Scholarlink

<p>Many interesting functions arising in applications map into Riemannian manifolds. We present an algorithm, using the manifold exponential and logarithm, for approximating such functions. Our approach uses techniques to approximate functions into linear spaces, applied in the tangent space of the manifold. Our main contribution consists of showing how to upper bound the final approximation error in the manifold in terms of the intermediate approximation error in the tangent space and a lower bound in the manifold’s sectional curvature. Furthermore, when the sectional curvature is nonnegative, such as for compact Lie groups, the error in the manifold is at least as small as the error in the tangent space. We implement the algorithm in a Julia package <Emphasis FontCategory="NonProportional">ManiFactor.jl</Emphasis> and apply it to two example problems.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Approximating maps into manifolds with lower curvature bounds

  • 期刊论文

  • 发表日期：

    2025年8月12日

查看全文
  • Simon Jacobsson,
  • Raf Vandebril,
  • Joeri Van der Veken,
  • Nick Vannieuwenhoven

摘要

Many interesting functions arising in applications map into Riemannian manifolds. We present an algorithm, using the manifold exponential and logarithm, for approximating such functions. Our approach uses techniques to approximate functions into linear spaces, applied in the tangent space of the manifold. Our main contribution consists of showing how to upper bound the final approximation error in the manifold in terms of the intermediate approximation error in the tangent space and a lower bound in the manifold’s sectional curvature. Furthermore, when the sectional curvature is nonnegative, such as for compact Lie groups, the error in the manifold is at least as small as the error in the tangent space. We implement the algorithm in a Julia package ManiFactor.jl and apply it to two example problems.