A Joint-Distribution Route to Fair Representations with Continuous Sensitive Attributes

arXiv:2608.10470v1 Announce Type: new Abstract: Fair representation learning with a continuous sensitive attribute S requires a representation Z that is statistically independent of S. Existing criteria, including generalized demographic parity, the expectation of integral probability metrics (EIPM), and mutual information, enforce this independence by averaging a per-value discrepancy between the conditional law P_{Z mid S=s} and the marginal P_Z over the law of S. This approach requires a nonparametric surrogate for the conditional law at each sensitive value. We propose evaluating independence through a single joint discrepancy dleft(P_{Z, S}, P_Z otimes P_Sright) between the joint law and the product of its marginals. We establish a disintegration identity; on decomposable witness classes it equals the conditional-integral functional that EIPM and generalized demographic parity instantiate. By reaching the same target without the conditional law, this discrepancy can be estimated directly from samples via a dependence statistic rather than conditional smoothing. We take the Hilbert-Schmidt independence criterion (HSIC) as an instance of the joint discrepancy d to investigate the statistical efficiency of replacing the conditional formulation. The HSIC estimator is a closed-form Oleft(n^2right) statistic that converges at the Oleft(n^{-1 / 2}right) rate, in contrast to the nonparametric Oleft(n^{-2 / 5}right) rate of the conditional-route estimators. We prove this instance is equivalent to the conditional maximum mean discrepancy (MMD) integral up to an explicit spectral tail. The corresponding algorithmic implementation, i.e., FRHSIC, attains fairness-accuracy tradeoffs comparable to conditional-route basel es while reducing per-epoch training time.

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Source: arXiv cs.LG | 2026-08-12