Allocation-Dependent Distortion for Fair Clustering Ensembles

Clustering ensembles combine multiple base partitions into a final consensus. When protected groups are present, the consensus must also account for how each group is distributed across the output clusters relative to the marginal cluster masses. We study allocation-dependent distortion, in which the conditional proportion of a group assigned to a cluster directly weights the distortion of that same group–cluster cell. The resulting Fair $k$-means Consensus Clustering (FKCC) objective incorporates group representation into consensus aggregation without enforcing profile alignment as a hard constraint. We show that the complete objective can move profile discrepancy in either direction. Fixed-distortion analysis characterizes the allocation response and identifies when quadratic group profiles coincide or remain close; it also explains the special role of the quadratic coefficient within the positive-power family. Center elimination yields an exact finite objective with monotone one-object descent. An algebraic finite–empirical identity holds for feasible induced assignments, while uniform estimation is established separately under a fixed-representation sampling model. Experiments on nine datasets show strong clustering utility and consistent matched reductions in profile discrepancy on the tested benchmarks. Anonymous code is available at https://anonymous.4open.science/r/method_code-F99B.