On Fairness of Medical Image Classification
with Multiple Sensitive Attributes via
Learning Orthogonal Representations

Notations. We consider group fairness in this work. While keeping the model utility for the privileged group, group fairness articulates the equality of some statistics like predictive rate between certain groups. Considering a binary classification problem with column vector inputs $x\in\mathcal{X}$, labels $y\in\mathcal{Y}=\{0,1\}$. Multi-sensitive attributes $a\in\mathcal{A}$ is vector of $m$ attributes sampled from the conjunction of binary attributes, i.e., Cartesian product of sensitive attributes $\mathcal{A}=\prod_{i\in[m]}A_{i}$^***$[m]=\{0,1,..,m\}$, where $A_{i}\in\{0,1\}$ is the $i$-th sensitive attribute. Our training data consist of tuples $\mathcal{D}=\{(x,y,a)\}$. We denote the classification model $f(x)=h_{T}(\phi_{T}(x))$ that predicts a class label $\widehat{y}$ given $x$, where $\phi_{T}:\mathcal{X}\mapsto\operatorname{{\mathbb{R}}}^{d}$ is a feature encoder for target embeddings, and $h_{T}:\operatorname{{\mathbb{R}}}^{d}\mapsto\operatorname{{\mathbb{R}}}$ is a scoring function. Similarly, we consider a sensitive attribute model $g(x)=\{h_{A_{1}}(\phi_{A}(x)),...,h_{A_{m}}(\phi_{A}(x))\}$ that predicts sensitive attributes associated with input $x$. Given the number of samples $n$, the input data representation is $X=[x_{1},\dots,x_{n}]$ and we denote the feature representation $Z_{T}=\phi_{T}(X)$, $Z_{A}=\phi_{A}(X)$ $\in\operatorname{{\mathbb{R}}}^{d\times n}$.

First, we focus on the column space of the target and the sensitive attribute representations. Column space orthogonality aims to learn target representations $Z_{T}$ that fulfill the following two aims: 1) have the least projection onto the sensitive space $\mathcal{S}_{A}$ and 2) preserve the representation power to predict $Y$.

Denote the target representation $Z_{T}=[\widetilde{z}_{T}^{1},\widetilde{z}_{T}^{2},\dots,\widetilde{z}_{T}^{n}]$ and the sensitive attribute representation $Z_{A}=[\widetilde{z}_{A}^{1},\widetilde{z}_{A}^{2},\dots,\widetilde{z}_{A}^{n}]$, where $\widehat{z}^{i}\in\operatorname{{\mathbb{R}}}^{d\times 1}$ is a column vector for $i\in[n]$, we represent the column space for $Z_{T}$ and $Z_{A}$ as $\operatorname{{\mathcal{S}}}_{T}={\rm span}(Z_{T})$ and $\operatorname{{\mathcal{S}}}_{A}={\rm span}(Z_{A})$ respectively. Aim 1 can be achieved by forcing $\operatorname{{\mathcal{S}}}_{T}=\operatorname{{\mathcal{S}}}_{A}^{\perp}$. Although both $\widetilde{z}_{T},\widetilde{z}_{A}\in\operatorname{{\mathbb{R}}}^{d}$, their coordinates may not be aligned as they are generated from two separate encoders. As a result, if $d\ll\infty$, then there is no straightforward way to achieve $\operatorname{{\mathcal{S}}}_{T}\perp\operatorname{{\mathcal{S}}}_{A}$ by directly constraining $\widetilde{z}^{i}_{T},\widetilde{z}^{j}_{A}$ (e.g., forcing $(\widetilde{z}^{i}_{T})^{\top}\widetilde{z}^{j}_{A}=0$). Aim 2 can be achieved by seeking a low-rank representation $\widetilde{\operatorname{{\mathcal{S}}}}_{A}$ for $\operatorname{{\mathcal{S}}}_{A}$, whose rank is $k$ such that $k\ll d$, because we have ${\rm rank}(\operatorname{{\mathcal{S}}}_{T})+{\rm rank}(\operatorname{{\mathcal{S}}}_{A})=d$ if $\operatorname{{\mathcal{S}}}_{T}=\operatorname{{\mathcal{S}}}_{A}^{\perp}$ holds. Then $\operatorname{{\mathcal{S}}}_{A}^{\perp}$ would be a high-dimensional space with sufficient representation power for target embeddings. This is especially important when we face multiple sensitive attributes, as the total size of the space is $d$, and increasing the number of sensitive attributes would limit the capacity of $\operatorname{{\mathcal{S}}}_{T}$ to learn predictive $\widetilde{z}_{T}$. To this end, we first propose to find the low rank sensitive attribute representation space $\widetilde{\operatorname{{\mathcal{S}}}}_{A}$, and then encourage $Z_{T}$ to be in $\widetilde{\operatorname{{\mathcal{S}}}}_{A}$’s complement $\widetilde{\operatorname{{\mathcal{S}}}}_{A}^{\perp}$.

