The Optimization Landscape of Carath'eodory Decomposition of Toeplitz Covariances

Toeplitz covariance estimation is a classical signal processing problem, yet the geometry of the Gaussian maximum-likelihood objective remains only partially understood. Recent Newton-type, majorization-minimization, and gradient-based methods often solve the nonconvex problem globally with enough samples, but also reveal a difficult computational landscape. We study this phenomenon through an overparameterized Carath'eodory representation, which parameterizes the covariance by the frequencies and amplitudes of steering vectors. A simple example shows that, for any uniform frequency grid, optimizing only the amplitudes can leave a positive error even in the population setting, which motivates optimizing the frequencies as well. Our main result proves that this nonlinear parameterization adds no stationary points: for any sample covariance, every stationary point with a positive definite covariance gives a stationary point of the Toeplitz maximum-likelihood problem. In the population setting, every such point therefore recovers the true covariance. We also prove that gradient descent converges to the true covariance in the population setting, and to the unique maximum likelihood estimate when the sample covariance is close enough to the true covariance. These results do not require overparameterization. Numerically, overparameterization speeds up gradient descent and gives a more accurate estimate for a fixed number of iterations.