This paper proposes a novel algorithm for inferring gene regulatory networks which makes use of cubature Kalman filter (CKF) and Kalman filter (KF) techniques in conjunction with compressed sensing methods. The gene network is described using a state-space model. A nonlinear model for the evolution of gene expression is considered, while the gene expression data is assumed to follow a linear Gaussian model. The hidden states are estimated using CKF. The system parameters are modeled as a Gauss-Markov process and are estimated using compressed sensing-based KF. These parameters provide insight into the regulatory relations among the genes. The Cramér-Rao lower bound of the parameter estimates is calculated for the system model and used as a benchmark to assess the estimation accuracy. The proposed algorithm is evaluated rigorously using synthetic data in different scenarios which include different number of genes and varying number of sample points. In addition, the algorithm is tested on the DREAM4 in silico data sets as well as the in vivo data sets from IRMA network. The proposed algorithm shows superior performance in terms of accuracy, robustness, and scalability.