The Kolmogorov Superposition Theorem can Break the Curse of Dimensionality When Approximating High Dimensional Functions
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arXiv
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| Format: | Preprint |
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2021
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| author | Lai, Ming-Jun Shen, Zhaiming |
| author_facet | Lai, Ming-Jun Shen, Zhaiming |
| contents | We explain how to use Kolmogorov Superposition Theorem (KST) to break the curse of dimensionality when approximating a dense class of multivariate continuous functions. We first show that there is a class of functions called Kolmogorov-Lipschitz (KL) continuous in $C([0,1]^d)$ which can be approximated by a special ReLU neural network of two hidden layers with a dimension independent approximation rate $O(1/n)$ with approximation constant increasing quadratically in $d$. The number of parameters used in such neural network approximation equals to $(6d+2)n$. Next we introduce KB-splines by using linear B-splines to replace the outer function and smooth the KB-splines to have the so-called LKB-splines as the basis for approximation. Our numerical evidence shows that the curse of dimensionality is broken in the following sense: When using the standard discrete least squares (DLS) method to approximate a continuous function, there exists a pivotal set of points in $[0,1]^d$ with size at most $O(nd)$ such that the rooted mean squares error (RMSE) from the DLS based on the pivotal set is similar to the RMSE of the DLS based on the original set with size $O(n^d)$. The pivotal point set is chosen by using matrix cross approximation technique and the number of LKB-splines used for approximation is the same as the size of the pivotal data set. Therefore, we do not need too many basis functions nor too many function values to approximate a high dimensional continuous function $f$. Hence, the study in this paper provides an approach for dimension reduction problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_09963 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Kolmogorov Superposition Theorem can Break the Curse of Dimensionality When Approximating High Dimensional Functions Lai, Ming-Jun Shen, Zhaiming Numerical Analysis We explain how to use Kolmogorov Superposition Theorem (KST) to break the curse of dimensionality when approximating a dense class of multivariate continuous functions. We first show that there is a class of functions called Kolmogorov-Lipschitz (KL) continuous in $C([0,1]^d)$ which can be approximated by a special ReLU neural network of two hidden layers with a dimension independent approximation rate $O(1/n)$ with approximation constant increasing quadratically in $d$. The number of parameters used in such neural network approximation equals to $(6d+2)n$. Next we introduce KB-splines by using linear B-splines to replace the outer function and smooth the KB-splines to have the so-called LKB-splines as the basis for approximation. Our numerical evidence shows that the curse of dimensionality is broken in the following sense: When using the standard discrete least squares (DLS) method to approximate a continuous function, there exists a pivotal set of points in $[0,1]^d$ with size at most $O(nd)$ such that the rooted mean squares error (RMSE) from the DLS based on the pivotal set is similar to the RMSE of the DLS based on the original set with size $O(n^d)$. The pivotal point set is chosen by using matrix cross approximation technique and the number of LKB-splines used for approximation is the same as the size of the pivotal data set. Therefore, we do not need too many basis functions nor too many function values to approximate a high dimensional continuous function $f$. Hence, the study in this paper provides an approach for dimension reduction problems. |
| title | The Kolmogorov Superposition Theorem can Break the Curse of Dimensionality When Approximating High Dimensional Functions |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2112.09963 |