H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205 USA
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∣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
d(PZ,S,PZ⊗PS) 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(n2) statistic that converges at the
O(n−1/2) rate, in contrast to the nonparametric
O(n−2/5) 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.