On Donoho's Log-Exp Subdivision Scheme: Choice of Retraction and Time-Symmetry
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In recent years a number of different approaches for adapting linear subdivision schemes to manifold-valued data were proposed. In this article, we study the following family: (Sx)2i+σ=fxi(∑ℓa2ℓ+σgxi(xi−ℓ)),σ=0,1,i∈Z; here f is a smooth retraction on M, g is the corresponding local inverse, and (a2ℓ+σ) is the mask of a linear subdivision scheme. This particular way of adapting linear subdivision schemes to manifold, in which the same base pointxi is used for both the odd and even rules, is a fundamental building block of the wavelet transform proposed in Ur Rahman et al. [Multiscale Model. Simul., 4 (2005), pp. 1201–1232]. This feature is not shared by the other ways proposed in the recent literature. In this article, we expose the rather subtle smoothness equivalence properties of the above S; here “smoothness equivalence property” refers to how much smoothness the nonlinear S inherits from the underlying linear scheme. We first prove that one always gets C2 equivalence between S and the linear scheme regardless of the choice of f. In contrast, if one wants just one more order of smoothness equivalence, then the choice of f matters. We show that C3 equivalence is guaranteed by a condition on the third order Taylor expansions of f. This condition is further proved to be genuinely geometric in the sense that it is invariant under change of coordinates. Our second main result shows that the most natural choice f=exp in a symmetric space setting always satisfies the condition. Consequently, any third order accurate approximation of the exponential map would satisfy the same condition. This provides the ground for replacing the exponential map by a more computationally efficient approximant. The difficulty is that such an approximant must also be chosen such that it is by itself also a retraction of the underlying symmetric space or Lie group. Fortunately, it is a well-studied problem in the area of numerical geometric integration; many computationally efficient approximations to the exponential map are available for different symmetric spaces. Finally, we discuss the effect of time-symmetry on smoothness.
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