FairSplit: Decomposing the Embedding Space for Fair Classification

AI summary

Overview

Research area: Fair machine learning — specifically in-processing methods for group fairness in classification, combining representation learning with kernel-based dependence measures.

Technical level: Advanced. The paper pairs an empirical benchmark study with a theorem that bounds the demographic parity gap using the Hilbert–Schmidt Independence Criterion (HSIC) and the learned dimensionality of a subspace; readers will benefit from familiarity with embeddings, HSIC, and group fairness metrics.

Scope: A single in-processing method, FairSplit, that splits a model's embedding into a prediction-relevant "Fair Space" and a discarded "Sensitive Space," with a theoretical guarantee and experiments spanning tabular data, face images, and large pretrained vision backbones.

What This Paper Is About

Fair classifiers must keep accuracy high while preventing predictions from depending on a protected attribute such as gender or race. Most existing in-processing methods regularize the model's predictions, and many are built only for binary labels and binary sensitive attributes. FairSplit instead acts on the representation: it learns a binary mask that routes some embedding dimensions into a Fair Space used for prediction and the rest into a Sensitive Space that absorbs sensitive information, so the classifier never sees the sensitive part. The authors prove that the demographic parity gap of the resulting classifier is upper-bounded by both the size of the Fair Space and the HSIC between the Fair Space and the sensitive attribute — two quantities the method explicitly minimizes.

Key Contributions

1. A representation-level fairness method. FairSplit decomposes the embedding into a Fair Space (used by the classifier) and a complementary Sensitive Space, and learns the dimensionality of the Fair Space jointly with the classifier rather than fixing it in advance.

2. A theoretical bound that covers multiclass and multi-group settings. Theorem 3.2 shows the continuous demographic parity gap is bounded by c_G · B · sqrt(K·d_f) · sqrt(HSIC(z_f, s)), with c_G ≤ sqrt(2)/p_min, for any number of classes K and groups G ≥ 2.

3. A closed-form mask update. Proposition 3.1 gives b*_j = V_j^(s) / (V_j^(y) + V_j^(s)), where V_j^(y) and V_j^(s) are per-dimension within-class and within-group variances, allowing the mask to be optimized alternately with the network instead of by gradient descent.

4. Empirical validation across binary, multiclass, intersectional, and foundation-model settings. Experiments on Adult, CelebA-A, UTKFace-AR, and COMPAS, plus a fine-tuned ViT-B/16 and three frozen backbones, with a capability comparison (Table 1) showing that most baselines are restricted to binary targets or binary sensitive attributes while FairSplit is not.

Main Findings

• Strong fairness gains on binary benchmarks. On CelebA-A, FairSplit reduces ΔDP from ERM's 52.8 to 18.3, ABCC from 35.9 to 9.4, MCDP from 53.4 to 19.6, ΔEOpp from 36.8 to 2.6, and ΔEOdd from 75.0 to 5.2, while AP moves from 84.5 to 80.4. On Adult, ΔDP drops from 19.4 to 1.0 and ABCC from 18.7 to 2.4, with AP 72.9 versus ERM's 76.7.

• Best ABCC across the board. FairSplit achieves the lowest ABCC on every dataset reported in Tables 2 and 3, and the lowest MCDP on CelebA-A. On Adult, MCDP is 19.8 against ERM's 36.6 — the paper explains that predicted scores there cluster in a narrow range below 0.5, so small per-seed differences inflate MCDP while ABCC stays small, which favors methods that regularize predictions directly.

• Competitive in multiclass and multi-group regimes. On UTKFace-AR (3 age classes, 3 racial groups), FairSplit reaches ΔDP 12.6, ABCC 9.1, MCDP 18.7, ΔEOpp 14.9, ΔEOdd 22.1, and F1 70.4, versus ERM's 27.9 / 27.1 / 36.2 / 29.1 / 35.9 / 73.8. The HSIC(Y,S) baseline attains a slightly lower ΔDP (11.9) but is worse on ABCC (12.0), MCDP (20.4), ΔEOpp (18.1), and ΔEOdd (24.1).

• Very low ABCC under intersectional imbalance. On COMPAS with a gender × age attribute of four groups and p_min = 0.05, FairSplit records ABCC 1.2 against 6.7 for the next-best method, with ΔDP 16.8 versus ERM's 35.2 and AP 66.8 versus 69.1.

• Works on large pretrained models without modification. Fine-tuning ViT-B/16 end-to-end on FairFace at 224×224, FairSplit lowers ΔDP from 23.0 to 9.5, more than halves every other fairness metric (ABCC 17.2 → 5.8, ΔEOpp 22.3 → 8.1, ΔEOdd 33.5 → 9.5), and retains 97% of ERM's AP (86.4 vs 88.9), using 552 of 768 embedding dimensions.

• Transfers across frozen foundation backbones. On CLIP ViT-L/14, CLIP ResNet-50, and DINOv2 ViT-L/14, FairSplit reduces ΔDP to 9.1, 12.2, and 9.1 respectively (from 21.4, 23.4, 23.5), with the mask adapting d_f to each embedding (174/256, 168/256, 166/256). The paper notes transfer across five pretrained encoders with dimensionalities from 512 to 2048 without per-architecture retuning.

• The fairness cost tracks the base-rate gap. Defining Δπ as the difference in base rates between groups (40.0 for CelebA-A, 19.8 for Adult, 25.9 for UTKFace-AR, 26.6 for COMPAS), the authors note that lowering ΔDP below Δπ forces a minimum error. In a controlled experiment varying only Δπ, FairSplit's AP cost rises from 0.9 to 6.7 as Δπ grows from 5 to 45 points.

