Optimal Neural Network Approximation for High-Dimensional Continuous Functions

• URL: http://arxiv.org/abs/2409.02363v2
• Date: Tue, 10 Sep 2024 13:40:15 GMT
• Title: Optimal Neural Network Approximation for High-Dimensional Continuous Functions
• Authors: Ayan Maiti, Michelle Michelle, Haizhao Yang,
• Abstract summary: We present a family of continuous functions that requires at least width $d$, and therefore at least $d$ intrinsic neurons, to achieve arbitrary accuracy in its approximation. This shows that the requirement of $mathcalO(d)$ intrinsic neurons is optimal in the sense that it grows linearly with the input dimension $d$.
• Score: 5.748690310135373
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Recently, the authors of Shen Yang Zhang (JMLR, 2022) developed a neural network with width $36d(2d + 1)$ and depth $11$, which utilizes a special activation function called the elementary universal activation function, to achieve the super approximation property for functions in $C([a,b]^d)$. That is, the constructed network only requires a fixed number of neurons to approximate a $d$-variate continuous function on a $d$-dimensional hypercube with arbitrary accuracy. Their network uses $\mathcal{O}(d^2)$ fixed neurons. One natural question to address is whether we can reduce the number of these neurons in such a network. By leveraging a variant of the Kolmogorov Superposition Theorem, our analysis shows that there is a neural network generated by the elementary universal activation function with only $366d +365$ fixed, intrinsic (non-repeated) neurons that attains this super approximation property. Furthermore, we present a family of continuous functions that requires at least width $d$, and therefore at least $d$ intrinsic neurons, to achieve arbitrary accuracy in its approximation. This shows that the requirement of $\mathcal{O}(d)$ intrinsic neurons is optimal in the sense that it grows linearly with the input dimension $d$, unlike some approximation methods where parameters may grow exponentially with $d$.

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