Regular families of kernels for nonlinear approximation

Hamm, Keaton; Ledford, Jeff

doi:10.1016/j.jmaa.2019.03.015

Cited by 8 publications

(6 citation statements)

References 36 publications

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“…More specifically, there is more comprehensive theoretical approximation bound results. For example, [16,25,26] established such results assuming that the true function is specified by a Besov space or a Sobolev ball. Our theoretical result of Section 3 assumes that the true function belongs to a Hölder space and approximation error bound is established for such class.…”

Section: Flexibilitymentioning

confidence: 99%

“…We have the universal approximation theorem for RBF‐net [53] that ensures approximation of any truth

f_{0} false(boldx false)

using the above

f false(boldx false)

with arbitrarily small approximation error for large enough

K

. Under some assumptions on the smoothness of

f_{0} false(boldx false)

, we can compute the upper bound to the approximation error for a given

K

[16, 25, 26, 45, 72, 74]. We use such upper bounds while establishing posterior contraction rates in Section 3.…”

Section: High Dimensional Multivariate Gaussian Rbf‐netmentioning

confidence: 99%

“…There is a substantially rich literature on approximation theory using RBF‐net where the true function is specified under various smoothness classes such as using a Besov space or a Sobolev ball [16, 25, 26]. Gaussian RBF‐net estimation also shares commonality with the literature on nonparametric density estimation using a location‐scale mixture of normals.…”

Section: Theoretical Supportmentioning

confidence: 99%

See 2 more Smart Citations

Multivariate Gaussian RBF‐net for smooth function estimation and variable selection

Roy

2021

Statistical Analysis

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Neural networks are routinely used for nonparametric regression modeling. The interest in these models is growing with ever-increasing complexities in modern datasets. With modern technological advancements, the number of predictors frequently exceeds the sample size in many application areas. Thus, selecting important predictors from the huge pool is an extremely important task for judicious inference. This paper proposes a novel flexible class of single-layer radial basis functions (RBF) networks. The proposed architecture can estimate smooth unknown regression functions and also perform variable selection. We primarily focus on Gaussian RBF-net due to its attractive properties. The extensions to other choices of RBF are fairly straightforward. The proposed architecture is also shown to be effective in identifying relevant predictors in a low-dimensional setting using the posterior samples without imposing any sparse estimation scheme. We develop an efficient Markov chain Monte Carlo algorithm to generate posterior samples of the parameters. We illustrate the proposed method's empirical efficacy through simulation experiments, both in high and low dimensional regression problems. The posterior contraction rate is established with respect to empirical 2 distance assuming that the error variance is unknown, and the true function belongs to a Hölder ball. We illustrate our method in a Human Connectome Project dataset to predict vocabulary comprehension and to identify important edges of the structural connectome.

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Section: Flexibilitymentioning

confidence: 99%

“…We have the universal approximation theorem for RBF‐net [53] that ensures approximation of any truth

f_{0} false(boldx false)

using the above

f false(boldx false)

with arbitrarily small approximation error for large enough

K

. Under some assumptions on the smoothness of

f_{0} false(boldx false)

, we can compute the upper bound to the approximation error for a given

K

[16, 25, 26, 45, 72, 74]. We use such upper bounds while establishing posterior contraction rates in Section 3.…”

Section: High Dimensional Multivariate Gaussian Rbf‐netmentioning

confidence: 99%

Section: Theoretical Supportmentioning

confidence: 99%

See 1 more Smart Citation

Multivariate Gaussian RBF‐net for smooth function estimation and variable selection

Roy

2021

Statistical Analysis

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show abstract

“…We have the universal approximation theorem for RBF-net (Park and Sandberg, 1991) that ensures approximation of any truth f 0 (x) using the above f (x) with arbitrarily small approximation error for large enough K. Under some assumptions on the smoothness of f 0 (x), we can compute the upper bound to the approximation error for a given K (Maiorov, 2003;Hangelbroek and Ron, 2010;DeVore and Ron, 2010;Hamm and Ledford, 2019). The matrix DΩD in (3) is non-negative definite for any d ∈ R p and Ω ∈ R p ++ , the space of correlation matrices.…”

Section: Modelingmentioning

confidence: 99%

Nonparametric Group Variable Selection with Multivariate Response for Connectome-Based Modeling of Cognitive Scores

Roy¹

2021

Preprint

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In this article, we study possible relations between the structural connectome and cognitive profiles using a multi-response nonparametric regression model under group sparsity. The aim is to identify the brain regions having significant effect on cognitive functioning. The cognitive profiles are measured in terms of seven cognitive test scores from NIH toolbox of cognitive battery. We consider age-adjusted cognitive scores in this article. The structural connectomes are represented by adjacency matrices. Most existing works consider the upper or lower triangular section of these adjacency matrices as predictors. An alternative characterization of the connectivity properties is available in terms of the nodal attributes. Although any single nodal attribute may not be adequate to represent a complex brain network, a collection of nodal centralities together can encode different patterns of connections in the brain network. In this article, we consider nine different attributes for each brain region as our predictors. Here, each node thus correspond to a collection of nodal attributes. Hence, these nodal graph metrics may naturally be grouped together for each node, motivating us to introduce group sparsity for feature selection. We propose Gaussian RBF-nets with a novel group sparsity inducing prior to model the unknown mean functions. The covariance structure of the multivariate response is characterized in terms of a linear factor modeling framework. For posterior computation, we develop an efficient Markov chain Monte Carlo sampling algorithm. We show that the proposed method performs overwhelmingly better than all its competitors. Applying our proposed method to a Human Connectome Project (HCP) dataset, we identify the important brain regions and nodal attributes for cognitive functioning, as well as identify interesting low-dimensional dependency structures among the cognition related test scores.

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“…Figure 1 in [85] visualizes these relations and tells us that "time or band-localization" is wider than "time or band-limitness". A subspace of O M is D, the space of "time-limited" Schwartz functions, and a subspace of O C is Z, the space of "band-limited" Schwartz functions [145], cf. Figure 1 in [85].…”

Section: Finite Entire Local and Regular Functionsmentioning

confidence: 99%

On the Reversibility of Discretization

Fischer

Stens

2020

Mathematics

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“Discretization” usually denotes the operation of mapping continuous functions to infinite or finite sequences of discrete values. It may also mean to map the operation itself from one that operates on functions to one that operates on infinite or finite sequences. Advantageously, these two meanings coincide within the theory of generalized functions. Discretization moreover reduces to a simple multiplication. It is known, however, that multiplications may fail. In our previous studies, we determined conditions such that multiplications hold in the tempered distributions sense and, hence, corresponding discretizations exist. In this study, we determine, vice versa, conditions such that discretizations can be reversed, i.e., functions can be fully restored from their samples. The classical Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem is just one particular case in one of four interwoven symbolic calculation rules deduced below.

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Regular families of kernels for nonlinear approximation

Cited by 8 publications

References 36 publications

Multivariate Gaussian RBF‐net for smooth function estimation and variable selection

Multivariate Gaussian RBF‐net for smooth function estimation and variable selection

Nonparametric Group Variable Selection with Multivariate Response for Connectome-Based Modeling of Cognitive Scores

On the Reversibility of Discretization

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