cs.LGOct 28, 2026

FairSplit: Decomposing the Embedding Space for Fair Classification

Authors: Andrii Shkabrii, Anna Beer, Ylli Sadikaj, Claudia Plant

Organizations: Faculty of Computer Science, University of Vienna, Vienna, Austria · Doctoral School Computer Science, University of Vienna, Vienna, Austria · Webster Vienna Private University, Vienna, Austria · Research Network Data Science, University of Vienna, Vienna, Austria

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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.

Figures & tables

Figure 1: Embedding spaces on CelebA-S , where the task is to predict whether a person is smiling and gender is the sensitive attribute, i.e., the protected characteristic on which predictions should not depend, visualized with t-SNE and colored by the target (left) and the sensitive attribute (right). Each side shows three panels: the Vanilla embedding, trained by empirical risk minimization (ERM) without fairness constraints, and the Fair Space and Sensitive Space of FairSplit . In the Vanilla (ERM) embedding, smiling / not smiling correlates with female / male . FairSplit instead classifies from the Fair Space, which is guided towards independence from the sensitive attribute.
MethodMulticlassG>2G>2
DRAlign [ 37 ]✗✗
DiffMCDP [ 28 ]✗✗
PRemover [ 31 ]✗✗
MMD-B-Fair [ 11 ]✗✗
FairMixup [ 8 ]♠✗
AdvDebias [ 55 ]♠♠
Table 1: Method capabilities. Support for multiclass targets and non-binary sensitive attributes ( G>2G>2 ). ✓ supported, ✗ binary only, ♠ indicates that it should be possible, but has not been tested in the respective papers.
Figure 2: Overview of FairSplit . An encoder maps each input to an embedding z\mathbf{z} , and a learned binary mask b\mathbf{b} splits it into a Fair Space zf=b⊙z\mathbf{z}_{f}=\mathbf{b}\odot\mathbf{z} and a Sensitive Space zs=(1−b)⊙z\mathbf{z}_{s}=(\mathbf{1}-\mathbf{b})\odot\mathbf{z} . Only the Fair Space is passed to the classifier. During training, HSIC regularization reduces the dependence between the Fair Space and the sensitive attribute and increases it in the Sensitive Space, which gives sensitive information a destination outside the classifier’s input. The mask is updated in closed form by comparing, per dimension, how strongly the embedding varies within classes and within sensitive groups, so that the size of the Fair Space is learned end-to-end rather than fixed in advance.
 CelebA-A ( Δπ=40.0\Delta\pi=40.0 )Adult ( Δπ=19.8\Delta\pi=19.8 )
 Δ\Delta DP ↓\downarrowABCC ↓\downarrowMCDP ↓\downarrowΔ\Delta EOpp ↓\downarrowΔ\Delta EOdd ↓\downarrowAP ↑\uparrowΔ\Delta DP ↓\downarrowABCC ↓\downarrowMCDP ↓\downarrowΔ\Delta EOpp ↓\downarrowΔ\Delta EOdd ↓\downarrowAP ↑\uparrow
ERM52.8 ±\pm 0.835.9 ±\pm 0.853.4 ±\pm 0.836.8 ±\pm 1.575.0 ±\pm 2.184.5 ±\pm 0.419.4 ±\pm 0.618.7 ±\pm 0.236.6 ±\pm 0.69.3 ±\pm 2.018.2 ±\pm 2.376.7 ±\pm 0.5
AdvDebias45.9 ±\pm 3.431.2 ±\pm 1.646.8 ±\pm 3.826.0 ±\pm 4.457.1 ±\pm 7.684.9 ±\pm 0.76.1 ±\pm 5.77.7 ±\pm 5.016.7 ±\pm 11.121.8 ±\pm 9.525.3 ±\pm 8.972.8 ±\pm 2.6
FairMixup37.2 ±\pm 1.119.5 ±\pm 0.939.5 ±\pm 1.219.5 ±\pm 2.439.8 ±\pm 2.583.1 ±\pm 0.11.1 ±\pm 0.64.6 ±\pm 0.517.2 ±\pm 2.432.6 ±\pm 2.836.0 ±\pm 3.372.9 ±\pm 0.4
DRAlign31.7 ±\pm 4.915.4 ±\pm 2.933.6 ±\pm 4.312.4 ±\pm 4.829.9 ±\pm 9.180.3 ±\pm 1.20.7 ±\pm 0.56.1 ±\pm 0.319.8 ±\pm 0.934.2 ±\pm 1.037.8 ±\pm 1.073.6 ±\pm 0.4
HSIC (Y,S)(Y,S)23.9 ±\pm 1.413.6 ±\pm 0.527.0 ±\pm 1.73.3 ±\pm 1.511.6 ±\pm 3.081.3 ±\pm 0.51.9 ±\pm 0.54.1 ±\pm 0.311.2 ±\pm 1.127.8 ±\pm 1.731.1 ±\pm 2.073.0 ±\pm 0.4
Table 2: Performance on binary settings. CelebA-A and Adult are binary classification tasks with gender as the sensitive attribute, and we measure utility by average precision (AP). For each method, we consider all configurations that reach at least 95%95\% of ERM’s validation AP and report the one with the lowest Δ\Delta DP, without further optimizing AP. Bold and underline mark the best and second-best results. Results are averaged over 5 seeds. Δπ\Delta\pi is the base-rate gap between groups ( Section 4.1 ).
