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Geodesic Interpolation on the Grassmann Manifold: GLERP and Recursive GIDER Interpolants

This paper introduces GLERP, a geodesic-based interpolation method for the Grassmann manifold that is invariant to basis ambiguity, and extends it into a recursive higher-order scheme called GIDER $_n$ that achieves optimal convergence rates while outperforming or matching existing tangent-space and projection-matrix approaches.

Original authors: Shingyu Leung

Published 2026-06-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shingyu Leung

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to draw a smooth line connecting a series of dots. In the flat world of a standard piece of paper (Euclidean space), this is easy: you just draw a straight line between them. But what if your dots aren't on a flat sheet? What if they are scattered on the surface of a giant globe? If you try to draw a straight line through the Earth's core to connect them, you miss the surface entirely. Instead, you must follow the curve of the globe, taking the shortest path along the surface. This is called a "geodesic."

This paper is about doing that same kind of "curved line drawing," but for a very specific and abstract type of data: subspaces.

The Problem: The "Basis" Confusion

Imagine you have a collection of photos. Each photo shows a specific group of people.

Photo A shows three people standing in a row.
Photo B shows the exact same three people, but they've swapped places or rotated slightly.

In the world of standard math, these look like two different sets of numbers. But in the world of this paper, they represent the same thing: the "group" or the "space" occupied by those three people. The specific order or orientation (the "basis") doesn't matter; only the group itself matters.

The challenge is: How do you smoothly transition from "Group A" to "Group B" without getting confused by the fact that the people inside the group might have shuffled their positions? If you just average the numbers (the standard way), you might accidentally create a "ghost group" that doesn't actually exist.

The Solution: GLERP (The Curved Line)

The authors introduce a tool called GLERP (Grassmann Linear Interpolation). Think of this as the "great circle" route for these groups.

The Analogy: Imagine you are a pilot flying from New York to London. You don't fly in a straight line through the Earth; you fly along the curve of the planet. GLERP is the mathematical equivalent of that flight path. It connects two "groups" (subspaces) by following the shortest, smoothest curve on the abstract "shape" where all possible groups live (called the Grassmann manifold).
The Magic: It ignores the confusing "shuffling" of the people inside the group. It only cares about the shape of the group itself.

The Upgrade: GIDER (The Smooth Curve)

GLERP is great for connecting just two points. But what if you have ten points and want a smooth, high-quality curve that passes through all of them?

The authors build a more advanced tool called GIDER.

The Analogy: Imagine you are building a roller coaster track. GLERP is just a single straight segment between two supports. GIDER is the process of taking those segments and smoothing them out into a perfect, high-speed curve that hits every single support point exactly.
How it works: It uses a clever recursive trick (like a fractal pattern). It takes two small GLERP curves, connects them with another GLERP curve, and repeats this process until it builds a complex, smooth path that fits the data perfectly.

Why It Matters (The "Basis Invariance")

The paper emphasizes that this method is basis-invariant.

The Metaphor: Imagine you are describing a room. You can describe it by listing the furniture in order: "Sofa, Table, Chair." Or you can say: "Chair, Sofa, Table."
A "naive" math method might think these are two different rooms because the order is different.
GLERP and GIDER are smart enough to know: "Wait, it's the same room, just described differently." They ignore the order and focus only on the room's layout. This makes the results stable and reliable, no matter how the data is organized.

The Results

The authors tested their new method against two other popular ways of doing this:

Tangent-Space Interpolation: This is like flattening the globe onto a map, drawing a line, and then rolling it back up. It works well if the area is small, but can get messy if the area is large.
Projection-Matrix Interpolation: This is like trying to draw the curve on a shadow cast by the objects. It's easy to do but sometimes distorts the shape.

The Verdict:
In smooth, well-behaved situations, the new GIDER method and the "flattened map" method (Tangent-Space) give almost identical results. However, GIDER has a special advantage: it doesn't need to pick a "center point" to flatten the map around. It builds the curve directly on the shape itself. This makes it a more natural and robust way to handle complex, curved data.

The Future: Avoiding "Wobbles"

The paper briefly mentions that if the data has a sharp kink or a sudden turn, high-order curves can sometimes get "wobbly" (oscillate). The authors suggest that their recursive GIDER structure could be adapted to create a "smart" version (called GENO) that automatically chooses the smoothest path and avoids these wobbles, similar to how a driver might choose a smoother lane to avoid a pothole.

In Summary:
This paper provides a new, mathematically rigorous way to draw smooth lines between complex data shapes (subspaces). It ensures that the "shuffling" of the data's internal parts doesn't mess up the result, and it builds these lines by following the natural curves of the mathematical space they live in, rather than forcing them into a flat, artificial grid.

Technical Summary: Geodesic Interpolation on the Grassmann Manifold

Problem Statement
The paper addresses the challenge of interpolating data that resides on the Grassmann manifold, $\text{Gr}(r, m)$ , the space of all $r$ -dimensional linear subspaces of $\mathbb{R}^m$ . In many scientific computing and data analysis applications, matrix-valued data represent subspaces rather than fixed bases. Consequently, two matrices $Y$ and $YR$ (where $R \in O(r)$ ) represent the same point on the manifold. Standard Euclidean interpolation methods fail in this context because they are not invariant under this right orthogonal action; interpolating matrix entries directly destroys the geometric structure. The goal is to develop an interpolation framework that is intrinsic to the manifold, basis-invariant, and capable of achieving high-order accuracy for smooth subspace-valued curves.

