Best approximation by linear combinations of characteristic functions of half-spaces

Kainen, Paul C.; Kůrková, Véra; Vogt, Andrew

doi:10.1016/s0021-9045(03)00072-8

Cited by 19 publications

(15 citation statements)

References 18 publications

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“…, was shown in [11]. If C is bounded and M = conv n H d , then by Theorem 3.1(iii) it is sufficient to prove that conv n H d is boundedly compact and closed.…”

Section: Convergence Of Minimizing Sequences Formed By Variable-basismentioning

confidence: 93%

Minimization of Error Functionals over Variable-Basis Functions

Kainen¹,

Kurková²,

Sanguineti³

2004

SIAM J. Optim.

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Abstract. Generalized Tikhonov well-posedness is investigated for the problem of minimization of error functionals over admissible sets formed by variable-basis functions, i.e., linear combinations of a fixed number of elements chosen from a given basis without a prespecified ordering. For variablebasis functions of increasing complexity, rates of decrease of infima of error functionals are estimated. Upper bounds are derived on such rates which do not exhibit the curse of dimensionality with respect to the number of variables of admissible functions. Consequences are considered for Boolean functions and decision trees. 1. Introduction. Functionals defined as distances from (target) sets are called error functionals. Minimization of such functionals occurs in optimization tasks arising in various areas such as decision processes, system identification, machine learning, and pattern recognition.In various applications, admissible solutions over which error functionals are minimized are functions depending on a large number of variables: for example, when routing strategies have to be devised for large-scale communication and transportation networks, when an optimal closed-loop control law has to be determined for a dynamical system with high-dimensional output measurement vector and a large number of decision stages, etc. In the last decades, complex optimization problems of this kind have been approximately solved by searching suboptimal solutions over ad- Neural networks can be studied in a more general context of variable-basis functions, which also include other nonlinear families of functions such as free-node splines and trigonometric polynomials with free frequencies [17]. Families of variable-basis functions are formed by linear combinations of a fixed number of elements chosen from a given basis without a prespecified ordering [16], [17].When admissible functions depend on a large number of variables, implementation of some procedures of approximate optimization may be infeasible due to the "curse of dimensionality" [3]. For example, when optimization is performed over linear combinations of fixed-basis functions, the number of basis functions required to

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“…, was shown in [11]. If C is bounded and M = conv n H d , then by Theorem 3.1(iii) it is sufficient to prove that conv n H d is boundedly compact and closed.…”

Section: Convergence Of Minimizing Sequences Formed By Variable-basismentioning

confidence: 93%

Minimization of Error Functionals over Variable-Basis Functions

Kainen¹,

Kurková²,

Sanguineti³

2004

SIAM J. Optim.

Self Cite

View full text Add to dashboard Cite

show abstract

“…(see [35]) For any positive integers m, d and any p ∈ [1, ∞),

italicspa n_{m} H_{d} = {\sum_{i = 1}^{m} ω_{i} ψ ({bold-italic v}_{i} \cdot x + c_{i})}

, with ω i ∈ ℛ v i ∈ ℛ d and c i ∈ ℛ, is approximately a compact subset of (ℒ p ([0, 1] d ), ‖·‖ p ).…”

Section: The Proposed Iterative Methodsmentioning

confidence: 99%

Single-Image Super-Resolution via an Iterative Reproducing Kernel Hilbert Space Method

Deng

Guo

Huang

2016

IEEE Trans. Circuits Syst. Video Technol.

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Image super-resolution, a process to enhance image resolution, has important applications in satellite imaging, high definition television, medical imaging, etc. Many existing approaches use multiple low-resolution images to recover one high-resolution image. In this paper, we present an iterative scheme to solve single image super-resolution problems. It recovers a high quality high-resolution image from solely one low-resolution image without using a training data set. We solve the problem from image intensity function estimation perspective and assume the image contains smooth and edge components. We model the smooth components of an image using a thin-plate reproducing kernel Hilbert space (RKHS) and the edges using approximated Heaviside functions. The proposed method is applied to image patches, aiming to reduce computation and storage. Visual and quantitative comparisons with some competitive approaches show the effectiveness of the proposed method.

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“…Theorem 2 (see [29]) For every positive integers m, d and every p ∈ [1, ∞), span m H d is approximately a compact subset of (L p ([0, 1] d ), · p ).…”

Section: Heaviside Functionmentioning

confidence: 99%

Single image super-resolution by approximated Heaviside functions

Deng

Guo

Huang

2016

Information Sciences

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Image super-resolution is a process to enhance image resolution. It is widely used in medical imaging, satellite imaging, target recognition, etc. In this paper, we conduct continuous modeling and assume that the unknown image intensity function is defined on a continuous domain and belongs to a space with a redundant basis. We propose a new iterative model for single image super-resolution based on an observation: an image is consisted of smooth components and non-smooth components, and we use two classes of approximated Heaviside functions (AHFs) to represent them respectively. Due to sparsity of the non-smooth components, a L1 model is employed. In addition, we apply the proposed iterative model to image patches to reduce computation and storage. Comparisons with some existing competitive methods show the effectiveness of the proposed method.

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Best approximation by linear combinations of characteristic functions of half-spaces

Abstract: It is shown that in L p-spaces with p 2 1 1) there exists a best approximation mapping to the set of functions computable by Heaviside perceptron networks with n hidden units however for p 2 (1 1) such best approximation is not unique and cannot be continuous.

Cited by 19 publications

References 18 publications

Minimization of Error Functionals over Variable-Basis Functions

Minimization of Error Functionals over Variable-Basis Functions

Single-Image Super-Resolution via an Iterative Reproducing Kernel Hilbert Space Method

Single image super-resolution by approximated Heaviside functions

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