This study explores spectral algorithms within Reproducing Kernel Hilbert Spaces, focusing on heat kernels and diffusion spaces. The convergence performance of spectral algorithms is analyzed with integral operator techniques to derive tight upper bounds for generalized norms. The research establishes minimax lower bounds, demonstrating the asymptotic optimality of conclusions in specific contexts. The study emphasizes the importance of considering low-dimensional manifolds embedded in higher-dimensional spaces for enhanced algorithm performance.
The authors propose a novel approach to spectral algorithms by incorporating the manifold structure of input data. They derive convergence rates and minimax lower bounds, showcasing the practical significance of their findings in high-dimensional approximation scenarios.
The content delves into the theoretical challenges of deriving convergence rates in "hard learning" scenarios and demonstrates the optimality of using heat kernels in spectral algorithms. By exploring specific kernel functions like heat kernels on manifolds, the study offers improved convergence rates and minimax results.
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by Weichun Xia,... at arxiv.org 03-07-2024
https://arxiv.org/pdf/2403.03669.pdfDeeper Inquiries