 UTKFace-AR ( Δπ=25.9\Delta\pi=25.9 )COMPAS ( Δπ=26.6\Delta\pi=26.6 )
 Δ\Delta DP ↓\downarrowABCC ↓\downarrowMCDP ↓\downarrowΔ\Delta EOpp ↓\downarrowΔ\Delta EOdd ↓\downarrowF1 ↑\uparrowΔ\Delta DP ↓\downarrowABCC ↓\downarrowMCDP ↓\downarrowΔ\Delta EOpp ↓\downarrowΔ\Delta EOdd ↓\downarrowAP ↑\uparrow
ERM27.9 ±\pm 1.427.1 ±\pm 2.636.2 ±\pm 2.429.1 ±\pm 5.235.9 ±\pm 3.073.8 ±\pm 0.735.2 ±\pm 3.317.9 ±\pm 1.858.8 ±\pm 4.524.9 ±\pm 4.755.4 ±\pm 8.769.1 ±\pm 2.4
AdvDebias28.7 ±\pm 3.021.1 ±\pm 1.932.9 ±\pm 2.532.7 ±\pm 7.838.5 ±\pm 7.974.7 ±\pm 0.826.4 ±\pm 7.713.9 ±\pm 3.147.8 ±\pm 10.929.5 ±\pm 5.744.7 ±\pm 12.468.4 ±\pm 2.1
HSIC (Y,S)(Y,S)11.9 ±\pm 1.312.0 ±\pm 0.820.4 ±\pm 2.418.1 ±\pm 2.724.1 ±\pm 3.570.3 ±\pm 1.120.5 ±\pm 3.16.7 ±\pm 0.735.8 ±\pm 4.628.3 ±\pm 9.236.1 ±\pm 9.867.9 ±\pm 2.3
FairSplit12.6 ±\pm 1.19.1 ±\pm 1.018.7 ±\pm 1.214.9 ±\pm 4.622.1 ±\pm 4.970.4 ±\pm 1.516.8 ±\pm 3.61.2 ±\pm 0.432.9 ±\pm 6.223.8 ±\pm 6.731.5 ±\pm 7.666.8 ±\pm 2.4
Table 3: Performance on multiclass and intersectional settings. UTKFace-AR has a 3-class age target and 3 racial groups, with F1 as utility. COMPAS has a binary target and an intersectional gender ×\times age attribute, with AP as utility. Methods restricted to binary targets or sensitive attributes are omitted ( Table 1 ). As in Table 2 , all configurations achieve at least 95%95\% of ERM’s validation utility, and bold and underline mark the best and second-best results. Results are averaged over 5 seeds. With more than two groups, Δπ\Delta\pi is the largest base-rate gap over pairs of groups and classes.
