[Paper Review] Diffusion bridges for stochastic Hamiltonian systems with applications to shape analysis

Alexis Arnaudon, Frank van der Meulen+2 more|arXiv (Cornell University)|Feb 3, 2020

Morphological variations and asymmetry16 references5 citations

TL;DR

This paper introduces a novel diffusion bridge sampling scheme for stochastic Hamiltonian systems in shape analysis and fluid dynamics, enabling effective path reconstruction and parameter inference from partially observed landmark data. By leveraging geometric structure and handling nonlinear configuration spaces, it generalizes inexact matching to stochastic settings and overcomes limitations of prior methods in sampling efficiency and applicability to singular solutions.

ABSTRACT

Stochastically evolving geometric systems are studied in geometric mechanics for modelling turbulence parts of multi-scale fluid flows and in shape analysis for stochastic evolutions of shapes of, e.g., human organs. Recently introduced models involve stochastic differential equations that govern the dynamics of a diffusion process $X$. In applications $X$ is only partially observed at times $0$ and $T>0$. Conditional on these observations, interest lies in inferring parameters in the dynamics of the diffusion and reconstructing the path $(X_t,\, t\in [0,T])$. The latter problem is known as bridge simulation. We develop a general scheme for bridge sampling in the case of finite dimensional systems of shape landmarks and singular solutions in fluid dynamics. This scheme allows for subsequent statistical inference of properties of the fluid flow or the evolution of observed shapes. It covers stochastic landmark models for which no suitable simulation method has been proposed in the literature, that removes restrictions of earlier approaches, improves the handling of the nonlinearity of the configuration space leading to more effective sampling schemes and allows to generalise the common inexact matching scheme to the stochastic setting.

Motivation & Objective

• Address the challenge of inferring dynamics and reconstructing unobserved paths in stochastic geometric systems where only initial and final states are observed.
• Overcome limitations of existing bridge simulation methods that fail for stochastic landmark models or impose restrictive assumptions on nonlinearity and geometry.
• Develop a general-purpose sampling scheme applicable to finite-dimensional shape landmark systems and singular solutions in fluid dynamics.
• Enable statistical inference on fluid flow properties and shape evolution by providing accurate, efficient path reconstruction under stochastic dynamics.
• Generalize the inexact matching framework—commonly used in deterministic shape analysis—to the stochastic setting, enhancing modeling fidelity for real-world applications.

Proposed method

• Propose a bridge sampling scheme based on stochastic differential equations (SDEs) governing the dynamics of landmark configurations in shape analysis.
• Utilize the geometric structure of Hamiltonian systems to ensure the sampling process respects the intrinsic nonlinear geometry of the configuration space.
• Construct conditional diffusion processes that are pinned at observed initial and final landmark configurations, enabling path reconstruction via bridge simulation.
• Integrate numerical schemes that preserve the symplectic and momentum-conserving structure of the underlying Hamiltonian dynamics to improve sampling accuracy.
• Adapt the inexact matching paradigm to stochastic settings by incorporating random perturbations in the dynamics, allowing for realistic modeling of uncertainty in shape evolution.
• Apply the method to both finite-dimensional landmark systems and singular solutions in fluid dynamics, demonstrating broad applicability across geometric mechanics.

Experimental results

Research questions

• RQ1 How can one efficiently simulate diffusion bridges in stochastic Hamiltonian systems with nonlinear configuration spaces, such as those arising in shape analysis?
• RQ2 What sampling scheme enables accurate reconstruction of unobserved paths in landmark-based shape evolution models with partial observations?
• RQ3 In what ways can the inexact matching framework be generalized to stochastic dynamics while preserving geometric and statistical consistency?
• RQ4 How does the proposed method improve sampling efficiency and accuracy compared to existing approaches in the presence of nonlinearity and singular solutions?
• RQ5 What are the implications of this bridge sampling method for statistical inference on fluid flow parameters and shape evolution dynamics?

Key findings

• The proposed bridge sampling scheme successfully handles stochastic landmark models for which no prior simulation method existed, filling a critical gap in the literature.
• The method improves sampling efficiency by respecting the nonlinear geometry of the configuration space, leading to more effective exploration of the path space.
• It generalizes the inexact matching approach to stochastic dynamics, enabling more realistic modeling of uncertainty in shape evolution and fluid flow.
• The scheme is applicable to both finite-dimensional landmark systems and singular solutions in fluid dynamics, demonstrating broad theoretical and practical relevance.
• By preserving the symplectic and momentum-conserving structure of the underlying Hamiltonian system, the method ensures physically consistent and stable path reconstructions.
• The approach enables robust statistical inference on parameters governing fluid flow or shape evolution, even when only partial observations are available at endpoints.