To deal with the fiber nonlinear impairments in coherent optical communication systems, this paper proposes a nonlinear compensation (NLC) algorithm based on deep neural network (DNN) and improved principal component analysis (IPCA) by using the triplets derived from the first-order perturbation solution of the nonlinear Schr?dinger equation. The simulation systems of a single-channel 32 GBaud polarization-division-multiplexing 16-ary quadrature amplitude modulation (PDM-16QAM) optical transmission system are built to verify the feasibility of the proposed NLC algorithm. Compared with the DNN-NLC scheme, the IPCA-DNN-NLC scheme reduces the computational complexity by 90.7% with only a 0.06 dB Q-factor penalty, which means that the new algorithm enables similar NLC performance with much lower complexity. Compared with the digital back propagation (DBP) scheme, the IPCA-DNN-NLC scheme realizes a 0.91 dB Q-factor improvement over 800 km transmission. The proposed scheme can work normally without prior knowledge of the link parameters, which is versatile and robust.