BackboneMethodAP ↑\uparrowΔ\Delta DP ↓\downarrowABCC ↓\downarrowΔ\Delta EOpp ↓\downarrowΔ\Delta EOdd ↓\downarrowdf/md_{f}/m
ViT-B/16ERM88.9 ±\pm 0.223.0 ±\pm 0.517.2 ±\pm 0.122.3 ±\pm 1.133.5 ±\pm 1.1768 / 768
(fine-tuned)FairSplit86.4 ±\pm 0.39.5 ±\pm 1.65.8 ±\pm 1.48.1 ±\pm 2.69.5 ±\pm 1.9552 / 768
CLIP ViT-L/14ERM96.5 ±\pm 0.121.4 ±\pm 0.920.0 ±\pm 0.64.9 ±\pm 0.523.6 ±\pm 2.4256 / 256
(frozen)FairSplit93.1 ±\pm 0.49.1 ±\pm 1.36.4 ±\pm 0.70.8 ±\pm 0.34.8 ±\pm 1.6174 / 256
CLIP ResNet-50ERM96.6 ±\pm 0.123.4 ±\pm 0.421.0 ±\pm 0.36.1 ±\pm 0.529.4 ±\pm 1.5256 / 256
(frozen)FairSplit93.6 ±\pm 0.412.2 ±\pm 2.47.9 ±\pm 1.50.9 ±\pm 0.58.5 ±\pm 5.0168 / 256
Table 4: FairSplit on large pretrained models (5 seeds, gender as the sensitive attribute). ERM uses the same architecture as FairSplit . ViT-B/16 is fine-tuned end-to-end on FairFace to predict whether a person is at least 30 years old. For the frozen backbones, the Fair Space is learned for predicting high cheekbones on CelebA. A new linear head is then trained on this fixed Fair Space to predict a different attribute, age. df/md_{f}/m is the size of the Fair Space relative to the full embedding.
Figure 3: Trade-off between performance and fairness (higher performance is better) across four datasets. Each row is one dataset (CelebA-A, Adult, UTKFace-AR, COMPAS) with its own y-axis (AP for binary classification, F1 for the multiclass UTKFace-AR), and each column is one fairness metric. ERM ( ★ ★ ) is a single point. The ideal model (✚) lies in the top-left corner of every panel. FairSplit matches or outperforms competing methods across most settings and achieves the best ABCC on every dataset in Tables 2 and 3 .
Figure 8
Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

SymbolDescription
Data and Labels
x,xi\mathbf{x},\mathbf{x}_{i}Input features
y,yiy,y_{i}Class label ∈{1,…,K}\in\{1,\ldots,K\}
y,yi\mathbf{y},\mathbf{y}_{i}One-hot encoding of class label
s,sis,s_{i}Sensitive attribute ∈{1,…,G}\in\{1,\ldots,G\}
s,si\mathbf{s},\mathbf{s}_{i}One-hot encoding of sensitive attribute
Appendix
Table 6: Summary of notation used in this paper.
NameModalityTargetSamplesKKGGTarget Dist. (%)Sens. Dist. (%)
AdultTabularIncome45,2222275.2 / 24.832.5 / 67.5
COMPASTabular2-year recidivism5,2782453.0 / 47.058.8 / 21.7 / 14.6 / 5.0
CelebA-AImageAttractive202,5992248.7 / 51.358.3 / 41.7
UTKFace-ARImageAge (3-class)23,7053332.2 / 44.1 / 23.742.5 / 19.1 / 38.4
FairFaceImageAge ≥30\geq 3097,6982254.2 / 45.853.0 / 47.0
Appendix
Table 7: Dataset statistics. KK and GG denote the number of target classes and sensitive groups, respectively. Target Dist. shows the class distribution, and Sens. Dist. shows the sensitive group distribution. FairFace is used in Section 4.3 , where each seed samples 60,00060{,}000 training and 10,00010{,}000 test images. Groups are ordered Female/Male (Adult, CelebA-A), White/Black/Other (UTKFace-AR), Male/Female (FairFace), and Male-Older/Male-Young/Female-Older/Female-Young (COMPAS).