Then, we study the row space of target and sensitive attribute representations. Row space orthogonality aims to learn target representations $Z_{T}$ that have the least projection onto the sensitive row space $\widehat{\operatorname{{\mathcal{S}}}}_{A}$. In other words, we want to ensure orthogonality on each feature dimension between $Z_{T}$ and $Z_{A}$. Denote target representation $Z_{T}=[\widehat{z}_{T}^{1};\widehat{z}_{T}^{2};\dots;\widehat{z}_{T}^{d}]$ and sensitive attribute representation $Z_{A}=[\widehat{z}_{A}^{1};\widehat{z}_{A}^{2};\dots;\widehat{z}_{A}^{d}]$, where $\widehat{z}^{i}\in\operatorname{{\mathbb{R}}}^{1\times n}$ is a row vector for $i\in[d]$. We represent row space for target representations and sensitive attribute representations as $\widehat{\operatorname{{\mathcal{S}}}}_{T}={\rm span}(Z_{T}^{\top})$ and $\widehat{\operatorname{{\mathcal{S}}}}_{A}={\rm span}(Z_{A}^{\top})$ correspondingly. As the coordinates (i.e., the index of samples) of $\widehat{z}_{A}$ and $\widehat{z}_{T}$ are aligned, forcing $\widehat{\operatorname{{\mathcal{S}}}}_{T}\perp\widehat{\operatorname{{\mathcal{S}}}}_{A}$ can be directly applied by pushing $\widehat{z}_{T}^{i}$ and $\widehat{z}_{A}^{j}$ to be orthogonal for arbitrary $i,j\in[d]$.

Unlike column space, the orthogonality here won’t affect the utility, since the row vector $\widehat{z}_{T}$ is not directly correlated to the target $y$. To be specific, we let pair-wise row vectors $Z_{T}=[\widehat{z}_{T}^{1},\widehat{z}_{T}^{2},\dots,\widehat{z}_{T}^{d}]$ and $Z_{A}=[\widehat{z}_{A}^{1},\widehat{z}_{A}^{2},\dots,\widehat{z}_{A}^{d}]$ have a zero inner dot-product. Then for any $i,j\in[d]$, we try to minimize $<\widehat{z}_{T}^{i},\widehat{z}_{A}^{j}>$. Here we slightly modify the orthogonality by extra subtracting the mean vector $\mu_{A}$ and $\mu_{T}$ from $Z_{A}$ and $Z_{T}$ respectively, where $\mu=\mathbb{E}_{i\in[d]}\widehat{z}^{i}\in\mathbb{R}^{1\times n}$. Then orthogonality loss will naturally be integrated into a covariance loss:

In this way, the resulting loss encourages each feature of $Z_{T}$ to be independent of features in $Z_{A}$ thus suppressing the sensitive-encoded covariances that cause the unfairness.

