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

• URL: http://arxiv.org/abs/2608.10470v1
• Date: Tue, 11 Aug 2026 04:27:24 GMT
• Title: A Joint-Distribution Route to Fair Representations with Continuous Sensitive Attributes
• Abstract summary: We evaluate independence through a single discrepancy $dleft(P_Z, S, P_Z otimes P_Sright)$ between the joint law and the product of its marginals.<n>We take the Hilbert-Schmidt independence criterion as an instance of the joint discrepancy $d$ to investigate the statistical efficiency of replacing the conditional formulation.
• Score: 12.805268849262243
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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 $d\left(P_{Z, S}, P_Z \otimes P_S\right)$ 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 $O\left(n^2\right)$ statistic that converges at the $O\left(n^{-1 / 2}\right)$ rate, in contrast to the nonparametric $O\left(n^{-2 / 5}\right)$ 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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