Algorithms are proposed to address the radar target detection problem of compressed sensing (CS) under the conditions of a low signal-to-noise ratio (SNR) and a low signal-to-clutter ratio (SCR) echo signal. The algorithms include a two-stage classification for radar targets based on compressive detection (CD) without signal reconstruction and a support vector data description (SVDD) one-class classifier. First, we present the sparsity of the echo signal in the distance dimension to design a measurement matrix for CD of the echo signal. Constant false alarm rate (CFAR) detection is performed directly on the CD echo signal to complete the first-order target classification. In simulations, the detection performance is similar to that of the traditional matched filtering algorithm, but the data rate is lower, and the necessary data storage space is reduced. Then, the power spectrum features are extracted from the data after the first-order classification and converted to the feature domain. The SVDD one-class classifier is introduced to train and classify the characteristic signals to complete the separation of the targets and the false alarms. Finally, the performance of the algorithm is verified by simulation. The number of false alarms is reduced, and the detection probability of the targets is improved.