Approximation by neural network operators of convolution type activated by deformed and parametrized half hyperbolic tangent function
Asiye Arif, Tugba Yurdakadim
TL;DR
The paper addresses the problem of function approximation on $\mathbb{R}$ using convolution-type neural network operators activated by a symmetrized, $q$-deformed and $\beta$-parametrized half-hyperbolic tangent. It constructs three operators $\mathtt{B}_n$, $\mathtt{B}_n^*$, and $\overline{\mathtt{B}_n}$ with a kernel drawn from the density $\Psi$ and proves quantitative convergence to the identity via the modulus of continuity, along with global smoothness preservation. The authors extend the analysis to derivatives and provide higher-order asymptotics, including expansions with $\mathtt{A}_n$ and $\mathtt{A}_n^*$ terms. They also study iterated versions $\mathtt{B}_n^{r}$, $\mathtt{B}_n^{*r}$, and $\overline{\mathtt{B}_n}^{r}$, showing that iteration maintains convergence rates (not worse than the base operators) and preserves continuity and boundedness. Overall, the work blends quantum/deformed activation schemes with positive linear operator theory to yield provable, rate-guaranteed neural-network-inspired approximation operators.
Abstract
Here, we introduce three kinds of neural network operators of convolution type which are activated by q-deformed and \b{eta}-parametrized half hyperbolic tangent function. We obtain quantitative convergence results to the identity operator with the use of modulus of continuity. Global smoothness preservation of our operators are also presented and the iterated versions of them are taken into the consideration.