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

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• 作者：Tom Duchamp ; Gang Xie ; Thomas Yu
• 关键词：Nonlinear subdivision ; Affine connection ; Retraction ; Exponential map ; Riemannian manifold ; Symmetric space ; Curvature ; Time ; symmetry ; 41A25 ; 26B05 ; 22E05 ; 68U05 ; 37N30 ; 58C25 ; 65D10
• 刊名：Foundations of Computational Mathematics
• 出版年：2013
• 出版时间：October 2013
• 年：2013
• 卷：13
• 期：5
• 页码：693-728
• 全文大小：1064KB
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• 作者单位：Tom Duchamp (1)
  Gang Xie (2)
  Thomas Yu (3)
  
  1. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195-4350, USA
  2. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China
  3. Department of Mathematics, Drexel University, 3141 Chestnut Street, 206 Korman Center, Philadelphia, PA, 19104, USA
  
• ISSN：1615-3383

文摘

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 -em class="a-plus-plus">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 -em class="a-plus-plus">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?, 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?.