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

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 PZ mid S=s and the marginal PZ 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(PZ, S, PZ otimes PSright) 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(n2right) statistic that converges at the Oleft(n-1 / 2right) rate, in contrast to the nonparametric Oleft(n-2 / 5right) 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.