In cognitive radio, spectrum sensing is a key component to detect spectrum holes (i.e., channels not used by any primary users). Collaborative spectrum sensing among the cognitive radio nodes is expected to improve the ability of checking complete spectrum usage states. Unfortunately, due to power limitation and channel fading, available channel sensing information is far from being sufficient to tell the unoccupied channels directly. Aiming at breaking this bottleneck, we apply recent matrix completion techniques to greatly reduce the sensing information needed. We formulate the collaborative sensing problem as a matrix completion subproblem and a joint-sparsity reconstruction subproblem. Results of numerical simulations that validated the effectiveness and robustness of the proposed approach are presented. In particular, in noiseless cases, when number of primary user is small, exact detection was obtained with no more than 8% of the complete sensing information, whilst as number of primary user increases, to achieve a detection rate of 95.55%, the required information percentage was merely 16.8%.