We explore several reduced-dimension multiuser detection (RD-MUD) structures that significantly decrease the number of required correlation branches at the receiver front-end, while still achieving performance similar to that of the conventional matched-filter (MF) bank. RD-MUD exploits the fact that the number of active users is typically small relative to the total number of users in the system and relies on ideas of analog compressed sensing to reduce the number of correlators. We first develop a general framework for both linear and nonlinear RD-MUD structures. We then present theoretical performance analysis for two specific detectors: the linear reduced-dimension decorrelating (RDD) detector, which combines subspace projection and thresholding to determine active users and sign detection for data recovery, and the nonlinear reduced-dimension decision-feedback (RDDF) detector, which combines decision-feedback orthogonal matching pursuit for active user detection and sign detection for data recovery. The theoretical performance results for both detectors are validated via numerical simulations.