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Multivariate Regression using Neural Networks and Sums of Separable Functions

Herath, Herath Mudiyanselage Indupama Umayangi
http://orcid.org/0000-0002-1063-1813
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1648166101093853

Abstract Details

2022, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
Currently, artificial neural networks are the most popular approach to machine learning problems such as high-dimensional multivariate regression. Methods using sums of separable functions, which grew out of tensor decompositions, are designed to represent functions in high dimensions and can be applied to high-dimensional multivariate regression. Here we compare the ability of these two methods to approximate function spaces in order to assess their relative expressive power. We show that a general neural network result can be translated into sums of separable functions if the activation function satisfies certain smoothness conditions. Comparatively, we show that it is possible to approximate any sums of separable function result with neural networks using the approximation of products of functions by deep neural networks. We identify general approximation schemes in both the single-layer and deep-layer settings that apply to both methods for approximating certain function classes. In particular, we show that sums of separable functions give the same error rates as neural networks for function classes such as Barron’s functions and band-limited functions. Inspired by deep neural networks, we also introduce deep layer sums of separable functions that shows similar results as deep neural networks for functions with compositional structure.
Dr. Martin Mohlenkamp (Advisor)
Dr. David Chelberg (Committee Member)
Dr. Qiliang Wu (Committee Member)
Dr. Todd Young (Committee Member)
147 p.
Mathematics
Artificial neural networks; sums of separable functions; multivariate regression problem

Recommended Citations

Citations

  • Herath, H. M. I. U. (2022). Multivariate Regression using Neural Networks and Sums of Separable Functions [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1648166101093853

    APA Style (7th edition)

  • Herath, Herath Mudiyanselage Indupama. Multivariate Regression using Neural Networks and Sums of Separable Functions. 2022. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1648166101093853.

    MLA Style (8th edition)

  • Herath, Herath Mudiyanselage Indupama. "Multivariate Regression using Neural Networks and Sums of Separable Functions." Doctoral dissertation, Ohio University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1648166101093853

    Chicago Manual of Style (17th edition)

ohiou1648166101093853
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This open access ETD is published by Ohio University and OhioLINK.