MetricNameBinaryMulticlass / Multi-group
AP ↑\uparrowAverage Precision∑r(Rr−Rr−1)Pr\sum_{r}(R_{r}-R_{r-1})P_{r}–
F1 ↑\uparrowF1 Score2PR/(P+R)2PR/(P+R)1K∑k2PkRk/(Pk+Rk)\frac{1}{K}\sum_{k}2P_{k}R_{k}/(P_{k}+R_{k})
Δ\Delta DP ↓\downarrowDemographic Parity∣ΔP(y^=2)∣|\Delta\mathbb{P}(\hat{y}{=}2)|max⁡k,g,g′∣ΔPg,g′(y^=k)∣\max_{k,g,g^{\prime}}|\Delta\mathbb{P}_{g,g^{\prime}}(\hat{y}{=}k)|
Δ\Delta EOpp ↓\downarrowEqual Opportunity∣ΔTPR∣|\Delta\mathrm{TPR}|max⁡k,g,g′∣ΔTPRg,g′(k)∣\max_{k,g,g^{\prime}}|\Delta\mathrm{TPR}_{g,g^{\prime}}^{(k)}|
Δ\Delta EOdd ↓\downarrowEqualized Odds∣ΔTPR∣+∣ΔFPR∣|\Delta\mathrm{TPR}|+|\Delta\mathrm{FPR}|max⁡k,g,g′(∣ΔTPRg,g′(k)∣+∣ΔFPRg,g′(k)∣)\max_{k,g,g^{\prime}}\big(|\Delta\mathrm{TPR}_{g,g^{\prime}}^{(k)}|+|\Delta\mathrm{FPR}_{g,g^{\prime}}^{(k)}|\big)
ABCC ↓\downarrowArea Between CDF Curves∫01∣ΔF(t)∣ dt\int_{0}^{1}|\Delta F(t)|\,\mathrm{d}tmax⁡k,g,g′∫01∣ΔFg,g′(k)(t)∣ dt\max_{k,g,g^{\prime}}\int_{0}^{1}|\Delta F_{g,g^{\prime}}^{(k)}(t)|\,\mathrm{d}t
Appendix
Table 8: Performance and fairness metrics. y^∈{1,…,K}\hat{y}\in\{1,\ldots,K\} is the predicted class, yy the true class, and s∈{1,…,G}s\in\{1,\ldots,G\} the sensitive attribute, with class 22 as the positive class for binary targets. PP and RR are precision and recall, and Pr,RrP_{r},R_{r} their values at the rr -th threshold. With Pg(⋅)=P(⋅∣s=g)\mathbb{P}_{g}(\cdot)=\mathbb{P}(\cdot\mid s=g) , TPRg(k)=Pg(y^=k∣y=k)\mathrm{TPR}_{g}^{(k)}=\mathbb{P}_{g}(\hat{y}{=}k\mid y{=}k) , FPRg(k)=Pg(y^=k∣y≠k)\mathrm{FPR}_{g}^{(k)}=\mathbb{P}_{g}(\hat{y}{=}k\mid y{\neq}k) , and Fg(k)F_{g}^{(k)} the CDF of the predicted probability of class kk in group gg , Δ(⋅)g,g′\Delta(\cdot)_{g,g^{\prime}} denotes the difference between groups gg and g′g^{\prime} . Binary metrics drop the indices.
HyperparameterBenchmarksViT-B/16Frozen backbones
OptimizerAdamAdamAdam
Learning rate10−310^{-3}10−410^{-4}3×10−43\times 10^{-4}
StepLR step size ( γ=0.1\gamma{=}0.1 )50100200
Training steps150300600
Warm-up steps (fairness methods)505050
Batch size1024 (tabular), 128 (image)1281024
Appendix
Table 9: Training hyperparameters shared by all methods. The mask moving-average rate applies to FairSplit only.
MethodParameterAdultCelebA-AUTKFace-ARCOMPAS
AdvDebiasλ\lambda220.20.20.350.3511
FairMixupλ\lambda0.80.811——
DRAlignλ\lambda0.50.50.20.2——
 β\beta0.010.010.010.01——
HSIC (Y,S)(Y,S)λ\lambda70705050300300200200
 σ\sigma11555511
Appendix
Table 10: Selected best hyperparameters per baseline method and dataset. Among the configurations whose validation AP (validation F1 for UTKFace-AR) reaches at least 95%95\% of ERM’s, we select the one with the lowest mean validation Δ\Delta DP (see Section 4 ). Cells marked “—” indicate the method is not applicable to the dataset’s setting.
