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In this work, we present a new method based on the compressive sensing (CS) theory to correct phase unwrapping (PhU) errors in the multitemporal sequence of interferograms exploited by advanced differential interferometric synthetic aperture radar (DInSAR) techniques to generate deformation time series. The developed algorithm estimates the PhU errors by using a modified $L_{1}$ -norm estimator applied to the interferometric network built in the temporal/spatial baseline plane. Indeed, in order to search the minimum $L_{1}$ -norm sparse solution, we apply the iterative reweighted least-squares method with an improved weight function that takes account of the baseline characteristics of the interferometric pairs. Moreover, we also introduce a quality function to identify those solutions that have no physical meaning. Although the proposed approach can be applied to different multitemporal DInSAR approaches, our analysis is tailored to the full-resolution small baseline subset (SBAS) processing chain that we properly modify to implement the proposed CS-based algorithm. To assess the performance of the developed technique, we carry out an extended experimental analysis based on simulated and real SAR data. In particular, we process two wide SAR datasets acquired by Sentinel-1 and COSMO-SkyMed constellations over central Italy between 2011 and 2019. The achieved experimental results clearly demonstrate the effectiveness of the developed approach in retrieving PhU errors and generating displacement time series related to strongly nonlinear deformation phenomena. Indeed, the developed CS-based technique significantly increases the number of detected coherent points and improves the accuracy of the retrieved deformation time series. [ABSTRACT FROM AUTHOR]