Robust Learning Meets Quasar-Convex Optimization: Inexact High-Order Proximal-Point Methods

Robust representation alignment can lead to nonsmooth and nonconvex objectives when paired features are corrupted or mismatched. We show that several realizable alignment models built from stabilized capped and anisotropic losses are strongly quasar-convex. Motivated by this geometry, we study Inexact HiPPA, an inexact high-order proximal-point method with regularization order $p>1$. Under summable model-value inexactness, bounded HiPPA sequences converge in objective value to the global minimum, while strong quasar-convexity makes boundedness and uniqueness automatic. Under local stability and suitable forcing of the inexactness, the method converges linearly for $1<p\le 2$ and superlinearly with $Q$-order $p-1$ for $p>2$. A capped-quadratic alignment model verifies these local assumptions explicitly. On a controlled CIFAR-100 feature experiment, the mean outer iteration count drops from $14.8$ for $p=2$ to $4$ for $p=3$ and $3$ for $p=4$. A second corrupted-feature experiment uses a computable residual-and-descent rule and shows the clearest benefit from capped $p=3$ fitting under severe replacement corruption.