Adaptive Rank Approximation for FFT-based Toeplitz Inversion in Structured Covariance Estimation

The inversion of Toeplitz matrices plays a vital role in time series forecasting, structured covariance estimation in machine learning and signal processing. Traditional methods, including FFT-based methods and Levinson recursion, suffer from computational inefficiencies when dealing with large-scale datasets. This research recommends an Adaptive Rank Approximation (ARA) method incorporated with FFT-based Toeplitz inversion, which selectively recollects the most significant singular values, dynamically minimizing the computational complexity to O(k log k), where k << n. The proposed approach optimally balances accuracy and efficiency by leveraging eigenvalue thresholding for adaptive rank selection. Experimental assessments demonstrate significant improvements in computation time and stability over existing methods. The results confirm the superiority of our method in large-scale numerical computations and structured matrix operations.