We study how to learn correlated preferences from aggregate choice data through inverse optimization. Our focus is the Cross-Moment Model (CMM), a representative-agent choice model that captures rich substitution patterns via first two moments while remaining tractable for downstream optimization problems such as bundle pricing. Despite these attractive properties, CMM has seen limited empirical use because estimation requires recovering a latent covariance matrix from market shares. We propose a suboptimality-loss framework for CMM estimation. Instead of matching probabilities directly, we measure how suboptimal the observed shares are for the forward CMM problem induced by candidate parameters. We show that this loss is convex in the utility coefficients and admits a difference-of-convex representation in the covariance matrix. Leveraging this structure, we develop an alternating algorithm that combines projected-gradient updates for the utility block with a majorization-minimization step for the covariance block. The resulting covariance subproblem is convex, and the projection onto the feasible covariance set admits a closed-form spectral formula. We establish descent and sublinear stationarity guarantees for the proposed scheme. On the statistical side, we provide a local loss-based identification result in the noiseless correctly specified setting, supported by a local secondorder analysis of the population loss, and a finite-sample guarantee for market-share prediction when observed shares are empirical frequencies. Numerical experiments show stable convergence under correct specification, strong predictive performance under misspecification, and robust improvements in realized pricing performance in a synthetic bundling setting where correlation matters.