Modern analog computing, by gaining momentum from nonvolatile resistive memory devices, deals with matrix computations. In-memory analog computing has been demonstrated for solving some basic but ordinary matrix problems in one step. Among the more complicated matrix problems, compressed sensing (CS) is a prominent example, whose recovery algorithms feature high-order matrix operations and hardware-unfriendly nonlinear functions. In light of the local competitive algorithm (LCA), here, we present a closed-loop, continuous-time resistive memory circuit for solving CS recovery in one step. Recovery of one-dimensional (1D) sparse signal and 2D compressive images has been experimentally demonstrated, showing elapsed times around few microseconds and normalized mean squared errors of 10-2. The LCA circuit is one or two orders of magnitude faster than conventional digital approaches. It also substantially outperforms other (electronic or exotically photonic) analog CS recovery methods in terms of speed, energy, and fidelity, thus representing a highly promising technology for real-time CS applications.