Source-linked AI summary
Simulation of Conditioned Semimartingales on Riemannian Manifolds
Mathias Højgaard Jensen, Stefan Sommer
TL;DR
Conditioned-process simulation on manifolds is challenging because transition-density scores are generally intractable. The paper replaces that drift with a scaled radial-vector-field guide and proves a change-of-measure relationship to the conditioned process. The resulting scheme is used for manifold bridge simulation, transition-density approximation, and diffusion-mean estimation.
Problem
On manifolds, the transition-density score needed to describe conditioned-process drift is generally intractable, while bridge sampling supports likelihood and Bayesian inference.
Source: §1 · p. 2Method
The scheme guides a manifold-valued semimartingale with a scaled radial vector field and uses a change-of-measure argument to relate it to the conditioned process.
Source: §1 · p. 1, §1.1 · p. 3, §4.1 · p. 11Results
The guided process has a law equivalent to that of the conditioned process before the terminal time.
Source: §4.1 · p. 11Takeaways & Limitations
The scheme is used to simulate bridges on two- and three-dimensional manifolds, approximate transition densities, and estimate diffusion means.
Source: §1 · p. 2, §1.2 · p. 4The radial-field convergence is uniform only away from the cut locus, where squared distance is not differentiable.
Source: §1.1 · p. 4
Abstract
from arXiv · showhide
We present a scheme for simulating conditioned semimartingales taking values in Riemannian manifolds. Extending the guided bridge proposal approach used for simulating Euclidean bridges, the scheme replaces the drift of the conditioned process with an approximation in terms of a scaled radial vector field. This handles the fact that transition densities are generally intractable on geometric spaces. We prove the validity of the scheme by a change of measure argument, and we show how the resulting guided processes can be used in importance sampling and for approximating the density of the unconditioned process. The scheme is used for numerically simulating bridges on two- and three-dimensional manifolds, for approximating otherwise intractable transition densities, and for estimating the diffusion mean of sampled geometric data.
1. Introduction
The paper extends guided bridge proposals from Euclidean diffusions to conditioned semimartingales on Riemannian manifolds, using a radial-vector-field drift in place of the generally intractable transition-density score. It motivates this scheme through bridge sampling’s role in inference and geometric statistics, while noting that the cut locus complicates the guiding field.
1. Introduction
- The proposed manifold scheme adds a drift approximating the transition-density score, and its guided process is absolutely continuous with respect to the target bridge law.The paper presents this as a generalization of Delyon and Hu’s Euclidean scheme.Source: §1 · p. 1
- Bridge sampling supports likelihood and Bayesian inference, and manifold applications include transition-density approximation and diffusion-mean estimation.The paper also situates bridge sampling in parameter and density estimation, including applications in shape analysis.Source: §1 · p. 2, §1.2 · p. 4
1.1. Guided proposals
- The manifold proposal uses the gradient of half the squared distance to the target as a radial guide, but this field is discontinuous at the cut locus.The cited heat-kernel convergence is uniform on compact subsets away from the cut locus, where squared distance is not differentiable.Source: §1.1 · pp. 3–4, Figure 2 · p. 3
1.3. Outline
- The paper outlines a change-of-measure derivation, a radial-process convergence argument, rigorous proofs, and numerical examples on two- and three-dimensional manifolds.The examples include heat-kernel approximation and iterative bridge sampling for diffusion means.Source: §1.3 · p. 4
1.4. Notation and conventions
- The setup assumes a smooth d-dimensional manifold, bounded differentiable diffusion fields and inverses, and that the process spends zero Lebesgue time in the target’s cut locus.The notation identifies the manifold-valued process as the projection of a horizontal semimartingale and sets the distance gradient to zero on the cut locus.Source: §1.4 · p. 5
2. Background
The background develops the geometric and stochastic tools used to represent manifold-valued processes and their bridges. It also explains why bridge simulation on manifolds often requires approximations to transition-density-dependent drifts.
