  Providing various machine learning (ML) applications in the real world,
concerns about discrimination hidden in ML models are growing, particularly in
high-stakes domains. Existing techniques for assessing the discrimination level
of ML models include commonly used group and individual fairness measures.
However, these two types of fairness measures are usually hard to be compatible
with each other, and even two different group fairness measures might be
incompatible as well. To address this issue, we investigate to evaluate the
discrimination level of classifiers from a manifold perspective and propose a
"harmonic fairness measure via manifolds (HFM)" based on distances between
sets. Yet the direct calculation of distances might be too expensive to afford,
reducing its practical applicability. Therefore, we devise an approximation
algorithm named "Approximation of distance between sets (ApproxDist)" to
facilitate accurate estimation of distances, and we further demonstrate its
algorithmic effectiveness under certain reasonable assumptions. Empirical
results indicate that the proposed fairness measure HFM is valid and that the
proposed ApproxDist is effective and efficient.
