arXiv:2505.05168v3 Announce Type: replace-cross Abstract: Under mild conditions, this paper derives a least-squares local linear Fr\'echet curve predictor for response and regressor evaluated in a separable Hilbert space. We obtain the conditions allowing the implementation of this local linear Fr\'echet functional predictor in the ambient L^{2}-space of vector functions, with values in the time-varying tangent space on a compact Riemannian manifold. An intrinsic local linear Fr\'echet curve predictor evaluated in such a manifold is secondly proposed, based on a weighted Fr\'echet mean approac
This is a theoretical arXiv paper, typical of ongoing academic research in advanced mathematics and machine learning, reflecting incremental progress in a specialized field.
For a strategic reader, this specific paper is not immediately important as it's highly theoretical and far from practical application, representing foundational research.
This publication does not immediately change any real-world conditions or technologies, but contributes to the mathematical foundation of potential future machine learning algorithms.
Further theoretical understanding of Fréchet curve regression in complex spaces is advanced.
This foundational work might eventually contribute to more robust or efficient algorithms for analyzing time-series data on non-Euclidean manifolds in niche applications.
Extremely long-term, this could subtly influence advanced AI model development for applications like medical imaging or robotics where data inherently exists on manifolds.
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