Discrimination mitigation within machine learning (ML) models could be complicated because multiple factors may be interwoven hierarchically and historically. Yet few existing fairness measures can capture the discrimination level within ML models in the face of multiple sensitive attributes (SAs). To bridge this gap, we propose a fairness measure based on distances between sets from a manifold perspective, named as 'Harmonic Fairness measure via Manifolds (HFM)' with two optional versions, which can deal with a fine-grained discrimination evaluation for several SAs of multiple values. Because directly computing HFM may be costly, to accelerate its subprocedure -- the computation of distances of sets, we further propose two approximation algorithms named 'Approximation of distance between sets for one sensitive attribute with multiple values (ApproxDist)' and 'Approximation of extended distance between sets for several sensitive attributes with multiple values (ExtendDist)' to respectively resolve bias evaluation of one single SA with multiple values and that of several SAs with multiple values. Moreover, we provide an algorithmic effectiveness analysis for ApproxDist under certain assumptions to explain how well it could work. The empirical results demonstrate that our proposed fairness measure HFM is valid and approximation algorithms (i.e. ApproxDist and ExtendDist) are effective and efficient.

翻译：机器学习（ML）模型中的歧视缓解可能较为复杂，因为多种因素可能在历史进程中以层级方式交织在一起。然而，现有公平性度量方法中，鲜有能够捕捉面对多个敏感属性（SAs）时模型内部歧视水平的。为弥补这一差距，我们提出一种基于流形视角下集合间距离的公平性度量方法，命名为“基于流形的调和公平性度量（HFM）”，并提供两个可选版本，该方法能够处理具有多取值的多个敏感属性的细粒度歧视评估。由于直接计算HFM可能代价高昂，为加速其子过程——集合间距离的计算，我们进一步提出了两种近似算法，分别命名为“针对单一多值敏感属性的集合间距离近似算法（ApproxDist）”和“针对多个多值敏感属性的扩展集合间距离近似算法（ExtendDist）”，以分别解决单一多值敏感属性和多个多值敏感属性的偏见评估问题。此外，我们在特定假设下为ApproxDist提供了算法有效性分析，以解释其可能的工作效果。实证结果表明，我们提出的公平性度量HFM是有效的，且近似算法（即ApproxDist和ExtendDist）是高效且有效的。