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In view of the poor performance of the traditional adaptive filtering algorithm in identifying sparse systems under impulse noise, an affine projection maximum correntropy criterion with compound inverse proportional function (APMCCCIPF) algorithm is proposed in this paper, and the compound inverse proportional function approaching l0 norm is introduced into the traditional affine projection maximum correntropy criterion (APMCC) algorithm. The fixed step‐size APMCCCIPF algorithm cannot obtain faster convergence speed and lower steady‐state error at the same time, and the most effective method to solve this contradiction is to change the fixed step‐size in the algorithm to a variable step‐size. Further, a variable step‐size factor is constructed in the APMCCCIPF algorithm, and a novel variable step‐size APMCCCIPF algorithm is proposed. A modified Gaussian function, which can resist impulse noise interference, is applied to adjust the step‐size change. Besides, the convergence and complexity of the proposed algorithms are analyzed. In the simulation of sparse system identification and echo cancellation under impulsive noise, the proposed algorithms not only achieve a lower and faster convergence rate, but also obtain a lower steady‐state error. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]