Approximation and learning of anisotropic and mixed smooth functions by deep ReLU neural networks
arXiv:2605.31152v1 Announce Type: cross Abstract: This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions. When the error is measured in $L^p([0,1]^d)$ norm and the approximator is a network with width $W$ and depth $L$, recent works have proven the supper approximation rate $\mathcal{O}((WL)^{-2s/d})$ for Besov space $\mathcal{B}^s_{q,r}([0,1]^d)$ under the Sobolev embedding condition $s/d>1/q-1/p$. In order to overcome the curse of dimensionality in this rate, we extent this result to anisotropic and mixed smooth function classes. We establish
This research continues the ongoing effort in theoretical AI to understand and improve neural network approximation capabilities, especially concerning the 'curse of dimensionality' in learning complex functions.
Improved theoretical understanding of deep neural networks directly impacts their practical efficiency and scalability, potentially accelerating AI development and application in complex domains.
The ability to efficiently approximate anisotropic and mixed smooth functions, overcoming the curse of dimensionality, could lead to more robust and powerful deep learning models for diverse data types.
- · AI researchers and developers
- · Industries relying on complex data analysis
- · Advanced computing infrastructure providers
- · Traditional statistical modeling approaches not leveraging deep learning's full
This theoretical advancement could lead to more efficient and accurate AI models for high-dimensional data.
Better models could expand the practical applications of AI into areas previously limited by computational complexity or data size.
The enhanced capability of AI to generalize from less data could reduce training costs and accelerate research cycles in various scientific fields.
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Read at arXiv cs.LG