In this paper, we propose an algorithm for downlink (DL) channel covariance matrix (CCM) estimation for frequency division duplexing (FDD) massive multiple-input multiple-output (MIMO) communication systems with base station (BS) possessing a uniform linear array (ULA) antenna structure. We make use of the inherent similarity between the uplink (UL) CCM and the DL CCM due to angular reciprocity. We consider a setting where the UL CCM is mapped to DL CCM by a mapping function. We first present a theoretical error analysis of learning a nonlinear embedding by constructing a mapping function, which points to the importance of the Lipschitz regularity of the mapping function for achieving high estimation performance. Then, based on the theoretical ground, we propose a representation learning algorithm as a solution for the estimation problem, where Gaussian RBF kernel interpolators are chosen to map UL CCMs to their DL counterparts. The proposed algorithm is based on the optimization of an objective function that fits a regression model between the DL CCM and UL CCM samples in the training dataset and preserves the local geometric structure of the data in the UL CCM space, while explicitly regulating the Lipschitz continuity of the mapping function in light of our theoretical findings. The proposed algorithm surpasses benchmark methods in terms of three error metrics as shown by simulations.
