Stable High-Order Interpolation on the Grassmann Manifold by Maximum-Volume Coordinates and Arnoldi Orthogonalization

TL;DR

This paper introduces a stable high-order interpolation method on the Grassmann manifold using maximum-volume coordinates and Arnoldi orthogonalization, reducing computational costs and improving stability.

Contribution

It combines MV coordinates with Arnoldi-orthogonalized polynomial bases to enhance stability and efficiency in high-order Grassmann manifold interpolation.

Findings

Achieves highly accurate high-degree polynomial interpolation

Reduces computational overhead compared to SVD-based methods

Provides theoretical stability bounds for the proposed framework

Abstract

High-order interpolation on the Grassmann manifold $\Gr(n, p)$ is often hindered by the computational overhead and derivative instability of SVD-based geometric mappings. To solve the challenges, we propose a stabilized framework that combines Maximum-Volume (MV) local coordinates with Arnoldi-orthogonalized polynomial bases. First, manifold data are mapped to a well-conditioned Euclidean domain via MV coordinates. The approach bypasses the costly matrix factorizations inherent to traditional Riemannian normal coordinates. Within the coordinate space, we use the Vandermonde-with-Arnoldi (V+A) method for Lagrange interpolation and its confluent extension (CV+A) for derivative-enriched Hermite interpolation. By constructing discrete orthogonal bases directly from the parameter nodes, the solution of ill-conditioned linear system is avoided. Theoretical bounds are established to verify the stability of the geometric mapping and the polynomial approximation. Extensive numerical experiments demonstrate that the proposed MV-(C)V+A framework can produce highly accurate approximation in high-degree polynomial interpolation.