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

arXiv: https://arxiv.org/abs/2608.10470 · HTML: https://mlsanity.com/paper/2608.10470/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 39 references that are in the map.
- First posted on arXiv 11 Aug 2026.

## Builds on (most cited first, 14 of 14)

- [Counterfactual Fairness](https://mlsanity.com/paper/1703.06856/) (2017)
- [Retiring Adult: New Datasets for Fair Machine Learning](https://mlsanity.com/paper/2108.04884/) (2021)
- [Learning Adversarially Fair and Transferable Representations](https://mlsanity.com/paper/1802.06309/) (2018)
- [The Variational Fair Autoencoder](https://mlsanity.com/paper/1511.00830/) (2015)
- [Flexibly Fair Representation Learning by Disentanglement](https://mlsanity.com/paper/1906.02589/) (2019)
- [The HSIC Bottleneck: Deep Learning without Back-Propagation](https://mlsanity.com/paper/1908.01580/) (2019)
- [Achieving Equalized Odds by Resampling Sensitive Attributes](https://mlsanity.com/paper/2006.04292/) (2020)
- [Characteristic and Universal Tensor Product Kernels](https://mlsanity.com/paper/1708.08157/) (2017)
- [Fair Kernel Learning](https://mlsanity.com/paper/1710.05578/) (2017)
- [Fairness without the sensitive attribute via Causal Variational Autoencoder](https://mlsanity.com/paper/2109.04999/) (2021)
- [Kernel Dependence Regularizers and Gaussian Processes with Applications to Algorithmic Fairness](https://mlsanity.com/paper/1911.04322/) (2019)
- [MMD-B-Fair: Learning Fair Representations with Statistical Testing](https://mlsanity.com/paper/2211.07907/) (2022)
- [Fair Representation Learning for Continuous Sensitive Attributes using Expectation of Integral Probability Metrics](https://mlsanity.com/paper/2505.06435/) (2025)
- [A Uniform Concentration Inequality for Kernel-Based Two-Sample Statistics](https://mlsanity.com/paper/2405.14051/) (2024)

## Abstract (arXiv)

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.