In this section, we introduce the overall training schema as shown in Fig. 2 (c). For the sensitive branch, we noticed that training a shared encoder can pose a risk of sensitive-related features being used for the target classification [4], and we pre-train separate $\left\{\phi_{A},h_{A_{1}},...,h_{A_{m}}\right\}$ for multiple sensitive attributes using the sensitive loss as $L_{sens}=\frac{1}{m}\sum_{i\in[m]}L_{A_{i}}$. Here we use cross-entropy loss as $L_{A_{i}}$ for the $i$-th sensitive attribute. Hence $p(z_{A}|x)$ and $p(a|z_{A})$ in Eq. (4) can be obtained. Then, the multi-sensitive space $\operatorname{{\mathcal{S}}}_{A}$ is constructed as in Section 2.2 over the training data. For the target branch, we use cross-entropy loss as our classification objective $L_{T}$ to supervise the training of $\phi_{T}$ and $h_{T}$ and estimate $p(z_{T}|x)$ and $p(y|z_{T})$ in Eq. (4) respectively. Here we do not make additional constraints to $L_{T}$, which means it can be replaced by any other task-specific losses. At last, we apply our column and row orthogonality losses $L_{corth}$ and $L_{rorth}$ to representations as introduced in Section 2.2 and Section 2.3 along with detached $\operatorname{{\mathcal{S}}}_{A}$ and $Z_{A}$ to approximate independence between $p(z_{A}|x)$ and $p(z_{T}|x)$. The overall target objective is given as:

where $\lambda_{c}$ and $\lambda_{r}$ are hyper-parameters to weigh orthogonality and balance fairness and utility.

Dataset. We adopt CheXpert dataset [7] to predict Pleural Effusion in chest X-rays, as it’s crucial for chronic obstructive pulmonary disease diagnosis with high incidence. Subgroups are defined based on the following binarized sensitive attributes: self-reported race and ethnicity, sex, and age. Note that data bias (positive rate gap) is insignificant in the original dataset (see Table 1, row ’original’). To demonstrate the effectiveness of bias mitigation methods, we amplify the data bias by (1) firstly dividing the data into different groups according to the conjunction of multi-sensitive labels; (2) secondly calculating the positive rate of each subgroup; (3) sampling out patients and increase each subgroup’s positive rate gap to 0.12 (see Table 1, row ‘augmented’). We resize all images to $224\times 224$ and split the dataset into a 15% test set, and an 85% 5-fold cross-validation set.

Quantitative results. We summarize quantitative comparisons in Table 2. It can be observed that all the bias mitigation methods can improve fairness compared to ERM [17] at the cost of utility. While ensuring considerable classification accuracy, FCRO achieves significant fairness improvement both jointly and individually, demonstrating the effectiveness of our representation orthogonality motivation. To summarize, compared with the best performance in each metric, FCRO reduced classification disparity on subgroups with joint $\operatorname*{\mathrm{\Delta}}_{\mathrm{AUC}}$ by 2.7% and joint $\operatorname*{\mathrm{\Delta}}_{\mathrm{ED}}$ by 2.3% respectively, and experienced 0.5% $\operatorname*{\mathrm{\Delta}}_{\mathrm{AUC}}$ and 0.4% $\operatorname*{\mathrm{\Delta}}_{\mathrm{ED}}$ boosts regarding the average of three sensitive attributes. As medical applications are sensitive to classification thresholds, we further show calibration curves with the mean and standard deviation of subgroups defined on the conjunction of multiple sensitive attributes in Fig. 3 (a). The vanilla ERM [17] suffers from biased calibration among subgroups. Fairness algorithms can help mitigate this, while FCRO shows the most harmonious deviation and the most trustworthy classification. Qualitative results. We present the class activation map [2] in Fig. 3 (b). We observe that the vanilla ERM [17] model tends to look for sensitive evidence outside the lung regions, e.g., breast, which threatens unfairness. FCRO focuses on the pathology-related part only for fair Pleural Effusion classification, which visually confirms the validity of our method.