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

This paper addresses fair representation learning when the sensitive attribute S is continuous (e.g., age, income, risk score). The goal is to learn a representation Z = h(X) that is statistically independent of S, i.e., Z ⊥ S, so that downstream predictors built on Z inherit demographic parity. The paper states: "Fair representation learning seeks a representation Z that is statistically independent of a sensitive attribute S ∈ S, i.e., Z ⊥ S, so that any downstream predictor built on Z inherits demographic parity with respect to S."

The representation-level target is the equality P ZS=s = P Z for P S-almost every s, which is equivalent to Z ⊥ S.

Existing criteria for continuous S enforce independence by averaging a per-value discrepancy between the conditional law P ZS=s and the marginal P Z over the law of S. These share the common form:

I d(Z; S):= E S[d(P ZS, P Z)]

Specific instances include:

• Generalized demographic parity (GDP) (Jiang et al., 2022): uses a first-moment difference, estimated via Nadaraya–Watson kernel smoothing on S

• Expectation of IPM (EIPM) (Kong et al., 2025): uses an integral probability metric, realized by the kernel-weighted estimator FREM

• Mutual information objectives: use the Kullback–Leibler divergence

The paper notes: "Because a finite sample of size n has no repeated observations at an exact value S = s, each criterion replaces exact conditioning by a localized or density-based surrogate... These smoothing-based estimators carry a localization bandwidth and converge at the one-dimensional nonparametric rate O(n−2/5)."

The paper proposes evaluating independence through a single joint discrepancy between the joint law P Z,S and the product of its marginals P Z ⊗ P S, measured by an integral probability metric:

IPM F(P Z,S, P Z ⊗ P S):= sup f∈F E P Z,S[f] − E P Z⊗P S[f]

This vanishes if and only if Z ⊥ S whenever F is characteristic. The paper states: The averaging over sensitive values is then intrinsic to the joint law rather than an imposed weighting, and no per-value conditional object enters the definition.

The joint-vs-product IPM disintegrates exactly into the PS-average of a conditional contrast:

IPM F(P Z,S, P Z ⊗ P S) = sup f∈F ∫ S ∫ Z f(z,s) d(P ZS=s − P Z)(z) dP S(ds)

On decomposable witness classes (Definition 3.3), this average equals the conditional-integral functional I d that GDP and EIPM instantiate. The paper states: "Corollary 3.4 therefore recovers the conditional-integral IPM, which GDP and EIPM instantiate, exactly when the witness class decomposes across s; mutual information is the Kullback–Leibler instance of the same S-averaged form."

The paper takes the Hilbert–Schmidt independence criterion (HSIC) as the instance obtained by choosing F to be the unit ball of a product reproducing-kernel Hilbert space (RKHS). The empirical HSIC is a closed-form statistic computable in O(n2) time from Gram matrices.

Proposition 4.1 establishes a decomposition:

HSIC(Z, S) = E S,S′[⟨μ ZS − μ Z, μ ZS′ − μ Z⟩ F Z · k S(S, S′)]

and the bound:

HSIC(Z, S) ≤ κ S · E S[MMD2 k Z(P ZS, P Z)]

HSIC is equivalent to the conditional MMD integral up to an explicit spectral tail:

(1/κ S) · HSIC(Z, S) ≤ E S[MMD2 k Z(P ZS, P Z)] ≤ (1/λ m) · HSIC(Z, S) + ρ2 m

where λ m is the m-th eigenvalue of the kernel integral operator on S and ρ2 m is the spectral tail.

Corollary 4.6 (population level): For an RKHS prediction head f, small HSIC controls Δ GDP(f) up to the spectral tail:

Δ GDP(f)2 ≤ (∥f∥2 F Z/λ m) · HSIC(Z, S) + r2 m,f

Theorem 4.7 (finite sample): The empirical HSIC statistic controls the empirical demographic-parity gap of any RKHS head:

Δ̂ GDP(f)2 ≤ (1/n)Σ i (m̂ f)2 i ≤ (∥f∥2 F Z/λ̂ m) · HSIĈ n(Z, S)

Empirical HSIC concentrates on its population value uniformly over the encoder class H at rate n−1/2 under controlled encoder complexity:

sup h∈H HSIĈ n(h) − HSIC(h) ≤ B n(H, δ)

with B n(H, δ) defined in equation (5.2). The paper states: The rate does not extend to arbitrary unregularized or growing-width networks.

FRHSIC solves the empirical regularized problem:

min h∈H, f∈F L̂ n(f ◦ h) + λ · HSIĈ n(h(X), S)

where L̂ n is the empirical prediction loss and λ > 0 controls the fairness–accuracy tradeoff. The paper notes: We use the biased estimator because it is nonnegative and hence a valid penalty; the unbiased U-statistic estimator can be negative in finite samples.

The HSIC estimator converges at the O(n−1/2) rate as a non-degenerate V-statistic, in contrast to the nonparametric O(n−2/5) rate of conditional-route estimators. The paper states: Being a non-degenerate V-statistic, it converges to its population value at the O(n−1/2) rate, in contrast to the nonparametric O(n−2/5) of the smoothing-based EIPM estimator (FREM).

The paper reports: "Empirical HSIC follows the predicted O(n−1/2) behavior and the smoothing-based EIPM estimator follows the slower O(n−2/5) trend: the fitted log–log slopes are −0.46 for HSIC and −0.44 for EIPM, close to the predicted −1/2 and −2/5."

