Deep neural networks on diffeomorphism groups for optimal shape reparameterization

Abstract

One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms.

arXiv comment36 pages, 11 figures. Accepted by BIT Numerical Mathematics, not yet published

References in corpus (6)

• Stable Architectures for Deep Neural Networks
• Constructing reparametrization invariant metrics on spaces of plane curves
• Shape Analysis on Lie Groups with Applications in Computer Animation
• Faa di Bruno's formula for Gateaux differentials and interacting stochastic population processes
• Regularity of Lie Groups
• Shape analysis on Lie groups and homogeneous spaces

Cited by in corpus (3)

• Designing Stable Neural Networks using Convex Analysis and ODEs
• Controllability and diffeomorphism groups on manifolds with boundary
• Sub-Riemannian Landmark Matching and its interpretation as residual neural networks