💻 computer science

Approximating maps into manifolds with lower curvature bounds

This paper introduces a Julia-based algorithm for approximating functions mapping into Riemannian manifolds that bounds the forward error using a lower bound on sectional curvature, ensuring performance no worse than the linear case when curvature is nonnegative.

Original authors: Simon Jacobsson, Raf Vandebril, Joeri van der Veken, Nick Vannieuwenhoven

Published 2026-01-27

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CC BY 4.0

Original authors: Simon Jacobsson, Raf Vandebril, Joeri van der Veken, Nick Vannieuwenhoven

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

The Big Idea: Mapping Curved Worlds with Flat Tools

Imagine you are trying to draw a map of a hilly landscape (a Riemannian manifold). You want to predict the terrain at every point based on a few sample measurements. However, your drawing tools (like rulers and graph paper) only work perfectly on flat surfaces (linear spaces). If you try to draw a curved mountain range on flat paper using standard math, your map will be distorted and inaccurate.

This paper presents a clever three-step recipe to fix that. It allows us to use our familiar, flat drawing tools to approximate functions that live on curved surfaces, while giving us a mathematical guarantee on how accurate the result will be.

The Three-Step Recipe

The authors propose a simple "template" to solve this problem:

Pick a Base Camp: Choose a specific point on the curved surface (let's call it $p$ ). Think of this as setting up a camp at the bottom of a mountain.
Unroll the Map (The Pull-Back): Take your curved data and "unroll" it onto a flat sheet of paper (the tangent space) attached to your base camp. In math terms, they use a tool called the logarithm map to flatten the curved data into a straight line. Now, you have a standard, flat problem that you can solve with any old approximation technique (like fitting a curve through points).
Roll it Back Up (The Push-Forward): Once you have your flat approximation, you need to put it back onto the mountain. You use the exponential map to "roll" your flat solution back onto the curved surface.

The Secret Sauce: The Curvature Guarantee

The tricky part is that when you roll the paper back up, small errors in your flat drawing might get stretched or squished. The paper's main breakthrough is figuring out exactly how much the error can grow.

They discovered that the "stretchiness" of the error depends entirely on the curvature of the mountain:

If the mountain is flat or curves "up" (like a sphere): The error in your final map will be no worse than the error in your flat drawing. It's safe!
If the mountain curves "down" (like a saddle or a Pringles chip): The error might get slightly bigger. However, the authors provide a specific formula to calculate exactly how much bigger, based on how "curvy" the surface is.

The Analogy: Imagine stretching a rubber sheet. If the sheet is flat, a small tear stays small. If the sheet is curved like a saddle, that same small tear might stretch out a bit more. The paper tells you exactly how much it will stretch based on the shape of the saddle.

Why This Matters (The "ManiFactor" Tool)

The authors didn't just do the math; they built a software tool called ManiFactor.jl (written in the Julia programming language) to do this automatically.

They tested it on two real-world examples:

Krylov Subspaces: Used in solving complex physics equations (like heat transfer in a rod). The data here lives on a space of "directions" (Grassmannian manifold).
Low-Rank Matrices: Used in compressing data or image processing. The data here lives on a space of "rank-1 matrices" (Segre manifold).

In both cases, their method worked perfectly. The actual error they measured matched the theoretical "safety limit" they calculated using their curvature formulas.

A Shortcut: The "Retraction"

Sometimes, calculating the exact "roll up" and "unroll" steps (exponential and logarithm maps) is too slow or too hard to compute. The paper also shows you can use a "shortcut" called a retraction.

Think of a retraction as a "good enough" approximation of the roll-up process. It's like using a quick sketch instead of a detailed blueprint. The paper proves that even if you use this shortcut, you can still calculate a safe error bound, and in many cases, it's much faster to compute without losing much accuracy.

Summary

Problem: How to approximate functions on curved surfaces using flat math tools.
Solution: Flatten the data, solve it, and roll it back up.
Guarantee: The error is controlled by the surface's curvature. If the surface curves up or is flat, the error is safe. If it curves down, the error is predictable.
Result: A working software package that applies this to complex engineering and math problems.

Title: Approximating maps into manifolds with lower curvature bounds

Problem Statement
Approximation theory traditionally focuses on functions mapping into linear spaces (e.g., $\mathbb{R}^n$ ). However, many applications in engineering, statistics, and physics involve functions whose values lie on Riemannian manifolds (e.g., orthogonal matrices, symmetric positive definite matrices, or subspaces). Naively applying standard linear approximation schemes component-wise fails to preserve the intrinsic geometric structure of these manifolds. While various methods exist for specific cases (such as manifold splines or regression), there is a need for a general framework that can leverage existing linear approximation techniques while providing rigorous, intrinsic error bounds that account for the manifold's geometry.