2.1. Manifold valued processes
- A frame is a basis of a tangent space, and the orthonormal frame bundle provides the setting for horizontal lifts and stochastic integration on the manifold.The frame bundle has dimension d^2 + d for a d-dimensional smooth manifold.Source: §2.1 · p. 5, §2.1.1 · p. 5
2.1.1. Riemannian geometry
- The frame bundle attaches a tangent-space basis to each manifold point; the orthonormal frame bundle restricts these to isometries.Normal neighborhoods use the exponential map to provide local coordinates where its inverse is defined.Source: §2.1.1 · pp. 5–6
2.1.2. Horizontal semimartingales - stochastic development
- Horizontal development maps Euclidean semimartingales to manifold-valued processes through the frame bundle, with the projection giving the process on the manifold.The construction is also described as rolling without slipping.Source: §2.1.2 · p. 6
2.1.3. Geometric Itô formula
- The geometric Itô formula applies to manifold-valued semimartingales using their horizontal lift and anti-development, and can also be expressed through gradients and Hessians.The inner product in that expression is the Riemannian inner product on the tangent space.Source: §2.1.3 · p. 6
2.1.4. Riemannian Brownian bridges
- Brownian bridge SDEs involve the transition density, which is generally not available in closed form on Riemannian manifolds.The Brownian bridge generator uses the Laplace-Beltrami operator, and related bridge constructions offer alternatives on some settings.Source: §2.1.4 · p. 7
2.1.5. Related constructions
- Semi-classical Brownian bridges use a generator involving the radial direction and typically assume a pole; Fermi bridges target a submanifold, while generalized bridges use a diverging time-dependent function.The pole assumption avoids the cut locus, and the semi-classical bridge's radial part has a Euclidean Brownian bridge distribution.Source: §2.1.5 · pp. 7–8
3. A Girsanov change of measure
Intractable transition densities make direct simulation of conditioned processes difficult. The section uses a Girsanov change of measure to construct a radial bridge process while preserving equivalence of measures before the terminal time.
3. A Girsanov change of measure
- When the transition density cannot be computed, the conditioned-process drift cannot be obtained directly, motivating simulation proposals related by an absolutely continuous change of measure.The passage identifies this as a common basis for several approximation methods.Source: §3 · p. 8
- Assuming a non-explosive manifold process and invertible diffusion coefficient, the change of measure defines a local martingale and Radon-Nikodym derivative.Novikov's condition ensures the changed measure is equivalent to the original measure on [0,T).Source: §3 · p. 8
- Under the changed measure, substituting the transformed Brownian motion into the SDE yields the radial bridge process.The added drift is expressed using the radial process.Source: §3 · pp. 8–9
- Away from the cut locus and starting point, the radial process has unit gradient norm, and the associated integral is a one-dimensional standard Brownian motion.The excluded set is polar for the process.Source: §3 · p. 9
4. Bridge simulation on manifolds
The paper constructs guided manifold-valued semimartingales by adding a radial drift to a process obtained through stochastic development. It proves equivalence with the conditioned process before the terminal time and gives a Brownian-motion specialization.
4. Bridge simulation on manifolds
- Stochastic development lifts a Euclidean semimartingale to the frame bundle, where adding a radial drift yields a guiding term that pushes the process toward its target.The construction assumes a strong solution to the development SDE.Source: §4 · pp. 9–10, §4.1 · p. 10
4.1. Guided bridges
- The projected guided process has a law equivalent to that of the conditioned process before T, and the radial bridge converges almost surely to the target point.The proof uses a change of measure valid on each interval [0,t] for t<T.Source: §4.1 · pp. 11–12
4.2. The case of Brownian motion
- For Brownian motion, the conditioned law is absolutely continuous with respect to the guided law on every interval before T, and the likelihood-ratio expression simplifies.The expression is supported on M\Cut(v), and Ls is geometric local time at Cut(v); in Euclidean space the likelihood ratio is 1.Source: §4.2 · p. 12
5. Radial part of manifold valued semimartingales
The section uses radial-process results and cut-locus geometry to analyze the guided bridge. Its radial drift pulls toward the endpoint, and the proposal bridge reaches that endpoint almost surely.