ParameterAdultCelebA-AUTKFace-ARCOMPASViT-B/16Frozen
λf\lambda_{f}0.0250.0250.010.010.10.10.010.010.010.010.010.01
λs\lambda_{s}0.050.050.0250.0250.050.050.20.20.0250.0250.050.05
ρf\rho_{f}0.40.40.50.51.31.30.30.31.51.51.01.0
ρs\rho_{s}0.0050.0050.50.50.10.10.0050.0052.02.00.50.5
σ\sigma225555225555
Appendix
Table 11: Selected FairSplit hyperparameters per dataset, with the same selection rule as in Table 10 . σ\sigma is fixed at the modality default (tabular vs. image). For the frozen backbones, the hyperparameters are selected on CLIP ViT-L/14 and reused unchanged on the other two backbones.
Figure 5: Bound of Theorem 3.2 and the demographic parity gap of the logits (top), and the demographic parity gap of the predicted classes (bottom), as the HSIC regularization strength ρf\rho_{f} varies. Shaded regions are the mean ±\pm std over 5 seeds, and the dashed line marks the reported ρf\rho_{f} .
  UTKFace-ARCelebA-A
EncodermmF1 ↑\uparrowΔ\Delta DP ↓\downarrowRed.dfd_{f}AP ↑\uparrowΔ\Delta DP ↓\downarrowRed.dfd_{f}
ResNet-1851273.8 →\to 70.427.9 →\to 12.655%23384.5 →\to 80.452.8 →\to 18.365%334
ResNet-3451273.8 →\to 69.128.9 →\to 12.856%24384.5 →\to 78.153.2 →\to 18.465%297
ResNet-50204874.8 →\to 72.227.9 →\to 14.947%100385.9 →\to 80.652.4 →\to 25.551%1230
EfficientNetV2-S128073.2 →\to 73.829.1 →\to 15.945%116184.9 →\to 82.650.6 →\to 31.238%777
MobileNetV4-S96070.4 →\to 67.528.5 →\to 13.951%58484.0 →\to 74.751.6 →\to 19.363%419
Appendix
Table 12: Encoder robustness on UTKFace-AR (3-class age, 3 racial groups) and CelebA-A. FairSplit uses the same hyperparameters for every encoder, and only the encoder is changed, with no per-architecture retuning. Since the hyperparameters are not tuned per encoder, the 95%95\% utility gate of the main tables is not applied here. Utility and Δ\Delta DP columns show ERM →\to FairSplit , and Red. is the relative reduction of Δ\Delta DP. Mean over 5 seeds.
  m=32m=32m=64m=64m=128m=128m=256m=256m=512m=512
Δ\Delta DP ↓\downarrowERM19.8 ±\pm 0.819.4 ±\pm 0.618.5 ±\pm 0.518.2 ±\pm 0.317.5 ±\pm 0.2
FairSplit3.3 ±\pm 1.21.0 ±\pm 0.93.2 ±\pm 2.61.3 ±\pm 0.52.5 ±\pm 2.4
AP ↑\uparrowERM76.8 ±\pm 0.576.7 ±\pm 0.577.0 ±\pm 0.577.3 ±\pm 0.477.5 ±\pm 0.4
FairSplit71.5 ±\pm 0.872.9 ±\pm 0.971.6 ±\pm 1.270.8 ±\pm 1.274.3 ±\pm 1.0
ABCC ↓\downarrowERM18.8 ±\pm 0.318.7 ±\pm 0.218.5 ±\pm 0.218.6 ±\pm 0.218.5 ±\pm 0.1
FairSplit0.9 ±\pm 0.22.4 ±\pm 0.71.5 ±\pm 0.30.8 ±\pm 0.24.3 ±\pm 0.5
Appendix
Table 13: Embedding-size ablation on Adult (5 seeds).
BackboneTaskMethodAP ↑\uparrowΔ\Delta DP ↓\downarrowABCC ↓\downarrowΔ\Delta EOpp ↓\downarrowΔ\Delta EOdd ↓\downarrowdf/md_{f}/m
CLIP ViT-L/14AERM93.7 ±\pm 0.027.8 ±\pm 0.426.0 ±\pm 0.215.8 ±\pm 0.418.3 ±\pm 1.0256 / 256
FairSplit89.3 ±\pm 0.413.1 ±\pm 1.29.0 ±\pm 0.61.2 ±\pm 0.712.1 ±\pm 1.0174 / 256
BERM96.5 ±\pm 0.121.4 ±\pm 0.920.0 ±\pm 0.64.9 ±\pm 0.523.6 ±\pm 2.4256 / 256
FairSplit93.1 ±\pm 0.49.1 ±\pm 1.36.4 ±\pm 0.70.8 ±\pm 0.34.8 ±\pm 1.6174 / 256
CLIP ResNet-50AERM94.5 ±\pm 0.027.1 ±\pm 0.226.3 ±\pm 0.114.0 ±\pm 0.515.7 ±\pm 0.5256 / 256
FairSplit90.8 ±\pm 0.715.4 ±\pm 1.310.5 ±\pm 0.91.5 ±\pm 1.210.9 ±\pm 0.6168 / 256
Appendix
Table 14: Transfer of the Fair Space from Task A ( High_Cheekbones ) to Task B ( Young ) on three frozen backbones (5 seeds). Task B uses a fresh linear head on the frozen representation.
