In the present paper, our purpose is to obtain a nonlinear approximation by using convergence in $\varphi$-variation. Angeloni and Vinti prove some approximation results considering linear sampling-type discrete operators. These types of operators have close relationships with generalized sampling series. By improving Angeloni and Vinti's one, we aim to get a nonlinear approximation which is also generalized by means of summability process. We also evaluate the rate of approximation under appropriate Lipschitz classes of $\varphi$-absolutely continuous functions. Finally, we give some examples of kernels, which fulfill our kernel assumptions.