A result relating convex n-widths to covering numbers with some applications to neural networks
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912749396688896 |
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| author | Baxter, Jonathan Bartlett, Peter |
| author_facet | Baxter, Jonathan Bartlett, Peter |
| contents | In general, approximating classes of functions defined over high-dimensional input spaces by linear combinations of a fixed set of basis functions or ``features'' is known to be hard. Typically, the worst-case error of the best basis set decays only as fast as $Θ\(n^{-1/d}\)$, where $n$ is the number of basis functions and $d$ is the input dimension. However, there are many examples of high-dimensional pattern recognition problems (such as face recognition) where linear combinations of small sets of features do solve the problem well. Hence these function classes do not suffer from the ``curse of dimensionality'' associated with more general classes. It is natural then, to look for characterizations of high-dimensional function classes that nevertheless are approximated well by linear combinations of small sets of features. In this paper we give a general result relating the error of approximation of a function class to the covering number of its ``convex core''. For one-hidden-layer neural networks, covering numbers of the class of functions computed by a single hidden node upper bound the covering numbers of the convex core. Hence, using standard results we obtain upper bounds on the approximation rate of neural network classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04912 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A result relating convex n-widths to covering numbers with some applications to neural networks Baxter, Jonathan Bartlett, Peter Machine Learning In general, approximating classes of functions defined over high-dimensional input spaces by linear combinations of a fixed set of basis functions or ``features'' is known to be hard. Typically, the worst-case error of the best basis set decays only as fast as $Θ\(n^{-1/d}\)$, where $n$ is the number of basis functions and $d$ is the input dimension. However, there are many examples of high-dimensional pattern recognition problems (such as face recognition) where linear combinations of small sets of features do solve the problem well. Hence these function classes do not suffer from the ``curse of dimensionality'' associated with more general classes. It is natural then, to look for characterizations of high-dimensional function classes that nevertheless are approximated well by linear combinations of small sets of features. In this paper we give a general result relating the error of approximation of a function class to the covering number of its ``convex core''. For one-hidden-layer neural networks, covering numbers of the class of functions computed by a single hidden node upper bound the covering numbers of the convex core. Hence, using standard results we obtain upper bounds on the approximation rate of neural network classes. |
| title | A result relating convex n-widths to covering numbers with some applications to neural networks |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2512.04912 |