Covariance Matrix Regularization for Banded Toeplitz Structure via Frobenius-Norm Discrepancy

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In many practical applications, the structure of covariance matrix is often blurred due to random errors, making the estimation of covariance matrix very difficult particularly for high-dimensional data. In this article, we propose a regularization method for finding a possible banded Toeplitz structure for a given covariance matrix A (e.g., sample covariance matrix), which is usually an estimator of the unknown population covariance matrix Σ. We aim to find a matrix, say B, which is of banded Toeplitz structure, such that the Frobenius-norm discrepancy between B and A achieves the smallest in the whole class of banded Toeplitz structure matrices. As a result, the obtained Toeplitz structured matrix B recoveries the underlying structure behind Σ. Our simulation studies show that B is also more accurate than the sample covariance matrix A when estimating the covariance matrix Σ that has a banded Toeplitz structure. The studies also show that the proposed method works very well in regularization of covariance structure.