Blind deconvolution of covariance matrix inverses for autoregressive processes
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Matrix C can be blindly deconvoluted if there exist matrices A and B such that C= A∗B, where ∗ denotes the operation of matrix convolution. We study the problem of matrix deconvolution in the case where matrix C is proportional to the inverse of the autocovariance matrix of an autoregressive process. We show that the deconvolution of such matrices is important in problems of Hankel structured low-rank approximation (HSLRA). In the cases of AR(1) and AR(2) models, we fully characterize the range of parameters where such deconvolution can be performed and provide construction schemes for performing deconvolutions. We also consider general stationary AR(p) processes, where we prove that the deconvolution C= A∗B does not exist if the matrix B is diagonal and its size is larger than p.
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