Fairness-aware Bayes Optimal Functional Classification

We thank the reviewer for acknowledging our theoretical contributions and recognising the importance of fairness in functional classification. Below, we address the weakness (W) point by point and further elaborate on the motivation behind our work. We respectfully hope the reviewer could consider raising the score in light of the additional motivation provided, comparisons to adapted methods, and clarifications of our model assumptions.

Practical motivation

In our paper, we focus on developing a novel and practically effective methodology for fairness-aware classification in functional data settings. This is an important and underexplored problem with significant real-world relevance. To the best of our knowledge, our proposed framework, Fair-FLDA, is the first to explicitly address fairness in the context of functional classification. Functional data pose unique challenges due to their infinite-dimensional nature, which makes it difficult or ineffective to directly apply fairness techniques developed for multivariate data.

Our motivation is primarily practical: in many applications involving functional data, such as biometric signals, longitudinal health records, or time-varying sensor data, ensuring fair and accurate classification is crucial. As demonstrated in Section 4, applying standard classification algorithms to such data can lead to substantial fairness violations (see the second and third columns in the right panel of Figure 2). In this response, we also examine a simple ``dimension reduction + standard fair classification" strategy: applying dimension reduction followed by fair multivariate classification. Additional baseline results using this approach are included in Tables 1 and 2 at the end of this response, along with detailed comparisons between Fair-FLDA and three post-processing and one pre-processing fairness methods originally designed for multivariate data. Across all experiments, Fair-FLDA consistently achieves lower misclassification rates while effectively mitigating disparity, highlighting its practical merits. In the revised version, we will provide a clearer explanation of the methodological and practical motivations that guide our study.

Theoretical contribution

While the main thrust of our contribution lies in methodology and empirical performance, we include theoretical analysis to support and validate the proposed approach. This analysis serves to reinforce the practical methodology rather than being the central focus of the paper. We appreciate your recognition of this theoretical component. Notably, theoretical guarantees for functional classification remain largely open in general settings, particularly when the underlying eigenspace is unknown. Our results address this challenging scenario while incorporating fairness constraints.

Writing for the second and third paragraphs (W1)

We appreciate your suggestions on our writing in the introduction. In the revised version, we will rewrite these two paragraphs to improve the logical flow and strengthen their connection. Specifically, in the third paragraph, we will refer to the real data example in Section 4 to demonstrate that existing classifiers exhibit substantial unfairness, thereby highlighting the practical importance of addressing fairness in functional data classification. We will also emphasise the lack of existing work in the literature for fair functional data classification to further motivate our contribution.

Assumptions on Gaussianity (W2)

Thank you for your question regarding our model assumptions. The classification problem for Gaussian processes has been extensively studied in the literature (e.g., [1], [2], [3]). In functional data analysis, it is well known that linear methods can often achieve optimal performance, largely due to the infinite dimensionality of the functional space (e.g., [1]). In some scenarios, linear classifiers can even lead to perfect classification (see Remark 2). When the data is generated from a Gaussian process, this optimality can be further understood through its connection to the structure of reproducing kernel Hilbert spaces (RKHS), as discussed in [2] and [3].

In our paper, the Gaussianity assumption offers us a theoretically tractable foundation for studying fairness in functional classification. Under this assumption, explicit expressions for the Radon–Nikodym derivative can be derived, which in turn facilitate the construction of the plug-in classifier, Fair-FLDA. To demonstrate the robustness of our method, we also include simulation results under non-Gaussian models in Appendix A.3, highlighting its practical performance beyond the Gaussian setting.

As our work is, to the best of our knowledge, the first to study fairness in functional data classification, we consider the Gaussian process model a natural and principled starting point. It provides a clean and interpretable platform for exploring fairness constraints in infinite-dimensional spaces. In future work, we plan to extend our methodology to more general, non-Gaussian settings.

Additional empirical evaluation (W3)

Our primary goal in this work is to introduce the first principled framework for fair functional classification with theoretical guarantees, providing the fairness community with a trustworthy algorithm. Our theoretical contributions are numerically justified through both synthetic and real datasets, along with additional experiments done in the rebuttal stage.

Since no prior work specifically targets fair functional classification, there are no widely established benchmark datasets in this context. We therefore use the publicly available dataset NHANES, which is frequently used in the functional data literature [4,5], to demonstrate the practical significance of our fair functional classifier. It is important to note that our method is broadly applicable to a wide range of functional data classification problems where fairness is a concern. For example, the Siena Scalp EEG dataset in the PhysioNet database contains EEG recordings from male and female subjects and can serve as a natural application of our fair functional classifier, where the response variable can be defined as the occurrence of a seizure.

