[Paper Review] Guarantees for Spectral Clustering with Fairness Constraints

Matthäus Kleindeßner, Samira Samadi+2 more|arXiv (Cornell University)|Jan 24, 2019

Human Mobility and Location-Based Analysis43 citations

TL;DR

The paper integrates Chierichetti et al.’s fairness notion into spectral clustering (unnormalized and normalized), provides algorithms, and proves recovery guarantees on a variant of the stochastic block model, plus empirical evidence on real data.

ABSTRACT

Given the widespread popularity of spectral clustering (SC) for partitioning graph data, we study a version of constrained SC in which we try to incorporate the fairness notion proposed by Chierichetti et al. (2017). According to this notion, a clustering is fair if every demographic group is approximately proportionally represented in each cluster. To this end, we develop variants of both normalized and unnormalized constrained SC and show that they help find fairer clusterings on both synthetic and real data. We also provide a rigorous theoretical analysis of our algorithms on a natural variant of the stochastic block model, where $h$ groups have strong inter-group connectivity, but also exhibit a "natural" clustering structure which is fair. We prove that our algorithms can recover this fair clustering with high probability.

Motivation & Objective

• Incorporate fairness constraints into spectral clustering so that each cluster reflects the demographic group proportions of the original data.
• Provide algorithms for both unnormalized and normalized spectral clustering with fairness constraints.
• Offer theoretical guarantees showing recovery of a fair clustering on a stochastic block model variant.
• Evaluate the proposed fair SC methods against standard SC on synthetic and real data.

Proposed method

• Extend spectral clustering by adding linear fairness constraints on the clustering encoding matrix H.
• Formulate fairness as F^T H = 0 and relax to H^T H = I_k, solving via eigen decomposition of a projected Laplacian.
• Provide Algorithm 2 for unnormalized SC with fairness constraints and Algorithm 3 for normalized SC with fairness constraints (and a discussion of their implementation via a nullspace projection).
• Use a variant of the stochastic block model to model fair ground-truth clusterings and analyze recovery guarantees.
• Apply k-means to the rows of the obtained embedding to recover the clustering.

Experimental results

Research questions

• RQ1 Can fairness constraints based on demographic group representation be incorporated into spectral clustering without sacrificing too much clustering quality?
• RQ2 Do fair variants of spectral clustering recover a fair ground-truth clustering in a stochastic block model that embodies both strong inter-group connectivity and a fair structure?
• RQ3 What are the computational and theoretical trade-offs between unnormalized and normalized fair spectral clustering?
• RQ4 How do the fair SC methods perform on real-world networks compared to standard SC?

Key findings

• Fairness constraints can be incorporated into SC via linear constraints on the embedding matrix H.
• The fair formulations lead to a relaxation that reduces to eigenproblems on a projected Laplacian, followed by k-means on rows of the embedding.
• The authors prove recovery guarantees for the fair clustering under a variant of the stochastic block model where ground-truth clustering is fair.
• Experiments show the fair SC methods achieve fairer clusterings than standard SC with objective values often close to standard SC.
• Algorithm 3 (normalized SC with fairness) generally outperforms Algorithm 2 (unnormalized SC with fairness) in terms of requiring smaller n for zero error and in empirical robustness.
• On real networks, fairness constraints tend to reduce balance gaps and maintain competitive RatioCut/NCut values.