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• W3100675395 abstract "We consider the theory of regression on a manifold using reproducing kernel Hilbert space methods. Manifold models arise in a wide variety of modern machine learning problems, and our goal is to help understand the effectiveness of various implicit and explicit dimensionality-reduction methods that exploit manifold structure. Our first key contribution is to establish a novel nonasymptotic version of the Weyl law from differential geometry. From this we are able to show that certain spaces of smooth functions on a manifold are effectively finite-dimensional, with a complexity that scales according to the manifold dimension rather than any ambient data dimension. Finally, we show that given (potentially noisy) function values taken uniformly at random over a manifold, a kernel regression estimator (derived from the spectral decomposition of the manifold) yields minimax-optimal error bounds that are controlled by the effective dimension." @default.
• W3100675395 created "2020-11-23" @default.
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• W3100675395 date "2020-01-01" @default.
• W3100675395 modified "2023-10-18" @default.
• W3100675395 title "Sample complexity and effective dimension for regression on manifolds" @default.
• W3100675395 hasPublicationYear "2020" @default.
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