• The guarantee is looser than the observed behavior. Because of the 1/p_min factor, the bound is loosest on COMPAS, yet empirical fairness does not degrade at the predicted rate — a gap the authors confirm persists across imbalance regimes.

Methodology in Plain English

An encoder maps each input to an m-dimensional embedding. A binary mask b then cuts that vector in two: the Fair Space b ⊙ z, which is the only thing the classifier sees, and the Sensitive Space (1 − b) ⊙ z, which is thrown away. Training combines three signals:

1. Structural compression. A loss pulls same-class examples together within the masked Fair Space and pulls same-sensitive-group examples together within the Sensitive Space, so class information is pushed into one set of dimensions and group information into the other.

2. HSIC regularization. HSIC is a kernel-based dependence measure that is zero exactly when two variables are independent (for characteristic kernels such as Gaussian RBF). The loss adds ρ_f · HSIC(Fair Space, sensitive attribute) to push the Fair Space toward independence, and subtracts ρ_s · HSIC(Sensitive Space, sensitive attribute) to keep sensitive information parked in the discarded part so it does not leak back.

3. Ordinary cross-entropy on the classifier, which sees only the Fair Space.

Because a binary mask cannot be trained by gradient descent, the authors alternate: fix the mask and update the network with Adam (centroids for the compression loss follow a moving average), then fix the representations and solve for the mask in closed form by comparing, per dimension, how much the embedding varies within classes versus within sensitive groups. The relaxed solution in [0,1] is smoothed with an exponential moving average and thresholded at 0.5. Evaluation follows the Han et al. in-processing fairness benchmark protocol, averaged over five seeds, and compares only configurations reaching at least 95% of ERM's validation utility (AP for binary tasks, F1 for multiclass), reporting ΔDP, ΔEOpp, ΔEOdd, ABCC, and MCDP.

Why This Matters

For research: The paper supplies something rare in fair classification — a guarantee that holds for multiclass targets and non-binary sensitive attributes, and that is stated for the downstream classifier operating on the learned representation rather than for a training-time penalty. It also imports HSIC-based embedding decomposition (previously used by H-SPLID for perturbation robustness) into fairness, and its bound is computable after training from the model and data.

Real-world applications:

• Hiring and credit scoring, where the label is often multiclass or ordinal and protected attributes are rarely binary.
• Medical risk prediction, where a single model must be audited across several demographic groups at once.
• Face-attribute and biometric systems built on frozen foundation models, where retraining the backbone is impractical but a lightweight projection and mask can be learned on top.
• Criminal justice risk assessment, illustrated by the COMPAS experiments with an intersectional gender × age attribute and a 0.05 minimum group prevalence.

Industry relevance: Many deployed systems are fine-tuned or prompt-tuned on top of large pretrained encoders. FairSplit's demonstrated ability to work on both fine-tuned ViT-B/16 and frozen CLIP/DINOv2 backbones, adapting the Fair Space size per embedding without architecture-specific retuning, maps directly onto that workflow. The requirement for sensitive attributes during training — with partial robustness to noisy or missing labels — is the main operational constraint.

Future Directions

• Extending the guarantee beyond linear heads. Theorem 3.2 assumes a linear classification head with bounded weights, which covers ResNets, ViTs, and MLPs used as heads but not nonlinear heads; the authors call this open.

• Closing the leakage gap. A nonlinear probe can still recover the sensitive attribute from the Fair Space, so dependence that HSIC (with the Gaussian kernel used in training) does not capture remains a target.

• Reducing reliance on sensitive annotations. FairSplit needs sensitive attributes during training; semi-supervised approaches for when annotations are unavailable are proposed as future work.

• Reconciling the bound with practice. The 1/p_min factor makes the guarantee loose under severe imbalance, while empirical fairness degrades far more slowly — understanding why, and tightening the analysis, is a natural follow-up.

• Privacy of the Sensitive Space. Because that space deliberately retains information about the sensitive attribute, the authors recommend protecting the model parameters in applications where the attribute must stay private.

Target Audience

Graduate students and researchers in fairness-aware machine learning, representation learning, and kernel methods who want both a practical in-processing algorithm and a provable fairness bound; applied scientists working with multiclass or multi-group fairness requirements, or deploying fairness interventions on top of frozen foundation models; and benchmark-oriented practitioners who will value the explicit capability table (Table 1) and the utility-threshold evaluation protocol.

Authors’ abstract

Fair classification faces a twofold challenge: maintaining high classification accuracy while limiting reliance on sensitive information. Our method FairSplit achieves this by learning a decomposition of the embedding space into a Fair Space used for prediction and a Sensitive Space that isolates sensitive information. During training, the model must simultaneously learn the classification task and separate fair from sensitive information. The Sensitive Space gives sensitive attributes a destination, so the Fair Space can be used for prediction without relying on them. This decomposition is guided via a learnable mask and the Hilbert-Schmidt Independence Criterion (HSIC) as a measure of statistical dependence. We provide theoretical guarantees showing that the demographic parity gap is upper-bounded by both (i) the learned dimensionality of the Fair Space and (ii) the HSIC between the Fair Space and sensitive attributes. FairSplit effectively handles multiclass tasks and non-binary sensitive attributes, overcoming key limitations of existing fair classification methods. Experiments on established fairness benchmarks show that FairSplit achieves competitive accuracy-fairness trade-offs. Code is available at https://github.com/shkabrii/fairsplit.