Spectrum sensing, which aims at detecting spectrum holes, is the precondition for the implementation of cognitive radio. Collaborative spectrum sensing among the cognitive radio nodes is expected to improve the ability of checking complete spectrum usage. Due to hardware limitations, each cognitive radio node can only sense a relatively narrow band. Consequently, the available channel sensing information is far from being sufficient for recognizing the wide range of unoccupied channels precisely. Aiming at breaking this bottleneck, we propose to apply matrix completion and joint sparsity recovery to reduce sensing and transmitting requirements and improve sensing results. Specifically, equipped with a frequency selective filter, each cognitive radio node senses linear combinations of multiple channel information and reports them the fusion center, where occupied channels are then decoded from the reports by novel matrix completion and joint sparsity recovery algorithms. As a result, t...