Error estimation in high dimensional space for stochastic collocation methods on arbitrary sparse samples

2013 ◽  
Author(s):  
Rick Archibald
Keyword(s):  
Error Estimation ◽  
Dimensional Space ◽  
Collocation Methods ◽  
High Dimensional ◽  
Stochastic Collocation ◽  
High Dimensional Space ◽  
Stochastic Collocation Methods ◽  
Sparse Samples

Neural networks trained with high-dimensional functions approximation data in high-dimensional space

2021 ◽  
pp. 1-12
Author(s):  
Jian Zheng ◽  
Jianfeng Wang ◽  
Yanping Chen ◽  
Shuping Chen ◽  
Jingjin Chen ◽  
...  
Keyword(s):  
Neural Networks ◽  
Dimensional Space ◽  
Data Distribution ◽  
High Dimensional ◽  
Sufficient Information ◽  
Sufficient Data ◽  
High Dimensional Space ◽  
Positive Effects ◽  
The Neural Networks ◽  
Using Data

Neural networks can approximate data because of owning many compact non-linear layers. In high-dimensional space, due to the curse of dimensionality, data distribution becomes sparse, causing that it is difficulty to provide sufficient information. Hence, the task becomes even harder if neural networks approximate data in high-dimensional space. To address this issue, according to the Lipschitz condition, the two deviations, i.e., the deviation of the neural networks trained using high-dimensional functions, and the deviation of high-dimensional functions approximation data, are derived. This purpose of doing this is to improve the ability of approximation high-dimensional space using neural networks. Experimental results show that the neural networks trained using high-dimensional functions outperforms that of using data in the capability of approximation data in high-dimensional space. We find that the neural networks trained using high-dimensional functions more suitable for high-dimensional space than that of using data, so that there is no need to retain sufficient data for neural networks training. Our findings suggests that in high-dimensional space, by tuning hidden layers of neural networks, this is hard to have substantial positive effects on improving precision of approximation data.

scholarly journals A System to Assess the Semantic Content of Student Essays

2001 ◽  
Vol 24 (3) ◽  
pp. 305-320 ◽  
Author(s):  
Benoit Lemaire ◽  
Philippe Dessus
Keyword(s):  
Latent Semantic Analysis ◽  
Semantic Analysis ◽  
Dimensional Space ◽  
Semantic Content ◽  
High Dimensional ◽  
High Dimensional Space

This paper presents Apex, a system that can automatically assess a student essay based on its content. It relies on Latent Semantic Analysis, a tool which is used to represent the meaning of words as vectors in a high-dimensional space. By comparing an essay and the text of a given course on a semantic basis, our system can measure how well the essay matches the text. Various assessments are presented to the student regarding the topic, the outline and the coherence of the essay. Our experiments yield promising results.

Stochastic Collocation Methods via $\ell_1$ Minimization Using Randomized Quadratures

2017 ◽  
Vol 39 (1) ◽  
pp. A333-A359 ◽  
Author(s):  
Ling Guo ◽  
Akil Narayan ◽  
Tao Zhou ◽  
Yuhang Chen
Keyword(s):  
Collocation Methods ◽  
Stochastic Collocation ◽  
Stochastic Collocation Methods

Effective approximation of high-dimensional space using neural networks

2021 ◽  
Author(s):  
Jian Zheng ◽  
Jianfeng Wang ◽  
Yanping Chen ◽  
Shuping Chen ◽  
Jingjin Chen ◽  
...  
Keyword(s):  
Neural Networks ◽  
Dimensional Space ◽  
High Dimensional ◽  
High Dimensional Space ◽  
Effective Approximation
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