Single Basepoint Subdivision Schemes for Manifold-valued Data: Time-Symmetry Without Space-Symmetry

• Published in: Foundations of computational mathematics Vol. 13; no. 5; pp. 693 - 728
• Main Authors: Duchamp, Tom, Xie, Gang, Yu, Thomas
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.10.2013; Springer; Springer Nature B.V
• Subjects: Applications of Mathematics; Computational mathematics; Computer Science; Curvature; Economics; Equivalence; Joints; Linear and Multilinear Algebras; Manifolds; Math Applications in Computer Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix Theory; Nonlinear theories; Numerical Analysis; Proximity; Space and time; Subdivisions; Tensors; Topological manifolds; United States; Retraction; 58C25; 68U05; Riemannian manifold; 37N30; 22E05; 41A25; Symmetric space; 65D10; Nonlinear subdivision; Exponential map; Time-symmetry; 26B05; Curvature; Affine connection
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-013-9144-1

This paper establishes smoothness results for a class of nonlinear subdivision schemes, known as the single basepoint manifold-valued subdivision schemes, which shows up in the construction of wavelet-like transform for manifold-valued data. This class includes the (single basepoint) Log–Exp subdivision scheme as a special case. In these schemes, the exponential map is replaced by a so-called retraction map f from the tangent bundle of a manifold to the manifold. It is known that any choice of retraction map yields a C 2  scheme, provided the underlying linear scheme is C 2 (this is called “ C 2  equivalence”). But when the underlying linear scheme is  C 3 , Navayazdani and Yu have shown that to guarantee C 3  equivalence, a certain tensor P f associated to f must vanish. They also show that P f vanishes when the underlying manifold is a symmetric space and f is the exponential map. Their analysis is based on certain “ C k  proximity conditions” which are known to be sufficient for C k  equivalence. In the present paper, a geometric interpretation of the tensor P f is given. Associated to the retraction map f is a torsion-free affine connection, which in turn defines an exponential map. The condition P f =0 is shown to be equivalent to the condition that f agrees with the exponential map of the connection up to the third order. In particular, when f is the exponential map of a connection, one recovers the original connection and P f vanishes. It then follows that the condition P f =0 is satisfied by a wider class of manifolds than was previously known. Under the additional assumption that the subdivision rule satisfies a time-symmetry , it is shown that the vanishing of P f implies that the C 4  proximity conditions hold, thus guaranteeing C 4  equivalence. Finally, the analysis in the paper shows that for k ≥5, the C k  proximity conditions imply vanishing curvature. This suggests that vanishing curvature of the connection associated to f is likely to be a necessary condition for C k equivalence for k ≥5.