Moreover, we have included more comparisons with other baseline methods to further demonstrate the superiority and practical relevance of our method for fair functional classification. We have incorporated additional baselines following a ``dimension reduction + standard fair classification" strategy. Specifically, we first apply functional principal component analysis (FPCA) to extract features, and then employ fair classification methods designed for multivariate data. These include three post-processing methods FPIR [6], PPF [7] and PPOT [8], and one pre-processing approach FUDS [6].

Results in Tables 1 and 2 show that our proposed fair classifier, Fair-FLDA, consistently achieves the lowest classification errors while effectively controlling disparity under the pre-specified levels. In contrast, the other four baseline methods exhibit higher classification errors. In particular, PPF shows poor disparity control when $\delta=0$, and FUDS fails to adequately control disparity on the NHANES dataset. These extensive numerical results highlight the superiority and practical necessity of our method for fair functional classification.

Overall, the proposed fair functional classifier, equipped with theoretical guarantees, fills an important gap in the literature on fair functional classification and contributes to advancing research in the fairness community. We hope these clarifications adequately address your concerns.

Table 1: Median classification error and DD over 500 runs under Gaussian with mean difference $\beta=1.5$ and sample size $n=1000$.

Table 2: Median classification error and DD over 500 runs under NHANES.

[1] Delaigle and Hall. Achieving near-perfect classification for functional data. JRSSB, 2012.

[2] Berrendero et al. On the use of reproducing kernel Hilbert spaces in functional classification. JASA, 2018.

[3] Torrecilla et al. Optimal classification of Gaussian processes in homo-and heteroscedastic settings. Statistics and Computing, 2020.

[4] Chang and McKeague, Empirical likelihood-based inference for functional means with application to wearable device data. JRSSB, 2022.

[5] Lin et al., Causal inference on distribution functions. JRSSB, 2023.

[6] Zeng et al., Bayes-optimal fair classification with linear disparity constraints via pre-, in-, and post-processing. arXiv:2402.02817.

[7] Chen et al., Post-hoc bias scoring is optimal for fair classification. ICLR 2024.

[8] Xian et al., Fair and optimal classification via post-processing. ICML 2023.

We thank the reviewer for appreciating the strong motivation of our work, the importance of the problem and the comprehensive exploration we conducted. Below, we address the weaknesses and limitations raised by the reviewer. We respectfully hope the reviewer considers raising the score given additional comparisons with alternative methods, and more clarifications and interpretation regarding the advantages of our approach.

Fairness in functional data classification

We appreciate your recognition of the importance of fairness. Our data application demonstrates that unfairness can indeed arise in functional classification settings, highlighting the practical relevance of this problem. This can be seen from Figures 4-6 in the appendix of our submission. Despite its significance, to the best of our knowledge, there is no previous work addressing unfairness in functional data classification. Our work bridges the gap by introducing a principled approach with theoretical guarantees.

Regarding the experimental studies, we provide additional comparisons with multivariate methods adapted to functional data, please see the response to your following point. Moreover, we emphasise that our approach is broadly applicable to a wide range of functional data classification problems where fairness is a concern. For example, the Siena Scalp EEG dataset in the PhysioNet database contains EEG recordings from male and female subjects and can serve as a natural application of our fair functional classifier, where the response variable can be defined as the occurrence of a seizure.

Comparisons with methods adapted to functional data

Thanks for the comment on the numerical evaluation. Since no existing fair classifiers can be directly applied to functional data, we adopt a ``dimension reduction + standard fair classifiers" strategy for comparison. Specifically, we first apply functional principal component analysis (FPCA) to extract features, and then employ fair classification methods designed for multivariate data. These include three post-processing methods FPIR [1], PPF [2] and PPOT [3], and one pre-processing approach FUDS [1]. Since the open-source code provided by [1] only supports evaluation using the DD measure, we currently report results under DD with the default parameters. In the revised version, we will include the complete numerical results under the other measures PD and DO.

Based on the numerical results, our proposed fair classifier, Fair-FLDA, consistently achieves the lowest classification errors while effectively controlling disparity under the pre-specified levels. In contrast, the other four baseline methods exhibit higher classification errors. In particular, PPF shows poor disparity control when $\delta=0$, and FUDS fails to adequately control disparity on the NHANES dataset. These extensive numerical results highlight the superiority and practical necessity of our method for fair functional classification.

