Fair Representation Learning Beyond Euclidean Geometry

Fair representation learning enforces that different demographic groups follow similar distributions in representation space, with similarity measured by a distributional discrepancy computed in Euclidean coordinates. This implicitly assumes the representation space is flat, which is an assumption violated whenever representations are embedded in curved spaces such as hyperbolic space, where the ambient Euclidean distance can differ arbitrarily from the geodesic distance the model actually uses. We show the mismatch changes the conclusion rather than the constant: two group distributions can be nearly matched under a Euclidean discrepancy while remaining well separated on the underlying manifold, so the resulting fairness guarantees do not transfer. We therefore measure discrepancy intrinsically, on Hadamard manifolds, non-positively curved spaces that include Euclidean and hyperbolic geometry as special cases and whose unique geodesics make projection well defined. Our Cartan–Hadamard Sliced-Wasserstein (CHSW) distance aggregates one-dimensional Wasserstein distances along geodesic projections, avoiding full optimal transport on the manifold while respecting its curvature, and is differentiable and cheap enough to serve as a training regularizer. We prove that minimizing it bounds demographic parity violation, and show on standard benchmarks that it improves fairness substantially at competitive accuracy.