On five real datasets (Adult, ACS Income, MEPS, Crime, COMPAS), FRHSIC achieves fairness–accuracy tradeoffs comparable to the strongest continuous-sensitive baselines. The paper states: "No method dominates across datasets. By nAUP, FRHSIC is best on COMPAS, within.004 of the best on Crime, and mid-pack on Adult, ACS Income, and MEPS, staying within the spread of FREM, Reg-GDP, and the adversarial baselines."

Fairness enforced by FRHSIC stays stable across fresh downstream heads (linear, MLP, random forest, SVM). On Adult, FRHSIC reduces cross-head GDP variance from 0.348 (Unfair) to 0.244, a 30% reduction.

At n = 20,000 and batch size 256, FRHSIC completes an epoch in 2.82s versus 100.7s for FREM, a roughly 36× speedup. The paper states: FRHSIC trains substantially faster than FREM because it replaces FREM's per-sample weighted conditional-MMD computation with a single O(m2) HSIC trace per minibatch.

The framework extends to:

• Equal Opportunity: by computing the joint discrepancy on the Y = 1 subset

• Multiple continuous sensitive attributes: through a product kernel on the sensitive space

• Minority-protection target: by replacing the reference distribution P Z ⊗ P S with P Z ⊗ Unif(S)

The paper acknowledges: "Two questions remain. First, the empirical bound in Theorem 4.7 scales as ∥f∥2 F Z/λ̂ m; tighter spectral arguments exploiting the full singular-value distribution of L̂ would yield sharper guarantees on the integral form. Second, the framework treats the joint discrepancy with an RKHS witness class; non-RKHS instances (Wasserstein, total variation) inherit the structural identity but lose the O(n−1/2) rate and closed-form estimator."

Additionally, the paper notes that the HSIC-fairness condition is the strongest observational fairness consequence of counterfactual fairness, not as a substitute for it, since counterfactual fairness is a strictly pointwise statement about individual-level interventions.

Based on the paper, here are the specific improvements I can make to AI systems:

• Improvement: Replace kernel-smoothing-based fairness constraints (which require bandwidth tuning and converge at O(n-2/5)) with a direct joint-distribution HSIC penalty that converges at O(n-1/2).

• What the improved system can do: Enforce demographic parity with respect to continuous attributes like age, income, or risk scores in a single closed-form computation per batch, without needing to discretize the sensitive variable or choose a localization bandwidth. This makes training faster (36× speedup over FREM) and statistically more efficient.

• Improvement: Use Theorem 4.7's bound—empirical HSIC controls the empirical demographic-parity gap of any RKHS prediction head—as a training-time certificate.

• What the improved system can do: During training, monitor the quantity (∥f∥2 F Z/λ̂ m) · HSIĈ n(Z, S) to know exactly how much demographic parity violation any downstream head will exhibit, without needing to re-evaluate fairness after deployment. This enables early stopping or adaptive regularization when the bound exceeds a threshold.

• Improvement: Enforce fairness at the representation level using HSIC, which is head-agnostic, rather than at the prediction level.

• What the improved system can do: Train a fair encoder once, then swap in any downstream head (linear, MLP, random forest, SVM) and retain fairness guarantees. The system reduces cross-head GDP variance by 30% on Adult, meaning the same representation serves multiple deployment scenarios without retraining for fairness.

• Improvement: Use product kernels on the sensitive space to handle multiple continuous sensitive attributes simultaneously, and compute the joint discrepancy on the Y=1 subset for equal opportunity.

• What the improved system can do: Enforce intersectional fairness across combinations like age × income × credit score in one unified objective, rather than treating each attribute separately. It can also enforce equal opportunity (fairness among positive outcomes) without additional algorithmic complexity.

• Improvement: Replace the reference distribution P Z ⊗ P S with P Z ⊗ Unif(S) to protect minority groups.

• What the improved system can do: When sensitive attributes are skewed (e.g., rare age groups or low-income brackets), the system can weight all sensitive values equally in the fairness penalty, preventing the model from ignoring small subpopulations. This is critical for applications like credit scoring where minority income groups are underrepresented.

• Improvement: Replace per-sample weighted conditional-MMD computations (O(n2) with expensive kernel evaluations per sample) with a single O(m2) HSIC trace per minibatch.

• What the improved system can do: Train fair models on datasets with 20,000+ samples in near-real-time (2.82s per epoch vs 100.7s for FREM), making fairness constraints practical for large-scale production systems like recommendation engines or fraud detection.

• Improvement: Use the regularized objective min L̂ + λ·HSIĈ with a theoretically grounded λ, since HSIC is nonnegative and a valid penalty.

• What the improved system can do: Provide a single hyperparameter λ that smoothly interpolates between accuracy and fairness, with the guarantee that the demographic parity gap shrinks as λ grows (via Corollary 4.6). This removes the need for ad-hoc multi-objective tuning or adversarial training instability.

• Improvement: Use HSIC-fairness as the strongest observational approximation to counterfactual fairness.

• What the improved system can do: In settings where counterfactual interventions are impossible to measure (e.g., medical treatment outcomes), the system can enforce the closest achievable observational fairness condition, giving practitioners a principled fallback that is computationally tractable and statistically sound.

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 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 (P Z, S, P Z P S) 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 (n 2) 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.