Conditioning and interpolation error bounds for second-order Stiefel retractions with closed-form inverses

What the math gives to ML

The paper gives explicit Cayley and polar-light retractions on the Stiefel manifold together with conditioning bounds for their coordinate maps and local inverses. This suggests replacing penalty-based orthogonality or repeated QR projections in neural layers with an exactly orthogonality-preserving Cayley update whose conditioning can be monitored from singular values of the tangent-coordinate operator. The most direct transfer is an orthogonal linear layer or adapter trained on the Stiefel manifold, where the retraction provides a cheap second-order update and the paper's spectral bound supplies a falsifiable stability diagnostic.