文献类型：期刊文献

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

作者：Duchamp, Tom[1];Xie, Gang[2];Yu, Thomas[3]

机构：[1]Univ Washington, Dept Math, Seattle, WA 98195 USA;[2]E China Univ Sci & Technol, Dept Math, Shanghai 200237, Peoples R China;[3]Drexel Univ, Dept Math, Philadelphia, PA 19104 USA

年份：2013

卷号：13

期号：5

起止页码：693

外文期刊名：FOUNDATIONS OF COMPUTATIONAL MATHEMATICS

收录：;EI(收录号：20134016808037);WOS:【SCI-EXPANDED(收录号:WOS:000324819700001)】；

基金：Tom Duchamp gratefully acknowledges the support and hospitality provided by the IMA during his visit from April to June 2011, when much of the work in this article was completed. Gang Xie's research was supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (No. 11101146).Thomas Yu's research was partially supported by the National Science Foundation grants DMS 0915068 and DMS 1115915. He is also indebted to a fellowship offered by the Louis and Bessie Stein family.

语种：英文

外文关键词：Nonlinear subdivision; Affine connection; Retraction; Exponential map; Riemannian manifold; Symmetric space; Curvature; Time-symmetry

摘要：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.