   Linear probeMLP probe   
BackboneTaskΔπ\Delta\piz\mathbf{z}zf\mathbf{z}_{f}zs\mathbf{z}_{s}zf\mathbf{z}_{f}df/md_{f}/mAP ↑\uparrowΔ\Delta DP ↓\downarrow
CLIP ViT-L/14Smiling1498.1 →\to 99.050.4 ±\pm 0.399.0 ±\pm 0.095.9 ±\pm 1.1167 / 25697.9 →\to 97.015.2 →\to 8.5
Young2498.5 →\to 99.061.0 ±\pm 6.999.0 ±\pm 0.089.0 ±\pm 1.991 / 25697.6 →\to 95.523.2 →\to 4.9
High_Cheekbones2598.4 →\to 99.054.7 ±\pm 2.199.0 ±\pm 0.097.2 ±\pm 0.9174 / 25693.7 →\to 89.327.8 →\to 13.1
Wavy_Hair3198.8 →\to 99.064.4 ±\pm 4.599.0 ±\pm 0.094.8 ±\pm 1.1115 / 25685.6 →\to 83.335.4 →\to 26.6
ResNet-18Attractive4085.7 →\to 94.888.2 ±\pm 2.494.8 ±\pm 0.294.5 ±\pm 0.3334 / 51284.5 →\to 80.452.8 →\to 18.3
Appendix
Table 15: Balanced accuracy (%, 5 seeds) of probes predicting the sensitive attribute from the full embedding z\mathbf{z} , the Fair Space zf\mathbf{z}_{f} , and the Sensitive Space zs\mathbf{z}_{s} of the same trained model, where random guessing scores 50%50\% . CLIP ViT-L/14 is frozen (Appendix G.3 ), and ResNet-18 is the CelebA-A model of Table 2 . For z\mathbf{z} , AP, and Δ\Delta DP, arrows show ERM →\to FairSplit with the same architecture. Lower probe accuracy means that ss is harder to predict.
 AdultUTKFace-ARCOMPAS
MethodLinearMLPLinearMLPLinearMLP
ERM71.7 ±\pm 0.879.1 ±\pm 0.969.4 ±\pm 0.868.2 ±\pm 0.945.1 ±\pm 0.948.5 ±\pm 0.6
AdvDebias80.0 ±\pm 2.580.1 ±\pm 1.970.0 ±\pm 1.266.4 ±\pm 1.848.7 ±\pm 0.547.8 ±\pm 0.7
FairMixup75.9 ±\pm 0.680.6 ±\pm 0.7————
DRAlign72.1 ±\pm 2.479.7 ±\pm 0.9————
HSIC (Y,S)(Y,S)72.0 ±\pm 1.678.6 ±\pm 2.564.8 ±\pm 1.862.2 ±\pm 1.844.1 ±\pm 0.747.5 ±\pm 0.6
Appendix
Table 16: Balanced accuracy (%, 5 seeds) of probes predicting the sensitive attribute from the input of each method’s classifier, the Fair Space zf\mathbf{z}_{f} for FairSplit and the full embedding z\mathbf{z} for the baselines. Random guessing scores 50%50\% on Adult, 33%33\% on UTKFace-AR, and 25%25\% on COMPAS. The probed models are those of Tables 2 and 3 . Lower probe accuracy means that ss is harder to predict, and the lowest value per column is in bold. A dash (—) marks baselines that support only binary targets and binary sensitive attributes, so they cannot be applied to UTKFace-AR (three age classes, three racial groups) or COMPAS (four groups).
