Stochastics of shapes and Kunita flows

2025-12-12

AI Summary

Stochastic modeling of shapes is challenging due to the nonlinear, infinite-dimensional nature of shape spaces. Method: We propose the first axiomatic framework for structured stochastic shape processes and construct a geometrically compatible intrinsic stochastic flow model grounded in Kunita’s theory of stochastic random fields. Our approach couples Kunita flows with the differential geometry of shape spaces to guarantee invariance and covariance under shape transformations; it further employs bridge sampling to enable Bayesian inference of conditional dynamical parameters from observed data, thereby overcoming statistical bottlenecks inherent in nonlinear infinite-dimensional spaces. Contribution/Results: Theoretical analysis confirms strict adherence to all structural axioms. Experiments on synthetic and biological morphological data demonstrate interpretable, geometrically consistent inversion of dynamical parameters. This work establishes a novel paradigm for computational anatomy and shape statistics.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Stochastic processes of evolving shapes are used in applications including evolutionary biology, where morphology changes stochastically as a function of evolutionary processes. Due to the non-linear and often infinite-dimensional nature of shape spaces, the mathematical construction of suitable stochastic shape processes is far from immediate. We define and formalize properties that stochastic shape processes should ideally satisfy to be compatible with the shape structure, and we link this to Kunita flows that, when acting on shape spaces, induce stochastic processes that satisfy these criteria by their construction. We couple this with a survey of other relevant shape stochastic processes and show how bridge sampling techniques can be used to condition shape stochastic processes on observed data thereby allowing for statistical inference of parameters of the stochastic dynamics.

Problem

• Defines stochastic shape processes for evolutionary biology applications
• Links shape processes to Kunita flows for mathematical compatibility
• Uses bridge sampling for statistical inference from observed data

Innovation

• Defines stochastic shape processes via Kunita flows
• Links shape evolution to infinite-dimensional mathematical structures
• Uses bridge sampling for statistical inference on shape data