5. Radial part of manifold valued semimartingales
- Barden and Le’s radial-process formula accounts for local-time contributions at zero and along the cut locus under the stated assumptions.The analysis describes the cut-locus geometry and the formula’s one-sided derivative and local-time terms.Source: §5 · pp. 12–13
5.1. Properties of the radial bridge
- The radial bridge’s drift always points toward the endpoint, acting as a pulling term that ensures convergence.The argument analyzes this behavior through bounds on the radial process.Source: §5.1 · p. 14
5.1.1. Bound for the radial bridge
- A radial-bridge bound holds up to exit from a regular domain under the theorem’s stated drift conditions.The bound applies for all t in [0, T).Source: §5.1.1 · pp. 14–15
5.1.2. Almost sure convergence
- Q-almost surely, the proposal bridge reaches its target, establishing the bridge property.The proof uses bounds on the radial bridge process to show its endpoint distance is zero.Source: §5.1.2 · p. 16
6. Proof of Theorem 4.3 and 4.4
The proof develops a pullback of the radial field and uses change of measure to establish the main results. It also derives the Brownian-case density expression and verifies convergence of the associated weight.
6.1. Pullback of radial field
- In frame coordinates, the covariant derivative becomes the standard derivative, supporting the pullback description of the radial field.The proof develops the corresponding SDE for the pulled-back field.Source: §6.1 · pp. 16–17
- The proof of Theorem 4.3 combines measure equivalence on [0, t], for each t < T, with the lemma-derived expressions to establish the result.Novikov’s condition ensures the measure equivalence.Source: §6.1 · p. 18
- The weight φ_t converges in L1 because it converges almost surely and the family is uniformly integrable.The passage states that E_Q[φ_t] is finite for each t in [0, T).Source: §6.1 · p. 18
- For Brownian driving noise, the proof rewrites the Radon–Nikodym derivative using an expression that includes the exponential-map Jacobian determinant.The Jacobian determinant is denoted by Θ.Source: §6.1 · pp. 18–19
7. Numerical experiments
The experiments illustrate radial-bridge simulations on two- and three-dimensional manifolds, use bridge sampling to estimate transition densities, and apply likelihood optimization to diffusion-mean estimation. They include paths conditioned to reach cut-locus points and density estimates on manifolds with no directly available closed form.
7. Numerical experiments
- The experiments apply the radial-bridge scheme to examples on two- and three-dimensional manifolds and to maximum-likelihood estimation of diffusion means.The numerical section includes bridge simulation, transition-density estimation, and diffusion-mean estimation.Source: §7 · p. 20
7.1. Simulation of bridge process on 2- and 3-dimensional manifolds
- On the torus, simulated radial bridges reach a cut-locus endpoint, while the cylinder examples show paths winding different ways to a cut-locus endpoint.The torus bridge drift follows the squared radial vector field multiplied by an increasing time-dependent scalar.Source: §7.1 · pp. 20–21
- On SO(3), the experiments illustrate radial-bridge paths from the identity matrix to a conditioned terminal value.The paths are represented by the left action on a basis of R3.Source: §7.1 · p. 21, Figure 5 · p. 21
7.2. Bridge simulation for density estimation on 2-Sphere
- On S2, estimated Brownian transition densities along a north-to-south geodesic are compared with a truncated heat-kernel expansion and two-dimensional Euclidean Brownian densities.The estimates are plotted for three time points; the passage gives T = 0.5, 1, 1.5, while the figure caption lists T = 0.5, 1, 2.Source: §7.2 · p. 22, Figure 6 · p. 22
- The scheme estimates heat kernels on an ellipsoid, where no closed-form solution is directly available, and on the embedded torus T2.The results are shown at times T = 1/2, 3/4, and 1.Source: §7.2 · p. 23, Figure 8 · p. 23, Figure 9 · p. 23
7.3. Bridge simulation for diffusion mean estimation
- Bridge sampling estimates the diffusion mean from 100 Brownian-motion samples on S2, with likelihood optimization converging in approximately 20 iterations.The initial mean guess converges to the true mean in the illustrated optimization.Source: §7.3 · p. 23, Figure 10 · p. 24