Methodology
The authors propose a general three-step template to approximate a map $f: \mathcal{R} \to \mathcal{M}$ , where $\mathcal{R}$ is a domain and $\mathcal{M}$ is a Riemannian manifold:

Linearization: Select a base point $p \in \mathcal{M}$ (heuristically chosen as the Karcher mean of sample points) to define a normal coordinate chart.
Pullback and Approximation: Map the function values to the tangent space $T_p\mathcal{M}$ using the manifold logarithm ( $\log_p$ ). This transforms the problem into approximating a vector-valued map $g = \log_p \circ f$ . Any standard linear approximation scheme (e.g., interpolation, regression, tensor decompositions) is applied to $g$ to produce an approximation $\hat{g}$ .
Pushforward: Map the approximation back to the manifold using the manifold exponential ( $\exp_p$ ), yielding the final approximant $\hat{f} = \exp_p \circ \hat{g}$ .

To analyze the error, the authors utilize Toponogov's theorem, a Riemannian comparison theorem. This theorem relates the geometry of geodesic triangles on the target manifold $\mathcal{M}$ to those on a model manifold $N$ with constant sectional curvature $H$ . By assuming a lower bound $H$ on the sectional curvature of $\mathcal{M}$ , the authors derive bounds on the intrinsic distance between the true function $f$ and the approximation $\hat{f}$ based on the error in the tangent space approximation.

Key Contributions

Theorem 3 (Main Contribution): The paper establishes a rigorous upper bound for the forward approximation error $d_{\mathcal{M}}(f(x), \hat{f}(x))$ $d_{M} (f (x), \hat{f} (x))$ in terms of the tangent space error $\epsilon = \|g(x) - \hat{g}(x)\|_p$ $ϵ = ∥ g (x) - \overset{g}{^} (x) ∥_{p}$ and the lower bound $H$ $H$ of the sectional curvature.
- If $H \geq 0$ (non-negative curvature), the error on the manifold is guaranteed to be no worse than the error in the tangent space ( $d_{\mathcal{M}} \leq \epsilon$ ).
- If $H < 0$ (negative curvature), the error is bounded by a function involving $\epsilon$ , $H$ , and the maximum norm $\sigma$ of the tangent vectors in the chart.
Extension to Retractions (Corollary 4): The framework is extended to cases where the exponential and logarithm maps are unknown or expensive, allowing the use of retractions (approximations of the exponential map) with additional error terms.
Condition Number Analysis (Proposition 6): The authors analyze the condition number of the map from the tangent space to the manifold, showing that it grows exponentially with the product of the tangent vector norm and the square root of the absolute curvature. This highlights the importance of choosing a base point $p$ that minimizes the norm of the tangent vectors (i.e., minimizing $\sigma$ ).
Implementation (ManiFactor.jl): The authors implement this framework in a Julia package, ManiFactor.jl. For the specific case of multivariate approximation, they employ tensorized Chebyshev interpolation combined with a Sequentially Truncated Higher-Order Singular Value Decomposition (ST-HOSVD) to handle the "curse of dimensionality" and construct data-sparse approximations.

Results
The paper validates the theoretical bounds through two numerical experiments:

Krylov Subspaces: Approximating a map into the Grassmannian manifold (representing subspaces) arising from a parametrized differential equation. The results confirm that the measured error adheres to the bound predicted by Theorem 3.
Dynamic Low-Rank Approximation: Approximating a map into the Segre manifold (rank-1 matrices). Again, the experimental error aligns with the theoretical bound.
Retraction Efficiency: In the Grassmannian example, the authors compared the standard exponential map against QR and polar retractions. While the error performance was virtually identical, the retractions offered a significant speedup in evaluation time (approximately 35-40% faster).

Significance and Claims
The paper claims that the approximability of maps into Riemannian manifolds is fundamentally linked to the manifold's curvature. The primary significance lies in providing a constructive, general-purpose algorithm that:

Allows the direct reuse of mature linear approximation techniques for manifold-valued data.
Provides explicit, intrinsic error bounds derived solely from a lower bound on sectional curvature, without requiring detailed knowledge of the specific manifold's global geometry beyond that bound.
Guarantees that for manifolds with non-negative curvature (such as compact Lie groups), the approximation error does not degrade compared to the linear case.

The authors note that while the choice of the linearization point $p$ is heuristic (based on the Karcher mean), the theoretical framework holds for any point where the image of the function fits within a geodesically convex normal coordinate chart. The work is presented as a step toward more robust reduced-order modeling and data analysis on manifolds, with the ManiFactor.jl package serving as a practical tool for these applications.