Titlebook: Regression and Fitting on Manifold-valued Data; Ines Adouani,Chafik Samir Textbook 2024 The Editor(s) (if applicable) and The Author(s), u

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,Spline Interpolation on?the?Special Orthogonal Group ,(,),us due to its inherent visualizability, enabling an intuitive understanding of the proposed algorithm. While . served as an excellent preliminary example to assess the concepts, the need to extend the work to more intricate manifolds becomes imperative to substantiate the complexity of the approach.

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,Spline Interpolation on?Stiefel and?Grassmann Manifolds,wever, a persistent challenge in many of these applications stems from the intricate geometric structures inherent in these manifolds?[4]. As real-world applications increasingly involve non-vector data, numerous algorithms for manifold embedding and manifold learning have been introduced to address

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,Spline Interpolation on the Manifold of?Probability Density Functions,ed set of observations .. Fitting a set of PDFs points constitutes a vital area of research in theoretical and computational statistics, with widespread applications in fields such as machine learning, medical imaging, computer vision, signal/video processing, and beyond

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Spline Interpolation on Other Riemannian Manifolds,emannian manifolds. Specifically, we focus on two such instances: the set of symmetric and positive-definite matrices (SPD), denoted as ., and hyperbolic spaces . characterized by constant negative curvature. These nonlinear spaces find wide-ranging applications where the demand for smooth interpola

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Introduction,loration extends to the generalization of the proposed Euclidean Bézier curve techniques to various examples of Riemannian manifolds. Such generalization involves an in-depth examination of the geometric properties of the Riemannian manifold.

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,Spline Interpolation and?Fitting in?,ion of an innovative method for solving the interpolation problem in . through the use of . Bézier splines. This approach adeptly navigates the complexities of fitting data in multiple dimensions, ensuring the desired continuity up?to the .th order and providing a nuanced and effective solution to this intricate problem.

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,Spline Interpolation on?Stiefel and?Grassmann Manifolds, these challenges. Recent efforts in this direction have focused on the development of essential geometric and statistical tools, including the Riemannian exponential map and its inverse, means, distributions, and geodesics?[5–7].

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