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Second-order differential operators, stochastic differential equations and Brownian motions on embedded manifolds

Du Nguyen, Stefan Sommer

TL;DR

The paper develops a unified embedded-coordinate framework for stochastic differential equations on manifolds, linking SDE generators to second-order differential operators and introducing second-order tangent vectors to describe manifold constraints. It derives a global-coordinate Laplace-Beltrami operator and constructs Riemannian Brownian motion via Itô and Stratonovich SDEs, with explicit treatments on matrix manifolds such as Lie groups, SPD, Stiefel, and Grassmann. The authors propose multiple numerical schemes—deterministic and stochastic projection, as well as retractive Euler–Maruyama methods and second-order tangent retractations—accompanied by a software library for practical implementation and validation. Experiments show long-time Brownian simulations converge to uniform distributions on compact manifolds, enabling efficient sampling on these spaces. The work provides concrete tools for diffusion-based methods on constrained spaces with embedded metrics, with potential impact on robotics, molecular dynamics, and sampling in high-dimensional geometric spaces.

Abstract

We specify the conditions when a manifold M embedded in an inner product space E is an invariant manifold of a stochastic differential equation (SDE) on E, linking it with the notion of second-order differential operators on M. When M is given a Riemannian metric, we derive a simple formula for the Laplace-Beltrami operator in terms of the gradient and Hessian on E and construct the Riemannian Brownian motions on M as solutions of conservative Stratonovich and Ito SDEs on E. We derive explicitly the SDE for Brownian motions on several important manifolds in applications, including left-invariant matrix Lie groups using embedded coordinates. Numerically, we propose three simulation schemes to solve SDEs on manifolds. In addition to the stochastic projection method, to simulate Riemannian Brownian motions, we construct a second-order tangent retraction of the Levi-Civita connection using a given E-tubular retraction. We also propose the retractive Euler-Maruyama method to solve a SDE, taking into account the second-order term of a tangent retraction. We provide software to implement the methods in the paper, including Brownian motions of the manifolds discussed. We verify numerically that on several compact Riemannian manifolds, the long-term limit of Brownian simulation converges to the uniform distributions, suggesting a method to sample Riemannian uniform distributions

Second-order differential operators, stochastic differential equations and Brownian motions on embedded manifolds

TL;DR

The paper develops a unified embedded-coordinate framework for stochastic differential equations on manifolds, linking SDE generators to second-order differential operators and introducing second-order tangent vectors to describe manifold constraints. It derives a global-coordinate Laplace-Beltrami operator and constructs Riemannian Brownian motion via Itô and Stratonovich SDEs, with explicit treatments on matrix manifolds such as Lie groups, SPD, Stiefel, and Grassmann. The authors propose multiple numerical schemes—deterministic and stochastic projection, as well as retractive Euler–Maruyama methods and second-order tangent retractations—accompanied by a software library for practical implementation and validation. Experiments show long-time Brownian simulations converge to uniform distributions on compact manifolds, enabling efficient sampling on these spaces. The work provides concrete tools for diffusion-based methods on constrained spaces with embedded metrics, with potential impact on robotics, molecular dynamics, and sampling in high-dimensional geometric spaces.

Abstract

We specify the conditions when a manifold M embedded in an inner product space E is an invariant manifold of a stochastic differential equation (SDE) on E, linking it with the notion of second-order differential operators on M. When M is given a Riemannian metric, we derive a simple formula for the Laplace-Beltrami operator in terms of the gradient and Hessian on E and construct the Riemannian Brownian motions on M as solutions of conservative Stratonovich and Ito SDEs on E. We derive explicitly the SDE for Brownian motions on several important manifolds in applications, including left-invariant matrix Lie groups using embedded coordinates. Numerically, we propose three simulation schemes to solve SDEs on manifolds. In addition to the stochastic projection method, to simulate Riemannian Brownian motions, we construct a second-order tangent retraction of the Levi-Civita connection using a given E-tubular retraction. We also propose the retractive Euler-Maruyama method to solve a SDE, taking into account the second-order term of a tangent retraction. We provide software to implement the methods in the paper, including Brownian motions of the manifolds discussed. We verify numerically that on several compact Riemannian manifolds, the long-term limit of Brownian simulation converges to the uniform distributions, suggesting a method to sample Riemannian uniform distributions
Paper Structure (28 sections, 15 theorems, 129 equations, 2 figures, 3 tables, 1 algorithm)

This paper contains 28 sections, 15 theorems, 129 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Outline
  4. Previous works
  5. Background
  6. Some linear algebra: trace, trace sums of bilinear operators and projections
  7. Riemannian geometry
  8. The Levi-Civita connection
  9. Tubular neighborhoods and retractions
  10. Second-order tangent vectors and second-order differential operators
  11. The Laplace-Beltrami operator as SOO and Riemannian Brownian motions
  12. Lifting of Brownian motion
  13. Laplace-Beltrami operators and Brownian motions on matrix manifolds
  14. Matrix Lie groups with left-invariant metrics
  15. Positive-definite matrices with the affine invariant metric
  16. ...and 13 more sections

Key Result

Lemma 1

Let $\{v_i\}_{i=1}^{\dim_{\mathcal{V}}}$ and $\{w_j\}_{j=1}^{\dim_{\mathcal{W}}}$ be orthonormal bases of two inner product spaces $\mathcal{V}, \mathcal{W}$. Let $\mathcal{E}$ and $\mathcal{F}$ be two other inner product spaces. Let $A, B, C$ be three linear operators, where $A$ maps $\mathcal{V}$

Figures (2)

  • Figure 1: Tubular neighborhood and $\mathcal{E}$-tubular retraction. The shaded region $D_{\mathcal{M}}$ is a tubular neighborhood of $\mathcal{M}$. The point $q\in D_{\mathcal{M}}$ could be retracted to $\mathcal{M}$ by the nearest point retraction $\pi_{_\perp}$. It may be more efficient to use another retraction $\pi$.
  • Figure 2: Tangent retraction. For small $|v|$, $(x, v)$ belongs to a tubular neighborhood of $\mathcal{M}$ in $T\mathcal{M}$, and can be retracted using $\pi_{_\perp}(x + v)$. It may be more efficient to use a different retraction $\mathfrak{r}(x, v)$.

Theorems & Definitions (29)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • ...and 19 more