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Manifold-valued function approximation from multiple tangent spaces

Hang Wang, Raf Vandebril, Joeri Van der Veken, Nick Vannieuwenhoven

TL;DR

This work addresses learning mappings from Euclidean inputs to manifold-valued outputs by introducing the multiple tangent spaces model (MTSM), which fuses several single-tangent-space models (STSM) via a weighted Fréchet mean. MTSM achieves a balance between STSM's computational efficiency and RMLS's broader geometric coverage by anchoring predictions at multiple points $p_j^*$ on $\mathcal{M}$, pulling back outputs to corresponding tangent spaces, learning local vector fields $\widehat{g}_j$, and aggregating via $\widehat{f}(x)=\mathrm{Exp}_{p_j^*}(\widehat{g}_j(x))$ followed by a Fréchet-mean fusion. The paper provides well-posedness, smoothness, and error guarantees, along with a practical offline/online algorithmic framework and extensive numerical experiments on SPD matrices, SO($3$), and parametric model order reduction benchmarks. Results indicate that MTSM offers competitive accuracy with favorable online costs, making it suitable for large-scale evaluation scenarios and MOR applications. Future work includes optimized anchor placement and adaptive sample selection to further tighten error bounds and reduce offline costs.

Abstract

Approximating a manifold-valued function from samples of input-output pairs consists of modeling the relationship between an input from a vector space and an output on a Riemannian manifold. We propose a function approximation method that leverages and unifies two prior techniques: (i) approximating a pullback to the tangent space, and (ii) the Riemannian moving least squares method. The core idea of the new scheme is to combine pullbacks to multiple tangent spaces with a weighted Fréchet mean. The effectiveness of this approach is illustrated with numerical experiments on model problems from parametric model order reduction.

Manifold-valued function approximation from multiple tangent spaces

TL;DR

This work addresses learning mappings from Euclidean inputs to manifold-valued outputs by introducing the multiple tangent spaces model (MTSM), which fuses several single-tangent-space models (STSM) via a weighted Fréchet mean. MTSM achieves a balance between STSM's computational efficiency and RMLS's broader geometric coverage by anchoring predictions at multiple points on , pulling back outputs to corresponding tangent spaces, learning local vector fields , and aggregating via followed by a Fréchet-mean fusion. The paper provides well-posedness, smoothness, and error guarantees, along with a practical offline/online algorithmic framework and extensive numerical experiments on SPD matrices, SO(), and parametric model order reduction benchmarks. Results indicate that MTSM offers competitive accuracy with favorable online costs, making it suitable for large-scale evaluation scenarios and MOR applications. Future work includes optimized anchor placement and adaptive sample selection to further tighten error bounds and reduce offline costs.

Abstract

Approximating a manifold-valued function from samples of input-output pairs consists of modeling the relationship between an input from a vector space and an output on a Riemannian manifold. We propose a function approximation method that leverages and unifies two prior techniques: (i) approximating a pullback to the tangent space, and (ii) the Riemannian moving least squares method. The core idea of the new scheme is to combine pullbacks to multiple tangent spaces with a weighted Fréchet mean. The effectiveness of this approach is illustrated with numerical experiments on model problems from parametric model order reduction.

Paper Structure

This paper contains 25 sections, 9 theorems, 46 equations, 7 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Riemannian geometry
  4. Fréchet mean
  5. Prior models for manifold-valued function approximation
  6. STSM: Single tangent space model
  7. RMLS: Riemannian moving least squares
  8. MTSM: Multiple tangent spaces model
  9. Well-posedness
  10. Smoothness
  11. Error bound
  12. Number of tangent spaces
  13. Algorithms
  14. Offline stage
  15. Online stage
  16. ...and 10 more sections

Key Result

Theorem 2.1

If there exists a point $p \in \mathcal{M}$ and a radius with $\mathcal{N}_\sigma$ equal to $\mathcal{M}$ or $\mathcal{B}_{2\sigma}(p)$, such that $y_1, \dots, y_{d} \in \mathcal{B}_{\sigma}(p)$, then the minimization in the definition of $\mathrm{avg}_{\mathcal{M}{}}$ in def_karcher_mean has a globally unique solution for all $\Phi \in \Delta^{d-1}$.

Figures (7)

  • Figure 1: A diagram of key geometric concepts, like the tangent space $T_p\mathcal{M}{}$ at $p \in \mathcal{M}{}$, the exponential map $\mathrm{Exp}_p$, the logarithmic map $\mathrm{Log}_p$, and a geodesic curve $\gamma_p(t)$.
  • Figure 1: An illustration of a $3$-MTSM. It shows three tangent spaces at the anchor points $p^*_1,\ p^*_2,\ p^*_3$ and the relevant vector-valued functions $\widehat{g}_{j}: \Omega_j \rightarrow T_{p^*_j}\mathcal{M}{}$. The approximation consists of computing the weighted Fréchet mean of $\mathrm{Exp}_{p^*_j}\widehat{g}_{j}(x)$.
  • Figure 1: The Wendland function $\varphi(d)$ and the smooth cutoff function $\varphi_j(d)$ with $\sigma_j^2=1$ and $c=\frac{1}{2}$.
  • Figure 2: The maximum relative error \ref{['eqn_relerr_matrix']} of the function \ref{['spd_fun']} approximated by RMLS, MRMLS, STSM and $3$-MTSM. The vertical axis is displayed in a logarithmic scale. The setup is described in \ref{['subsec:spdfun']}.
  • Figure 3: The relative error \ref{['eqn_relerr_matrix']} for the $\mathcal{SO}(3)$-valued function \ref{['eq:rota']}, approximated in different domains by different models. The scale used is the same for all panels in subfigure (a), and likewise for subfigure (b). The setup is described in \ref{['subsubsec:SO3']}.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Theorem 2.1: Uniqueness Afsari2011
  • Theorem 3.1: Error bound of STSM simon2024approx
  • Corollary 3.2
  • Proof 1
  • Definition 4.1: Multiple tangent space model
  • Proposition 4.2: Well-posedness
  • Proof 2
  • Remark 4.3
  • Theorem 4.4
  • Remark 4.5
  • ...and 7 more