{"paper":{"title":"On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.NA"],"primary_cat":"stat.ML","authors_text":"Yunfei Yang","submitted_at":"2024-09-02T02:26:01Z","abstract_excerpt":"This paper studies the problem of how efficiently functions in the Sobolev spaces $\\mathcal{W}^{s,q}([0,1]^d)$ and Besov spaces $\\mathcal{B}^s_{q,r}([0,1]^d)$ can be approximated by deep ReLU neural networks with width $W$ and depth $L$, when the error is measured in the $L^p([0,1]^d)$ norm. This problem has been studied by several recent works, which obtained the approximation rate $\\mathcal{O}((WL)^{-2s/d})$ up to logarithmic factors when $p=q=\\infty$, and the rate $\\mathcal{O}(L^{-2s/d})$ for networks with fixed width when the Sobolev embedding condition $1/q -1/p