Table 1: The median classification error and median DD measure over 500 repetitions under the Gaussian setting with mean difference $\beta=1.5$ and sample size $n=1000$.

Table 2: The median classification error and median DD measure over 500 repetitions under NHANES data.

Presentation of numerical experiments

Thank you for your suggestions on our numerical presentations. In the revised version, we will rewrite Section 4 to update figures to include axis labels and legends, and clarify the experimental setup to provide more insights into our results.

Due to the restrictions of the rebuttal format, we will talk through Figure 2 in the original submission here as an example for clarification. In Figure 2 of the original paper, we compare the performance of Fair-FLDA in terms of both misclassification error and fairness control as the disparity threshold $\delta$ varies. Focusing on the left panel of Figure 2, the orange dots represent the FLDA classifier, which ignores fairness constraints and serves as the baseline in the comparison. The blue stars and pink triangles are the results for Fair-FLDA and Fair-FLDA$_c$, respectively. Focusing on the misclassification error plots in the first column, we observe that in the fair-impacted regime (i.e. $\delta$ is small), the error decreases as $\delta$ increases, and eventually approaches the baseline as the fairness constraint is relaxed. To evaluate fairness performance, the second and third columns report disparity measures under various values of $\delta$. Specifically, points below the grey dashed line indicate that the fairness constraint is satisfied. We show that FLDA consistently fails to meet these fairness requirements, while Fair-FLDA maintains the desired median disparity level, and Fair-FLDA$_c$ ensures that the disparity remains below $\delta$ with at least 95% probability. We have also included results on model misspecification in Appendix A.3 to illustrate the robustness of Fair-FLDA.

Usage of calibration data

The calibration data are primarily introduced here for technical convenience to bring independence among samples in our theoretical studies. In practice, our Algorithm 1 can be implemented by executing both S1 and S2 on the whole dataset. In all numerical experiments in Section 4, we mimic the effect of sample splitting via a cross-fitting approach detailed in Appendix A.1, where two classifiers are trained by alternating the roles of data used for model estimation and threshold calibration, and then averaged. To further illustrate the effect of sample splitting, we include additional numerical results on both simulated and real datasets in Table 3 below. Although the reduction in sample size from data splitting slightly increases the misclassification error, the unfairness measure under NoSplit is usually higher than the threshold $\delta$ due to the dependence of the data used in training and calibration.

Table 3: Comparison of data splitting for Fair-FLDA on the real and simulated datasets. Results are reported as the median over 500 iterations. NoSplit: the results of Fair-FLDA applied without data splitting.

[1] Zeng et al., Bayes-optimal fair classification with linear disparity constraints via pre-, in-, and post-processing. arXiv:2402.02817.

[2] Chen et al., Post-hoc bias scoring is optimal for fair classification. ICLR 2024.

[3] Xian et al., Fair and optimal classification via post-processing. ICML 2023.

We thank the reviewer for recognising the importance of fairness-aware functional classification and our first contribution in this area. Below, we address the weaknesses (W) raised by the reviewer.

1. Methodological and theoretical novelty (W1)

Thank you for your appreciation of our work. To the best of our knowledge, this paper is the first to study fairness with functional data. While there is a rich body of literature on fairness-aware classification for multivariate data, most existing methods do not extend naturally to functional data due to the infinite dimensionality of the function space. Rather than the posterior probabilities $\mathbb{P}(Y=1|A=a, X=x)$ considered in most of the previous papers, our methodology fully respects the functional nature of data by constructing the classifier based on the Radon--Nikodym derivative $\mathrm{d}P_{a,1}(x)/\mathrm{d}P_{a,0}$, which is a more natural functional used for functional classification. Theoretically, we provide not only a fairness guarantee of the designed classifier Fair-FLDA in Theorem 3, but also an explicit convergence rate of excess risk in Theorem 5.

To further demonstrate the advantages of our approach, we have included additional numerical results in which dimension reduction using functional principal component analysis (FPCA) is first performed, followed by the application of existing post- and pre-processing fairness methods. Specifically, we first apply FPCA to extract features, and then employ fair classification methods designed for multivariate data. These include three post-processing methods FPIR [1], PPF [2] and PPOT [3], and one pre-processing approach FUDS [1].

Results in Tables 1 and 2 show that our proposed fair classifier, Fair-FLDA, consistently achieves the lowest classification errors while effectively controlling disparity under the pre-specified levels. In contrast, the other four baseline methods exhibit higher classification errors. In particular, PPF shows poor disparity control when $\delta=0$, and FUDS fails to adequately control disparity on the NHANES dataset. These extensive numerical results highlight the superiority and practical necessity of our method for fair functional classification.