Methodology
The authors propose a framework built upon the intrinsic geometry of the Grassmann manifold, specifically utilizing the exponential and logarithm maps to define geodesic paths.

GLERP (Grassmann Linear Interpolation):
The foundational building block is GLERP, a Grassmann analogue of Spherical Linear Interpolation (SLERP). For two nearby subspaces $Y_0$ and $Y_1$ , GLERP is defined as:
$\text{GLERP}(Y_0, Y_1; \tau) = \text{Exp}_{Y_0}(\tau \text{Log}_{Y_0}(Y_1)), \quad 0 \le \tau \le 1$
This follows the constant-speed geodesic joining the two points. The method relies on the singular value decomposition (SVD) of the tangent vector derived from the logarithm map to compute the interpolation explicitly.
GIDER $_n$ (Recursive Geodesic Interpolation):
To achieve higher-order accuracy, the authors define a recursive family of interpolants, GIDER $_n$ , based on Neville's algorithm. In the standard Euclidean Neville algorithm, affine combinations are used; here, every affine step is replaced by a GLERP operation.
$I^k_j(t) = \text{GLERP}\left(I^{k-1}_j(t), I^{k-1}_{j+1}(t); \lambda^k_j(t)\right)$
where $\lambda^k_j(t)$ is the standard barycentric coordinate. The final interpolant is $I^n_0(t)$ . This construction allows for geodesic extrapolation (where the parameter $\lambda$ may fall outside $[0, 1]$ ) provided the subspaces remain within the injectivity neighborhood (i.e., principal angles remain strictly less than $\pi/2$ ).
Error Measurement:
The paper emphasizes basis-invariant error metrics. It utilizes:
- Intrinsic Grassmann Distance: Based on principal angles $\theta_j$ , $d_{\text{Gr}} = (\sum \theta_j^2)^{1/2}$ .
- Projection-Matrix Error: Based on the Frobenius norm of the difference between orthogonal projectors, $\|P - \hat{P}\|_F$ .
- Procrustes-Aligned Error: Minimizing the Frobenius distance over all orthogonal transformations.

Key Contributions

Definition of GLERP: The paper formally introduces GLERP as the constant-speed geodesic interpolation on $\text{Gr}(r, m)$ and proves it is second-order accurate for smooth curves under standard normal-neighborhood assumptions.
Recursive GIDER $_n$ Construction: The authors define a recursive, higher-order interpolation scheme that exactly matches $n+1$ data points. They prove that for $C^{n+1}$ curves, GIDER $_n$ achieves local $(n+1)$ -st order accuracy.
Admissibility Conditions: A quantitative sufficient condition is provided to ensure that the recursive construction (including intermediate extrapolations) remains within the injectivity neighborhood, preventing the principal angles from approaching the cut locus ( $\pi/2$ ). This is guaranteed for sufficiently dense sampling of smooth curves.
Comparative Analysis: The paper provides a detailed comparison between GIDER $_n$ $_{n}$ , Tangent-Space Interpolation (TSI), and Projection-Matrix Interpolation (PROJ).
- TSI maps data to a single reference tangent space, interpolates, and maps back. It is efficient but depends on the choice of reference.
- PROJ interpolates projectors in the ambient space and projects back via eigenvectors. It is extrinsic and basis-invariant but sensitive to spectral gaps.
- GIDER $_n$ is fully intrinsic and recursive, avoiding commitment to a single tangent space, making it suitable for ENO (Essentially Non-Oscillatory) extensions.

Results
Numerical experiments on $\text{Gr}(2, 5)$ using a smooth, non-geodesic curve confirm the theoretical convergence rates:

For interpolation order $n$ , the error scales as $O(h^{n+1})$ .
GIDER $_n$ and TSI yield nearly indistinguishable errors in smooth, local regimes, confirming that both reduce locally to polynomial interpolation in tangent coordinates.
PROJ achieves the same formal order of accuracy but exhibits larger error constants in the tested examples, consistent with its extrinsic nature.
The results highlight that intrinsic distance measurements can become sensitive to floating-point roundoff at very high orders (due to the $\arccos$ calculation of principal angles near 1), whereas projection-matrix errors provide a more stable diagnostic for high-order convergence.

Significance and Claims
The paper claims to establish a rigorous geometric framework for interpolating subspace data that respects the manifold structure. Its primary significance lies in:

Basis Invariance: Providing a method that is mathematically consistent regardless of the matrix representative chosen for a subspace.
Recursive Geodesic Structure: Offering a direct manifold analogue of Neville interpolation that does not rely on a fixed reference chart, which is crucial for handling data that may not be globally representable in a single tangent space.
Foundation for ENO: The recursive structure naturally extends to Grassmann ENO (GENO) procedures, where candidate stencils are selected based on intrinsic geometric variation to suppress oscillations near non-smooth features.

The authors note that while GIDER $_n$ and TSI perform similarly in smooth local tests, the recursive geodesic nature of GIDER $_n$ offers a distinct structural advantage for adaptive and non-linear reconstruction problems where a single reference tangent space may be insufficient. The paper concludes by outlining the potential for GENO extensions as a direction for future work.