Vibration signals captured from faulty mechanical components are often associated with transients which are significant for machinery fault diagnosis. However, the existence of strong background noise makes the detection of transients a basis pursuit denoising (BPD) problem, which is hard to be solved in explicit form. With sparse representation theory, this paper proposes a novel method for machinery fault diagnosis by combining the wavelet basis and majorization-minimization (MM) algorithm. This method converts transients hidden in the noisy signal into sparse coefficients; thus the transients can be detected sparsely. Simulated study concerning cyclic transient signals with different signal-to-noise ratio (SNR) shows that the effectiveness of this method. The comparison in the simulated study shows that the proposed method outperforms the method based on split augmented Lagrangian shrinkage algorithm (SALSA) in convergence and detection effect. Application in defective gearbox fault diagnosis shows the fault feature of gearbox can be sparsely and effectively detected. A further comparison between this method and the method based on SALSA shows the superiority of the proposed method in machinery fault diagnosis.