We propose an unsupervised compressed sensing (CS)-based framework to compress, recover, and cluster neural action potentials. This framework can be easily integrated into high-density multi-electrode neural recording VLSI systems. Embedding spectral clustering and group structures in dictionary learning, we extend the proposed framework to unsupervised spike sorting without prior label information. Additionally, we incorporate group sparsity concepts in the dictionary learning to enable the framework for multi-channel neural recordings, as in tetrodes. To further improve spike sorting success rates in the CS framework, we embed template matching in sparse coding to jointly predict clusters of spikes. Our experimental results demonstrate that the proposed CS-based framework can achieve a high compression ratio (8:1 to 20:1), with a high quality reconstruction performance (>8 dB) and a high spike sorting accuracy (>90%).