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- W4310413607 abstract "Let $Omega = [0,1]^d$ be the unit cube in $mathbb{R}^d$. We study the problem of how efficiently, in terms of the number of parameters, deep neural networks with the ReLU activation function can approximate functions in the Sobolev spaces $W^s(L_q(Omega))$ and Besov spaces $B^s_r(L_q(Omega))$, with error measured in the $L_p(Omega)$ norm. This problem is important when studying the application of neural networks in a variety of fields, including scientific computing and signal processing, and has previously been completely solved only when $p=q=infty$. Our contribution is to provide a complete solution for all $1leq p,qleq infty$ and $s > 0$, including asymptotically matching upper and lower bounds. The key technical tool is a novel bit-extraction technique which gives an optimal encoding of sparse vectors. This enables us to obtain sharp upper bounds in the non-linear regime where $p > q$. We also provide a novel method for deriving $L_p$-approximation lower bounds based upon VC-dimension when $p < infty$. Our results show that very deep ReLU networks significantly outperform classical methods of approximation in terms of the number of parameters, but that this comes at the cost of parameters which are not encodable." @default.
- W4310413607 created "2022-12-10" @default.
- W4310413607 creator A5072176149 @default.
- W4310413607 date "2022-11-25" @default.
- W4310413607 modified "2023-10-18" @default.
- W4310413607 title "Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spaces" @default.
- W4310413607 doi "https://doi.org/10.48550/arxiv.2211.14400" @default.
- W4310413607 hasPublicationYear "2022" @default.
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