Two universality results for polynomial reproducing kernels

Abstract Online learning is a classical algorithm for optimization problems. Due to its low computational cost, it has been widely used in many aspects of machine learning and statistical learning. Its convergence performance depends heavily on the step size. In this paper, a two-stage step size is proposed for the unregularized online learning algorithm, based on reproducing Kernels. Theoretically, we prove that, such an algorithm can achieve a nearly min–max convergence rate, up to some logarithmic term, without any capacity condition.

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Reproducing kernels of Sobolev spaces on ℝd and applications to embedding constants and tractability

Analysis and Applications ◽

10.1142/s0219530518500094 ◽

2018 ◽

Vol 16 (05) ◽

pp. 693-715 ◽

Cited By ~ 2

Author(s):

Erich Novak ◽

Mario Ullrich ◽

Henryk Woźniakowski ◽

Shun Zhang

Keyword(s):

Sobolev Space ◽

Sobolev Spaces ◽

Reproducing Kernel ◽

Absolute Error ◽

Reproducing Kernels ◽

Worst Case ◽

Reproducing Kernel Formula ◽

Positive Integers ◽

Integration Errors ◽

Exponentially Small

The standard Sobolev space [Formula: see text], with arbitrary positive integers [Formula: see text] and [Formula: see text] for which [Formula: see text], has the reproducing kernel [Formula: see text] for all [Formula: see text], where [Formula: see text] are components of [Formula: see text]-variate [Formula: see text], and [Formula: see text] with non-negative integers [Formula: see text]. We obtain a more explicit form for the reproducing kernel [Formula: see text] and find a closed form for the kernel [Formula: see text]. Knowing the form of [Formula: see text], we present applications on the best embedding constants between the Sobolev space [Formula: see text] and [Formula: see text], and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in [Formula: see text], whereas worst case integration errors of algorithms using [Formula: see text] function values are also exponentially small in [Formula: see text] and decay at least like [Formula: see text]. This yields strong polynomial tractability in the worst case setting for the absolute error criterion.

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A Unified Reproducing Kernel Method and Error Estimation for Solving Linear Differential Equation with Functional Constraints

Mathematical Problems in Engineering ◽

10.1155/2015/823264 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9

Author(s):

Xinjian Zhang ◽

Xiongwei Liu

Keyword(s):

Differential Equation ◽

Approximate Solution ◽

Linear Differential Equation ◽

Error Estimation ◽

Reproducing Kernel ◽

Kernel Method ◽

Inner Product ◽

Reproducing Kernel Method ◽

Linear Differential ◽

Polynomial Reproducing

A unified reproducing kernel method for solving linear differential equations with functional constraint is provided. We use a specified inner product to obtain a class of piecewise polynomial reproducing kernels which have a simple unified description. Arbitrary order linear differential operator is proved to be bounded about the special inner product. Based on space decomposition, we present the expressions of exact solution and approximate solution of linear differential equation by the polynomial reproducing kernel. Error estimation of approximate solution is investigated. Since the approximate solution can be described by polynomials, it is very suitable for numerical calculation.

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