Our results resolve an open question by Gillen et al. (2018) by showing that online learning under an unknown individual fairness constraint is possible even without assuming a strong parametric form of the

Section A provides a proof that isometry preserves angles. Section D lists the grid considered for hyper-parameters. T is an isometry iff it preserves inner products. Suppose T is an isometry. Conversely, if T preserves inner products, then nullT (v w),T ( v w) null = null v w,v w null, which implies null T ( v w)null = null v w null, and since T is linear, nullT (v) T ( w) null = null v w null .

Recovering a low-CP-rank tensor from noisy linear measurements is a central challenge in high-dimensional data analysis, with applications spanning tensor PCA, tensor regression, and beyond. We exploit the intrinsic geometry of rank-one tensors by casting the recovery task as an optimization problem over the Segre manifold, the smooth Riemannian manifold of rank-one tensors. This geometric viewpoint yields two powerful algorithms: Riemannian Gradient Descent (RGD) and Riemannian Gauss-Newton (RGN), each of which preserves feasibility at every iteration. Under mild noise assumptions, we prove that RGD converges at a local linear rate, while RGN exhibits an initial local quadratic convergence phase that transitions to a linear rate as the iterates approach the statistical noise floor. Extensive synthetic experiments validate these convergence guarantees and demonstrate the practical effectiveness of our methods.

We will employ a convenient "empirical measure" notation to concisely represent finite-sample and population quantities in the analysis.