable spectral clustering under group fairness by introducing Fair-SMW, a Lagrangian–Sherman–Morrison–Woodbury reformulation that enforces the linear fairness constraint $F^T H = 0$ through a projection-based approach and three matrix variants $G_{ ext{sym}}$, $G_{ ext{rw}}$, and $G_{ ext{aff}}$. It derives a reduced eigenproblem via a null-space basis and then leverages the SMW identity to obtain a computationally efficient surrogate that yields larger eigen-gaps, reducing eigensolver restarts while preserving proportional representation in clusters. Empirical results on SBM and real networks (e.g., LastFM, FacebookNet, Deezer, German) show substantial runtime savings with competitive or improved balance compared to state-of-the-art methods like S-Fair-SC, particularly in sparse graphs where solver iterations dominate. The work provides a practical, robust framework for fair, scalable clustering and motivates further exploration of spectral properties to guide algorithm design.

Abstract

Recent research has focused on mitigating algorithmic bias in clustering by incorporating fairness constraints into algorithmic design. Notions such as disparate impact, community cohesion, and cost per population have been implemented to enforce equitable outcomes. Among these, group fairness (balance) ensures that each protected group is proportionally represented within every cluster. However, incorporating balance as a metric of fairness into spectral clustering algorithms has led to computational times that can be improved. This study aims to enhance the efficiency of spectral clustering algorithms by reformulating the constrained optimization problem using a new formulation derived from the Lagrangian method and the Sherman-Morrison-Woodbury (SMW) identity, resulting in the Fair-SMW algorithm. Fair-SMW employs three alternatives to the Laplacian matrix with different spectral gaps to generate multiple variations of Fair-SMW, achieving clustering solutions with comparable balance to existing algorithms while offering improved runtime performance. We present the results of Fair-SMW, evaluated using the Stochastic Block Model (SBM) to measure both runtime efficiency and balance across real-world network datasets, including LastFM, FacebookNet, Deezer, and German. We achieve an improvement in computation time that is twice as fast as the state-of-the-art, and also flexible enough to achieve twice as much balance.

Alternatives to the Laplacian for Scalable Spectral Clustering with Group Fairness Constraints

TL;DR

The paper tackles scalable spectral clustering under group fairness by introducing Fair-SMW, a Lagrangian–Sherman–Morrison–Woodbury reformulation that enforces the linear fairness constraint through a projection-based approach and three matrix variants , , and . It derives a reduced eigenproblem via a null-space basis and then leverages the SMW identity to obtain a computationally efficient surrogate that yields larger eigen-gaps, reducing eigensolver restarts while preserving proportional representation in clusters. Empirical results on SBM and real networks (e.g., LastFM, FacebookNet, Deezer, German) show substantial runtime savings with competitive or improved balance compared to state-of-the-art methods like S-Fair-SC, particularly in sparse graphs where solver iterations dominate. The work provides a practical, robust framework for fair, scalable clustering and motivates further exploration of spectral properties to guide algorithm design.

Abstract

Recent research has focused on mitigating algorithmic bias in clustering by incorporating fairness constraints into algorithmic design. Notions such as disparate impact, community cohesion, and cost per population have been implemented to enforce equitable outcomes. Among these, group fairness (balance) ensures that each protected group is proportionally represented within every cluster. However, incorporating balance as a metric of fairness into spectral clustering algorithms has led to computational times that can be improved. This study aims to enhance the efficiency of spectral clustering algorithms by reformulating the constrained optimization problem using a new formulation derived from the Lagrangian method and the Sherman-Morrison-Woodbury (SMW) identity, resulting in the Fair-SMW algorithm. Fair-SMW employs three alternatives to the Laplacian matrix with different spectral gaps to generate multiple variations of Fair-SMW, achieving clustering solutions with comparable balance to existing algorithms while offering improved runtime performance. We present the results of Fair-SMW, evaluated using the Stochastic Block Model (SBM) to measure both runtime efficiency and balance across real-world network datasets, including LastFM, FacebookNet, Deezer, and German. We achieve an improvement in computation time that is twice as fast as the state-of-the-art, and also flexible enough to achieve twice as much balance.
Paper Structure (19 sections, 11 theorems, 76 equations, 3 figures, 4 algorithms)

This paper contains 19 sections, 11 theorems, 76 equations, 3 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spectral Clustering
  4. Fairness Constraint
  5. New Formulation for a Fairness Constraint
  6. Previously Proposed Algorithms
  7. Unnormalized Fair-SC
  8. Normalized Fair-SC
  9. S-Fair-SC
  10. Fair-SMW
  11. Choice of G
  12. Theorems for the eigenvalues of the symmetric G
  13. Theorems for the eigenvalues of the non-symmetric G
  14. Time Complexity
  15. Experiments
  16. ...and 4 more sections

Key Result

Lemma 2.1

For $s \in [h]$, we introduce an indicator vector $f^{(s)} \in \mathbb{R}^{n}$ such that $f^{(s)}_i = 1$ if $i \in \bm{V}_s$ and $f^{(s)}_j = 0$ if $j \notin \bm{V}_s$. We then set the clustering of $\bm{V}$ as follows: Then for every $\ell \in [k]$, we have that

Figures (3)

  • Figure 1: Example of $GF=0$
  • Figure 1: (Left) Balance, (Center) Algorithm Run Time, (Right) Eigensolver Run Time
  • Figure 2: Comparison of eigenvalue gaps, runtimes, and scalability for SMW and S-Fair-SC methods across different datasets and graph sizes.

Theorems & Definitions (22)

  • Lemma 2.1: Fairness constraints as linear constraint on $H$
  • Proof 1
  • Lemma 2.2: Linear Dependency of $\hat{f}_i^{(s)}$
  • Proof 2
  • Theorem 4.1
  • Proof 3
  • Theorem 4.2
  • Proof 4
  • Theorem 4.3
  • Proof 5
  • ...and 12 more