Table 1: The median classification error and median DD measure over 500 repetitions under the Gaussian setting with mean difference $\beta=1.5$ and sample size $n=1000$.

Table 2: The median classification error and median DD measure over 500 repetitions under NHANES data.

2. Diversity and generalizability (W2)

Thank you for the comment. In our work, we use the publicly available NHANES dataset as a representative data example to demonstrate the effectiveness of our method. It is worth noting that our approach is broadly applicable to a wide range of functional data classification problems where fairness is a concern. For example, the Siena Scalp EEG dataset in the PhysioNet database contains EEG recordings from male and female subjects and can serve as a natural application of our fair functional classifier, where the response variable can be defined as the occurrence of a seizure. Overall, our method offers a principled and broadly applicable tool for fair classification in functional data analysis. We will include a relevant discussion on this point in the revised version.

[1] Zeng et al., Bayes-optimal fair classification with linear disparity constraints via pre-, in-, and post-processing. arXiv:2402.02817.

[2] Chen et al., Post-hoc bias scoring is optimal for fair classification. ICLR 2024.

[3] Xian et al., Fair and optimal classification via post-processing. ICML 2023.

We thank the reviewer for the appreciation of our theoretical contributions and the strong motivation for studying fair functional classification. Below we address the weaknesses (W), questions (Q) and limitations (L) raised by the reviewer. We respectfully hope the reviewer could consider raising the score given the additional comparisons to adapted methods, the clarified advantages of our method and extensions to scenarios with missing or multiple sensitive attributes.

1. Additional empirical evaluation (W1, Q4, L1)

Comparisons with standard fairness methods adapted to functional data. To the best of our knowledge, our work is the first to derive the Bayes optimal fair classifier and to establish explicit convergence rates in the context of functional data. There are no existing fair functional classifiers available for direct comparison.

As suggested, we have incorporated additional baselines following a "dimension reduction + standard fair classification" strategy. Specifically, we first apply functional principal component analysis to extract features, and then employ fair classification methods designed for multivariate data. These include three post-processing methods FPIR [1], PPF [2] and PPOT [3], and one pre-processing approach FUDS [1], with default parameters as in the open source code of [1].

Results in Tables 1 and 2 show that our proposed fair classifier, Fair-FLDA, consistently achieves the lowest classification errors while effectively controlling disparity under the pre-specified levels. In contrast, the other four baseline methods exhibit higher classification errors. In particular, PPF shows poor disparity control when $\delta=0$, and FUDS fails to adequately control disparity on the NHANES dataset. These extensive numerical results highlight the superiority and practical necessity of our method for fair functional classification.

Table 1: Median classification error and DD over 500 runs under Gaussian with mean difference $\beta=1.5$ and sample size $n=1000$.

Table 2: Median classification error and DD over 500 runs under NHANES.

Synthetic/real datasets. In the original submission, we conducted extensive simulations under various model configurations to illustrate the effects of sample sizes, alignment of mean difference, model misspecification and perfect classification. These synthetic experiments provide insights into the practical implications of our theoretical convergence rates. Detailed results are deferred to the Supplementary Material due to space constraints.

Since no prior work specifically targets fair functional classification, there are no widely established benchmark datasets in this context. We therefore use the NHANES dataset, a common dataset in functional data studies [4, 5], to demonstrate the practical significance of our fair functional classifier. It is important to note that our method is broadly applicable; for instance, it can be used on the Siena Scalp EEG dataset in the PhysioNet database to classify seizures while ensuring gender fairness.

2. Advantages of our method over simpler alternatives (W2, Q1, L2)

We clarify our advantages from both theoretical and numerical aspects. Theoretically, our method fully respects the functional nature of the data by deriving the explicit form of the Bayes optimal fair classifier in Eq(4), which naturally motivates its practical approximation via truncation in Algorithm 1. We establish that the excess risk converges to zero and provide an explicit convergence rate in Theorem 5. In contrast, simpler alternatives such as "dimensional reduction + standard fair classifiers" lack any theoretical guarantees. Numerically, through extensive comparisons with FPIR, PPF PPOT and FUDS on both synthetic and real data, our method consistently achieves the lowest classification error while effectively controlling disparity, significantly outperforming these simpler alternatives.

