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Perfect compressed sensing (CS) recovery can be achieved when a certain basis space is found to sparsely represent the original signal. However, due to the diversity of the signals, there does not exist a universal predetermined basis space that can sparsely represent all kinds of signals, which results in an unsatisfying performance. To improve the accuracy of recovered signal, this paper proposes an adaptive basis CS reconstruction algorithm by minimizing the rank of an accumulated matrix (MRAM), whose eigenvectors approximate the optimal basis sparsely representing the original signal. The accumulated matrix is constructed to efficiently exploit the second-order statistical property of the signal's autocorrelations. Based on the theory of matrix completion, MRAM reconstructs the original signal from its random projections under the observation that the constructed accumulated matrix is of low rank for most natural signals such as periodic signals and those coming from an autoregressive stationary process. Experimental results show that the proposed MRAM efficiently improves the reconstruction quality compared with the existing algorithms.