Source-linked AI summary
Conditioning and interpolation error bounds for second-order Stiefel retractions with closed-form inverses
Rasmus Jensen, Ralf Zimmermann
TL;DR
Interpolation on the Stiefel manifold needs efficient coordinate maps with reliable error behavior and computable inverses. The paper analyzes polar-light and canonical Cayley retractions, derives conditioning-based error bounds, and applies them to Hermite interpolation. The retractions are well-conditioned forward, while inverse maps require local domains and, potentially, isometric data centering; retraction-based interpolation is competitive with Riemannian normal coordinates.
Problem
The paper addresses how to obtain explicit and stable interpolation error bounds on the Stiefel manifold when using second-order retractions with closed-form inverses.
Method
The paper analyzes the conditioning of polar-light and canonical Cayley retractions and inverses, uses isometric data centering, and applies the maps to Hermite interpolation.
Results
The forward retractions are well-conditioned, with condition-number bounds K = 2 for canonical Cayley and K = 3 for polar-light, while inverse maps can be arbitrarily ill-conditioned.
Takeaways & Limitations
Retraction-based interpolation preserves Euclidean interpolation order and can yield smaller errors than Riemannian normal coordinates, although coordinate choice is problem-dependent.
Takeaways & Limitations
The inverse canonical Cayley and polar-light retractions are only locally defined and require conditions such as avoiding eigenvalue −1; data conditioning can also require preprocessing.
Abstract
from arXiv · showhide
Retractions provide a computationally efficient alternative to the Riemannian exponential and logarithm maps for practical data-processing tasks on manifolds. In particular, second-order retractions with closed-form inverse are well-suited for interpolation problems on manifolds. On the Stiefel manifold of orthogonal frames, there are only two retractions of this type: the Cayley retraction, which is second-order accurate under the canonical metric, and the recently proposed polar-light retraction, which is second-order accurate under the Euclidean metric. In this paper, we study the properties of these maps in the context of interpolation on the Stiefel manifold. To obtain explicit interpolation error bounds, we examine the conditioning of the retraction maps and their inverses. We show that the retractions are well-conditioned, and we derive interpolation error bounds similar to those of classical Euclidean interpolation. The inverse retractions are not well-conditioned in general, and we discuss how data can be mapped via an isometric group action to ensure stable computations. As with all retractions on compact manifolds, the inverse canonical Cayley retraction and the inverse polar-light retraction exist only locally, and we construct normal neighborhoods around any point in which either the inverse Cayley retraction or the invese polar-light retraction are guaranteed to be computable. As an application of the retraction maps, we consider Hermite interpolation, where the objective is to reproduce both sampled function values and derivative information. A numerical example demonstrates that retraction-based interpolation is competitive with classical methods based on Riemannian normal coordinates.
1. Introduction.
The paper develops interpolation methods on the Stiefel manifold using second-order retractions and their inverses, focusing on conditioning, explicit error bounds, and Hermite interpolation. It combines retraction-based coordinates with data centering and numerical evaluation.
- Motivation: Classical manifold interpolation maps data to a tangent space, applies a Euclidean interpolation scheme, and maps the result back to the manifold.This approach supports Lagrange, Hermite, and spline methods.
- Related work: Riemannian normal coordinates support interpolation error bounds through distance estimates based on sectional curvature.Prior work establishes preservation of asymptotic order under suitable tangent-space interpolation schemes.
- Contributions: The paper derives explicit interpolation error bounds for the polar-light and canonical Cayley retractions and their inverses.Forward mappings are well-conditioned, whereas inverse mappings can be arbitrarily ill-conditioned.
- Contributions: An isometric data-centering scheme exploits the homogeneity of the Stiefel manifold to move data into a numerically favorable local domain.The paper also constructs explicit normal neighborhoods where both inverse retractions are guaranteed to be well-defined.
- Hermite interpolation: Hermite interpolation maps derivative information with differentials of the inverse retractions rather than finite-difference approximation.The paper avoids the costly Fréchet derivative of the matrix logarithm by modifying the polar-light formulation.
- Evaluation: The paper includes a numerical experiment and compares retraction-based interpolation with Riemannian normal-coordinate methods.The study is organized around background, error analysis, Hermite interpolation, and numerical evaluation.
2. Background.
The background introduces matrix and Stiefel-manifold geometry, Cayley and polar-light coordinate maps, group-action preprocessing, and conditioning concepts. These constructions provide closed-form inverses and the coordinate framework used for interpolation analysis.
- Matrix preliminaries: Cayley transformations and inverses approximate the matrix exponential and logarithm with second-order Taylor accuracy.These approximations motivate their use as structure-preserving alternatives.
- Polar-light retraction: Polar-light coordinates define a second-order retraction under the Euclidean metric and provide a closed-form inverse.The matrix exponential distinguishes this map from the standard polar factor retraction.
- Group action: Homogeneous group actions move any base point to a reference point and transform tangent vectors into block coordinates.The transformation can be implemented efficiently using a low-rank modification of the identity.
- Cayley retraction: The Cayley retraction replaces the matrix exponential with the Cayley transformation and is second-order under the canonical metric.Its inverse requires only inversion of a p×p matrix.