MethodHeadAP/F1 ↑\uparrowΔ\Delta DP ↓\downarrowABCC ↓\downarrowMCDP ↓\downarrowΔ\Delta EOpp ↓\downarrowΔ\Delta EOdd ↓\downarrowdfd_{f}
CelebA-A
ERMLinear84.5 ±\pm 0.452.8 ±\pm 0.835.9 ±\pm 0.853.4 ±\pm 0.836.8 ±\pm 1.575.0 ±\pm 2.1512
FairSplitLinear80.4 ±\pm 1.118.3 ±\pm 1.19.4 ±\pm 1.319.6 ±\pm 1.82.6 ±\pm 0.35.2 ±\pm 1.7334
ERMNon-linear85.3 ±\pm 0.453.7 ±\pm 1.937.5 ±\pm 1.254.0 ±\pm 2.037.3 ±\pm 3.075.6 ±\pm 4.9512
FairSplitNon-linear81.7 ±\pm 0.523.1 ±\pm 3.613.5 ±\pm 1.524.3 ±\pm 3.34.4 ±\pm 2.211.0 ±\pm 5.0369
UTKFace-AR
Appendix
Table 17: FairSplit with linear and non-linear classification heads on zf\mathbf{z}_{f} (5 seeds). The utility metric is AP on CelebA-A and F1 on UTKFace-AR. For each dataset and head, bold marks the better of ERM and FairSplit .
 Δ\Delta DP ↓\downarrowABCC ↓\downarrowMCDP ↓\downarrowΔ\Delta EOpp ↓\downarrowΔ\Delta EOdd ↓\downarrowAP ↑\uparrow
ERM54.0 ±\pm 0.434.2 ±\pm 0.854.4 ±\pm 0.542.4 ±\pm 1.879.3 ±\pm 1.684.1 ±\pm 0.3
AdvDebias52.3 ±\pm 2.735.1 ±\pm 1.153.6 ±\pm 1.935.4 ±\pm 8.873.3 ±\pm 7.085.4 ±\pm 0.3
HSIC (Y,S)(Y,S)19.4 ±\pm 1.910.2 ±\pm 1.123.1 ±\pm 2.82.3 ±\pm 1.27.2 ±\pm 4.080.7 ±\pm 0.8
FairSplit18.2 ±\pm 1.69.7 ±\pm 1.022.8 ±\pm 2.81.8 ±\pm 1.37.1 ±\pm 3.580.7 ±\pm 0.7
Appendix
Table 18: CelebA-A with an imposed 95/595/5 group imbalance in the training data (5 seeds). Each method keeps its hyperparameters from Tables 10 and 11 . Best in bold , second-best underlined .
Δπ\Delta\piAP ↑\uparrowΔ\Delta DP ↓\downarrowAP cost ↓\downarrow
ERMFairSplitERMFairSplit
597.5 ±\pm 0.096.6 ±\pm 0.34.0 ±\pm 0.20.7 ±\pm 0.50.9
1597.5 ±\pm 0.096.4 ±\pm 0.214.4 ±\pm 0.56.6 ±\pm 0.71.1
2597.6 ±\pm 0.195.3 ±\pm 0.324.7 ±\pm 0.312.6 ±\pm 1.12.3
3597.8 ±\pm 0.193.9 ±\pm 0.335.4 ±\pm 0.317.4 ±\pm 1.63.9
4598.0 ±\pm 0.191.3 ±\pm 0.445.7 ±\pm 0.320.3 ±\pm 1.96.7
Appendix
Table 19: Varying only the base-rate gap Δπ\Delta\pi (frozen CLIP ViT-L/14, Smiling , 5 seeds).
Figure 6: Hyperparameter ablation on CelebA-A (5 seeds). We vary λf,λs,ρf,ρs\lambda_{f},\lambda_{s},\rho_{f},\rho_{s} (left to right) while fixing the other three at their defaults (dashed line). Rows: AP, parity gaps ( Δ\Delta DP, ABCC, MCDP), error-rate gaps ( Δ\Delta EOpp, Δ\Delta EOdd), and the learned Fair Space dimensionality dfd_{f} . Fairness improvements are driven primarily by ρf\rho_{f} , and the method is robust to λf\lambda_{f} . Increasing λs\lambda_{s} and ρs\rho_{s} shrinks dfd_{f} .
