The Internet of Things paradigm heavily relies on a network of a massive number of machine-type devices (MTDs) that monitor various phenomena. Consequently, MTDs are randomly activated at different times whenever a change occurs. In general, fewer MTDs are simultaneously activated across the network, resembling targeted sampling in compressed sensing. Therefore, signal recovery in machine-type communications is addressed through joint user activity detection and channel estimation algorithms built using compressed sensing theory. However, most of these algorithms follow a two-stage procedure in which a channel is first estimated and later mapped to find active users. This approach is inefficient because the estimated channel information is subsequently discarded. To overcome this limitation, we introduce a novel covariance-learning matching pursuit (CL-MP) algorithm that bypasses explicit channel estimation. Instead, it focuses on estimating the indices of the active users greedily. Simulation results presented in terms of probability of miss detection, exact recovery rate, and computational complexity validate the proposed technique's superior performance and efficiency.