- Coordinate structure: The polar-light and canonical Cayley retractions are presented as the only known second-order Stiefel retractions with closed-form inverses.Their local-coordinate representations relate Stiefel points to skew(p) × R^(n−p)×p.
- Conditioning: The polar decomposition separates a matrix into orthogonal and symmetric positive-definite factors, whose conditioning differs substantially.The positive-definite factor is well-conditioned, while the orthogonal factor becomes ill-conditioned near rank deficiency.
3. Error analysis for data processing and interpolation error bounds.
The paper analyzes a three-step retraction-based interpolation pipeline on the Stiefel manifold, deriving conditioning results and explicit error bounds for polar-light and Cayley coordinates. Retractions preserve interpolation order with controlled amplification, while inverse maps require locality and, potentially, isometric data centering for stable computation.
- Interpolation pipeline: Retraction-based interpolation maps manifold data to local coordinates, applies Euclidean interpolation, and maps the result back to the manifold.The parameterization choice determines the interpolant and its error constant.
- Conditioning: The polar-light coordinate map requires polar decomposition of the upper block and matrix-logarithm evaluation, whose stability depends on singular values and eigenvalue placement.The polar factor becomes problematic near rank deficiency or when eigenvalues approach the negative real axis.
- Conditioning: An isometric group action moves data to a neighborhood of the canonical point E while preserving pairwise Euclidean distances.The transformation can be constructed with Householder QR and evaluated in O(np^2) FLOPS.
- Error bounds: The forward polar-light map satisfies ∥φE(X) −φE(Y)∥F ≤ 3 · ∥X −Y ∥F for sufficiently close local coordinates.This yields a bound relating coordinate interpolation errors to manifold errors.
- Error bounds: The resulting manifold interpolation error preserves the Euclidean interpolation order, with constants amplified by at most a factor of 3.Classical Lagrange and Hermite error bounds can therefore be transferred under the stated neighborhood conditions.
- Inverse retractions: Inverse retractions are only locally defined: Cayley inversion excludes eigenvalue −1, while polar-light inversion requires the corresponding polar factor to exclude −1.Explicit normal neighborhoods guarantee computability, and the inverse polar-light construction generalizes across St(n, p).
4. Hermite interpolation.
The paper develops Hermite interpolation on the Stiefel manifold by mapping point and derivative data through retraction-based local coordinates, then applying Euclidean Hermite interpolation. A numerical experiment compares Cayley and polar-light coordinates with Riemannian normal coordinates across two Stiefel dimensions.
- Hermite interpolation setup: Hermite interpolation combines sampled function values and derivative information at two points using standard cubic Hermite polynomials.The manifold procedure maps both point and derivative data into local coordinates before interpolation.
- Retraction-based coordinates: Point data are mapped to local coordinate matrices, while derivative data are mapped bijectively using the differential of the coordinate map.The coordinate construction uses a chart around a reference point and maps the interpolated result back to the Stiefel manifold.
- Data centering: When data lie outside the chosen neighborhood, an isometric group action moves the data and derivative information into a neighborhood where the Hermite procedure applies.The final interpolant is obtained by applying the inverse group action after interpolation.
- Inverse Cayley coordinates: The inverse Cayley formulation maps two point and derivative samples into a selected reference tangent space before applying the Euclidean Hermite scheme.Using the tangent space at either sampled point is permitted; the example selects the tangent space at the first point.
- Numerical example: The numerical example interpolates a QR-derived Stiefel curve on St(80, 40) and St(800, 40) using polar-light, canonical Cayley, and Riemannian normal coordinates.For n ≫ p, polar-light outperforms Cayley and normal coordinates in manifold interpolation error; near n = 2p = 80, Cayley has the smallest relative errors and polar-light the largest.
5. Concluding remarks.
The concluding analysis establishes local computability and conditioning guarantees for the two inverse retractions, then relates their coordinate conditioning to interpolation-error amplification. The numerical example shows that retraction-based interpolation can outperform Riemannian normal coordinates, although the best coordinates depend on the problem.
- Inverse-map conditions: Both inverse retractions require that the relevant matrix block avoid the eigenvalue −1 condition.For the polar-light and canonical Cayley retractions, this requirement concerns the polar factor ˆU^T U and the corresponding upper diagonal block.
- Conditioning and stabilization: A group action can move data to a neighborhood of E whose upper diagonal block is perfectly conditioned, countering inverse-retraction ill-conditioning.This provides a practical stabilization mechanism for computations involving inverse retractions.
- Normal neighborhoods: Explicit normal neighborhoods are constructed around every Stiefel point where both inverse retractions are guaranteed to be well-defined.These neighborhoods are images of open tangent-space balls under the Riemannian exponential map, with conservative radii specified in the paper.
- Interpolation-error bounds: K = 2 for the canonical Cayley retraction and K = 3 for the polar-light retraction bound asymptotic amplification of Euclidean interpolation error on the manifold.The bounds connect local coordinate interpolation accuracy to the resulting manifold interpolation error.
- Numerical conclusion: Retraction-based interpolation can yield smaller errors than Riemannian normal coordinates, but the optimal coordinate choice is problem-dependent.The numerical conclusion is comparative rather than universal across the tested settings.