Covariance Estimation with Piecewise-Smooth Regularization

Covariance estimation for ordered variables is often built on a global regularity assumption, yet many covariance surfaces are only locally smooth: dependence evolves gradually within regimes and changes slope at a small number of locations. This paper addresses the estimation of such piecewise-smooth covariance matrices under a positive semidefinite constraint. We formulate this structure as sparsity of masked second divided differences, where the mask leaves the central variance coordinate of each stencil unpenalized, preserving near-diagonal dependence while allowing marginal variances to vary freely. The estimator combines a Frobenius data-fidelity term with a folded-concave penalty, and we develop an efficient ADMM-based algorithm that provably converges to a first-order stationary point. Under sub-Gaussian sampling, we further establish a finite-sample oracle inequality showing that the error decomposes into the oracle approximation error and a selection cost of order $\log p/n$ per unknown slope change. Simulations and an electricity-load forecasting task confirm that the method captures local slope changes that global smoothness models miss.