Covariance matrix estimation techniques require high acquisition costs that challenge the sampling systems storing and transmission capabilities. For this reason, various acquisition approaches have been developed to simultaneously sense and compress the relevant information of the signal using random projections. However, estimating the covariance matrix from the random projections is an ill-posed problem that requires further information about the data, such as sparsity, low rank, or stationary behavior. Furthermore, this approach fails using high compression ratios. Therefore, this paper proposes an algorithm based on the projected gradient method to recover a low-rank or Toeplitz approximation of the covariance matrix. The proposed algorithm divides the data into subsets projected onto different subspaces, assuming that each subset contains an approximation of the signal statistics, improving the inverse problems condition. The error induced by this assumption is analytically derived along with the convergence guarantees of the proposed method. Extensive simulations show that the proposed algorithm can effectively recover the covariance matrix of hyperspectral images with high compression ratios (8-15% approx) in noisy scenarios. Additionally, simulations and theoretical results show that filtering the gradient reduces the estimators error recovering up to twice the number of eigenvectors.
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