Sobolev Space
A class of function spaces in functional analysis in which functions and their weak derivatives up to a specified order all belong to an Lp space. Sobolev spaces are widely used in the theory of partial differential equations and function approximation.
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The Sobolev space H^s can be equivalently described as the space of functions whose Peter–Weyl expansion coefficients remain square-integrable after being weighted by the s-th power of the eigenvalues; the larger s is, the smoother the function and the faster the high-frequency coefficients decay.
In the deterministic setting without Dropout, the optimal approximation rate for Sobolev space approximation is benchmarked by a parameter complexity of ε^{-d/n}, as studied by Barron, DeVore, and others.
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The Sobolev space H^s can be equivalently described as the space of functions whose Peter–Weyl expansion coefficients remain square-integrable after being weighted by the s-th power of the eigenvalues; the larger s is, the smoother the function and the faster the high-frequency coefficients decay.
50%UnverifiedIn the deterministic setting without Dropout, the optimal approximation rate for Sobolev space approximation is benchmarked by a parameter complexity of ε^{-d/n}, as studied by Barron, DeVore, and others.
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