The study focuses on accurately extracting non-periodic fault pulses from rolling bearings operating at variable speeds. Traditional methods face difficulties due to the non-periodic nature of the pulses. To address this, a sparse representation method is proposed by utilizing flexible analytic wavelet transform (FAWT) as dictionary and non-convex regularization. This method effectively matches and extracts the non-periodic fault pulses, while avoiding underestimation of large pulses. A multi-objective optimization model using angular domain correlation kurtosis and harmonic-to-noise ratio of envelope order spectrum is constructed and solved using the NSGA-II algorithm. The density estimation strategy is proposed to find the optimal parameter combination on the Pareto front. Experimental of simulation signal results show that the proposed method can accurately extract non-periodic fault pulses.