Almost optimal estimates for approximation and learning by radial basis function networks

Lin, Shao-Bo; Liu, Xia; Rong, Yuanhua; Zhao, Xu

doi:10.1007/s10994-013-5406-z

Cited by 30 publications

(25 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The approximation rate O(N −2α/d ) is tight in terms of N and increasing L cannot improve the approximation rate in N. The success of deep Figure 1: A summary of existing and our new results on the approximation rate of ReLU FNNs for continuous functions. Existing results [18,25,40,44,62,65,67] are applicable in the areas in , , and ; our new result is suitable for almost all areas when L ≥ 2.…”

Section: Introductionmentioning

confidence: 71%

“…For another example, given an approximation error ε, [54] proved the existence of a ReLU FNN with a constant but still unknown number of layers approximating a C β function within the target error. These works can be divided into two cases: 1) FNNs with varying width and only one hidden layer [18,25,40,65] (visualized by the region in in Fig. 1); 2) FNNs with a fixed width of O(d) and a varying depth larger than an unknown number L [44,67] (represented by the region in in Fig.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Deep Network Approximation Characterized by Number of Neurons

Shen¹

2020

CICP

View full text Add to dashboard Cite

This paper develops simple feed-forward neural networks that achieve the universal approximation property for all continuous functions with a fixed finite number of neurons. These neural networks are simple because they are designed with a simple and computable continuous activation function σ leveraging a triangularwave function and a softsign function. We prove that σ-activated networks with width 36d(2d + 1) and depth 11 can approximate any continuous function on a d-dimensioanl hypercube within an arbitrarily small error. Hence, for supervised learning and its related regression problems, the hypothesis space generated by these networks with a size not smaller than 36d(2d + 1) × 11 is dense in the space of continuous functions. Furthermore, classification functions arising from image and signal classification are in the hypothesis space generated by σ-activated networks with width 36d(2d + 1) and depth 12, when there exist pairwise disjoint closed bounded subsets of R d such that the samples of the same class are located in the same subset.

show abstract

Section: Introductionmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Deep Network Approximation Characterized by Number of Neurons

Shen¹

2020

CICP

View full text Add to dashboard Cite

show abstract

“…Refs. [158][159][160][161][162], this technique has only been applied to limited prognostic studies. iii.…”

Section: Artificial Neural Network (Ann)mentioning

confidence: 99%

A review on prognostic techniques for non-stationary and non-linear rotating systems

Kan

Tan

Mathew

2015

Mechanical Systems and Signal Processing

305

153

View full text Add to dashboard Cite

a b s t r a c tThe field of prognostics has attracted significant interest from the research community in recent times. Prognostics enables the prediction of failures in machines resulting in benefits to plant operators such as shorter downtimes, higher operation reliability, reduced operations and maintenance cost, and more effective maintenance and logistics planning. Prognostic systems have been successfully deployed for the monitoring of relatively simple rotating machines. However, machines and associated systems today are increasingly complex. As such, there is an urgent need to develop prognostic techniques for such complex systems operating in the real world. This review paper focuses on prognostic techniques that can be applied to rotating machinery operating under nonlinear and non-stationary conditions. The general concept of these techniques, the pros and cons of applying these methods, as well as their applications in the research field are discussed. Finally, the opportunities and challenges in implementing prognostic systems and developing effective techniques for monitoring machines operating under nonstationary and non-linear conditions are also discussed.

show abstract

“…Given the importance of this issue, establishing the universal approximation ability of LS-SVR as a universal approximation, like fuzzy systems and neural networks, is significant and should be demonstrated. In this study, the universal approximation theorem for Gaussian kernel is briefly reviewed and meticulous details of the universal approximation theorems for the other kernels which are used in the literature are referred to [38][39][40].…”

Section: Universal Approximation Property Of Ls-svrmentioning

confidence: 99%