3. Extensions to scenarios with missing or multiple sensitive attributes (W3, Q3, L3, L4)

Our framework in Section 2.2 can be naturally extended to settings where the sensitive feature is unavailable at testing, as well as to multi-class classification problems. We provide a sketch of methods in this response, and we will incorporate this addition in the revision.

Missing sensitive attributes. The extension consists of three steps.

Step 1. A key ingredient in our framework, when sensitive features are available, is that both the misclassification error $R$ and disparity measure $D$ are linear in classifiers $f:\mathcal{X}\times \mathcal{A}\rightarrow [0,1]$. To extend the framework to settings where sensitive attributes are unavailable at testing, it is necessary to show that $R$ and $D$ remain linear to classifiers $f:\mathcal{X} \rightarrow [0,1]$, which is solely defined on $\mathcal{X}$. Following a similar idea as in the proof of Prop.1, it can be verified that DO, PD and DD are still linear and of the form $D(f) = \int_{\mathcal{X}}f(x)w_{X, D}(x)dP_{X|Y=0}(x)$, where $w_{X,D}:\mathcal{X} \rightarrow \mathbb{R}$ is a weight function.

Step 2. By the generalised Neyman--Pearson lemma and a similar argument to the one used in the proof of Theorem 2, we can derive an explicit formula for $\delta$-fair Bayes optimal classifier of the form $f_{D}^\star(x) =\mathbf{1}[\pi_{1} \frac{dP_{X|Y=1}}{dP_{X|Y=0}}(x) -\pi_{0} \geq \tau^\star w_{x, D}(x)].$

Step 3. Construct a plug-in classifier. Further to the estimation in Algorithm 1, an additional non-parametric estimator can be used to estimate the probability of A given X and Y.

Multi-class classification. Consider predictive disparity as an example. For a multi-class sensitive attribute $a \in \mathcal{A} = [1, \ldots, |\mathcal{A}|]$, motivated by the proof of Theorem 2 in our paper and Theorem 4.8 in [1], we conjecture that the generalised Neyman-Pearson lemma leads to $f_{PD}^\star(x,a) = \mathbf{1} [\frac{dP_{a,1}}{dP_{a,0}}(x) \geq \frac{\pi_{a,0}+\tau^\star_a}{\pi_{a,1}}]$, where the threshold $\tau^\star_a$ is selected in a way similar to Eq(3) in the paper.

4. Eigenspace estimation and computational efficiency (W4, Q2, L3)

Eigenspace estimation plays a fundamental role in FDA. We justify the performance of eigenspace estimation from both theoretical and numerical perspectives. In our paper, the theoretical guarantees for eigenfunction and eigenvalue estimation have been established in Lemmas 35 and 36, respectively. These results are optimal, matching the minimax rate established in [6] up to poly-logarithmic factors.

To support our theory, we provide further simulation results in Table 3, where Fair-FLDA refers to the proposed classifier in Section 4; Truth uses true eigenfunctions and eigenvalues; Fourier replaces estimated eigenfunctions with Fourier basis and eigenvalues with covariance projection scores. Overall, disparity control is comparable across methods, with all meeting their fairness criteria. Comparing the results of Fair-FLDA with Truth, we see that the misclassification errors of the proposed classifiers are even smaller than those obtained without eigenspace estimation. This improvement is attributed to the data-adaptive nature of the estimated eigenfunctions, which captures more variance than the fixed true basis. Substituting the estimated eigenfunctions with the pre-specified Fourier basis leads to a noticeable increase in misclassification errors.

Regarding computational complexity, the eigensystem is estimated using a discretisation-based approximation in implementation, with an overall complexity of $O(K^2J)$, where $K$ is the number of evaluated points and $J$ is the truncation level. Due to the inherent smoothness of functional data, choosing $J$ in 50–100 will provide a sufficiently accurate approximation.

Table 3: Median classification error and DO over 500 runs under Gaussian with mean diff $\beta=1.5$ and sample size $n=1000$.

[1] Zeng et al., Bayes-optimal fair classification with linear disparity constraints via pre-, in-, and post-processing. arXiv:2402.02817.

[2] Chen et al., Post-hoc bias scoring is optimal for fair classification. ICLR 2024.

[3] Xian et al., Fair and optimal classification via post-processing. ICML 2023.

[4] Chang and McKeague, Empirical likelihood-based inference for functional means with application to wearable device data. JRSSB, 2022.

[5] Lin et al., Causal inference on distribution functions. JRSSB, 2023.

[6] Wahl. Lower bounds for invariant statistical models with applications to principal component analysis. Ann. Inst. H. Poincaré Probab. Statist, 2022.