In the typical vibration signal of a reciprocating air compressor, multi-source nonlinear characteristics are exhibited and are often drowned out in background noise, which leads to a lack of robustness in traditional feature analysis methods and difficulty in effective extraction. To address this issue, an algorithm based on ABC-SGMD is proposed in this paper. The Symplectic Geometry Mode Decomposition (SGMD), which is optimized with the Artificial Bee Colony algorithm (ABC), is utilized to decompose the signal, and a multi-feature fusion model is constructed for fault feature extraction. The extracted features are then input into the Self-Adaptive Evolutionary Extreme Learning Machine (SaDE-ELM), and a fault diagnosis model based on ABC-SGMD and SaDE-ELM is established. Ultimately, the signals of reciprocating air compressors and experimental data are used to demonstrate the applicability of the method. The results manifest that this framework has superiority in handling nonlinear and non-stationary signals.