Smooth isotonic covariances

We consider the problem of estimating the covariance matrix of a high-dimensional random vector in the scarce data setting, where the number of samples is less than or comparable to the dimension. The sample covariance matrix is a poor choice in this setting, and a variety of structural assumptions have been considered in the literature: covariance selection models with sparse precision matrices, low-rank models (PCA and factor analysis), sparse plus low-rank, and even multi-scale structures. We consider another type of structure, which plays an important role in several applications, where the random vectors can be `indexed' over a low-dimensional manifold, and the covariance matrix has smoothness and monotonicity properties over the manifold. These assumptions appear in applications as diverse as modeling the noise covariance in sensor-array networks, and in interest-rate modeling in computational finance. We describe how these assumptions can be enforced in a convex optimization framework using semidefinite programming (SDP) and first order proximal gradient methods, and motivate expected sample complexity requirements. We apply our approach in the interest rate modeling setting.