Efficient recovery of principal components from compressive measurements with application to Gaussian mixture model estimation

There has been growing interest in performing signal processing tasks directly on compressive measurements, e.g. low-dimensional linear measurements of signals taken with Gaussian random vectors. In this paper, we present a highly efficient algorithm to recover the covariance matrix of high-dimensional data from compressive measurements. We show that, as the number of data samples increases, the eigenvectors (principal components) of the empirical covariance matrix of a simple matrix-vector multiplication of the compressive measurements converge to the true principal components of the original data. Also, we investigate the perturbation of eigenvalues of the covariance matrix under random projection of the data to find conditions for approximate recovery of them. Furthermore, we introduce an important application of our proposed method for efficient estimation of the parameters of Gaussian Mixture Models from compressive measurements. We present experimental results demonstrating the performance and efficiency of our proposed algorithms.