This paper tackles fair clustering when people belong to many intersecting sensitive groups, like race and gender together. Standard fair-clustering methods struggle when multiple sensitive attributes create 2^q possible subgroups, because some subgroups become very small and the optimization gets slow or unstable. The authors propose COVA-FC (COVAriance-based Fair Clustering), a new algorithm that turns a subgroup fairness goal into a covariance-based objective that is easier to optimize with gradients.

The authors first define a subgroup-fairness gap that measures how much cluster membership probabilities differ between each subgroup and the overall population. They then derive an exact surrogate for this gap based on empirical covariance. Intuitively, this covariance measures how much knowing a person’s subgroup helps predict their cluster label. For a fixed hard clustering, the paper proves the covariance surrogate matches the original subgroup gap exactly.

To make training practical, the paper relaxes two discrete parts of the problem. It replaces hard cluster labels with soft assignments — probabilities that each point belongs to each cluster — so the loss becomes differentiable. It also replaces the discrete “worst” subgroup search with a smooth convex combination over observed subgroups. Those relaxations yield a continuous objective that can be optimized with gradient methods and that allows parallel per-instance updates. The resulting algorithm is called COVA-FC.

Why this matters: the proposed formulation handles many subgroups without building huge linear programs or suffering numerical instability. The authors point out that prior penalty methods can blow up when tiny subgroups appear and that hard-assignment methods struggle to finely match subgroup proportions. In experiments on benchmark datasets (as reported by the authors), COVA-FC achieves competitive trade-offs between clustering cost and fairness while improving computational efficiency compared to existing baselines. The framework can also be extended to jointly control marginal fairness — fairness for each sensitive attribute on its own — alongside subgroup fairness.