VariantAP/F1 ↑\uparrowΔ\Delta DP ↓\downarrowABCC ↓\downarrowMCDP ↓\downarrowΔ\Delta EOpp ↓\downarrowΔ\Delta EOdd ↓\downarrowdfd_{f}
CelebA-A
ERM84.5 ±\pm 0.452.8 ±\pm 0.835.9 ±\pm 0.853.4 ±\pm 0.836.8 ±\pm 1.575.0 ±\pm 2.1512
No Sensitive Space80.7 ±\pm 0.726.6 ±\pm 1.416.1 ±\pm 1.026.8 ±\pm 1.05.5 ±\pm 0.918.4 ±\pm 2.8512
Unregularized ( λs=ρs=0\lambda_{s}{=}\rho_{s}{=}0 )80.1 ±\pm 0.822.2 ±\pm 2.412.4 ±\pm 1.722.8 ±\pm 1.61.9 ±\pm 0.99.1 ±\pm 2.4391
Unregularized + λs+\,\lambda_{s}79.9 ±\pm 1.220.1 ±\pm 2.710.7 ±\pm 1.821.6 ±\pm 2.02.4 ±\pm 1.38.1 ±\pm 2.5366
Unregularized + ρs+\,\rho_{s}80.9 ±\pm 0.621.4 ±\pm 1.811.2 ±\pm 1.022.4 ±\pm 1.21.4 ±\pm 2.26.8 ±\pm 2.8371
Appendix
Table 20: Component ablation on CelebA-A and UTKFace-AR (5 seeds). The utility metric is AP on CelebA-A and F1 on UTKFace-AR. The best value per metric within each dataset is in bold . † One of the five seeds collapses to a constant predictor.
 AP ↑\uparrowΔ\Delta DP ↓\downarrowABCC ↓\downarrow
ERM84.5 ±\pm 0.452.8 ±\pm 0.835.9 ±\pm 0.8
Kernel family ( σ=5\sigma=5 )
Gaussian RBF80.4 ±\pm 1.118.3 ±\pm 1.10 9.4 ±\pm 1.3
Matérn 3/23/280.5 ±\pm 1.518.6 ±\pm 2.910.0 ±\pm 2.1
Laplacian78.1 ±\pm 2.535.4 ±\pm 6.616.4 ±\pm 4.2
Gaussian RBF bandwidth
Appendix
Table 21: HSIC kernel family and Gaussian RBF bandwidth on CelebA-A (5 seeds). Only the kernel or σ\sigma changes, and ERM is reported as a reference without fairness regularization.
MethodAdult (s)vs. ERMCelebA-A (s)vs. ERM
ERM11.41.00 ×\times27.71.00 ×\times
AdvDebias10.60.93 ×\times28.61.03 ×\times
FairMixup12.41.09 ×\times35.81.29 ×\times
DRAlign13.81.21 ×\times48.61.76 ×\times
HSIC (Y,S)(Y,S)11.91.05 ×\times28.41.03 ×\times
DiffMCDP16.31.43 ×\times28.81.04 ×\times
Appendix
Table 22: Wall-clock training time per run (in seconds), averaged over 5 seeds on a single NVIDIA A100. “vs. ERM” is the multiplicative cost relative to the ERM baseline.
ConditionAP ↑\uparrowΔ\Delta DP ↓\downarrowABCC ↓\downarrow
ERM84.5 ±\pm 0.452.8 ±\pm 0.835.9 ±\pm 0.8
FairSplit (clean)80.4 ±\pm 1.118.3 ±\pm 1.10 9.4 ±\pm 1.3
Noisy 10%82.8 ±\pm 0.728.3 ±\pm 2.616.2 ±\pm 1.5
Noisy 30%85.5 ±\pm 0.253.1 ±\pm 0.931.5 ±\pm 0.9
Noisy 50%85.6 ±\pm 0.354.0 ±\pm 0.432.2 ±\pm 0.6
Missing 10%81.0 ±\pm 0.921.8 ±\pm 1.911.6 ±\pm 1.2
Appendix
Table 23: Robustness on CelebA-A under noisy and missing sensitive attributes (5 seeds). “Noisy pp ” flips each sensitive label with probability pp . “Missing pp ” drops the sensitive label for a fraction pp of samples (these samples contribute only to the cross-entropy loss). “Shuffled” randomly permutes all sensitive labels. ERM and clean FairSplit are reported as references.
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