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Some Comparisons of Complexity in Dictionary-Based and Linear Computational Models

Gnecco, Giorgio and Kůrková, Věra and Sanguineti, Marcello Some Comparisons of Complexity in Dictionary-Based and Linear Computational Models. Neural Networks , 24 (2). pp. 171-182. ISSN 0893-6080 (2011)

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Neural networks provide a more flexible approximation of functions than traditional linear regression. In the latter, one can only adjust the coefficients in linear combinations of fixed sets of functions, such as orthogonal polynomials or Hermite functions, while for neural networks, one may also adjust the parameters of the functions which are being combined. However, some useful properties of linear approximators (such as uniqueness, homogeneity, and continuity of best approximation operators) are not satisfied by neural networks. Moreover, optimization of parameters in neural networks becomes more difficult than in linear regression. Experimental results suggest that these drawbacks of neural networks are offset by substantially lower model complexity, allowing accuracy of approximation even in high-dimensional cases. We give some theoretical results comparing requirements on model complexity for two types of approximators, the traditional linear ones and so called variable-basis types, which include neural networks, radial, and kernel models. We compare upper bounds on worst-case errors in variable-basis approximation with lower bounds on such errors for any linear approximator. Using methods from nonlinear approximation and integral representations tailored to computational units, we describe some cases where neural networks outperform any linear approximator.

Item Type: Article
Identification Number: 10.1016/j.neunet.2010.10.002
Uncontrolled Keywords: Linear approximation schemes; Variable-basis approximation schemes; Model complexity; Worst-case errors; Neural networks; Kernel models
Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Research Area: Computer Science and Applications
Depositing User: Giorgio Gnecco
Date Deposited: 17 Sep 2013 13:06
Last Modified: 17 Sep 2013 13:06
URI: http://eprints.imtlucca.it/id/eprint/1778

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