Quantum geometric-entropic optimization for customer lifetime value prediction: convergence theory and an empirical study on transactional retail data

Predicting customer churn from transactional data is a central prob- lem in management science, with direct implications for retention strategy, revenue forecasting, and resource allocation. This paper introduces Quantum Geometric-Entropic Optimization (Q-GEO), a framework that integrates Geometric-Entropic Optimization – com- bining Riemannian gradient methods with entropy-regularized opti- mal transport – into the training of variational quantum kernels for classification. The algorithm operates on a parameter mani- fold equipped with a Fisher-Wasserstein metric and incorporates Sinkhorn-type projections to enforce distributional coherence on the quantum feature space. We establish three theoretical contribu- tions: (i) a convergence theorem for Q-GEO-trained variational quan- tum kernels under a combined Polyak–Łojasiewicz and Sinkhorn contraction framework, yielding linear convergence in the Rieman- nian condition number plus geometric contraction of the Sinkhorn residual; (ii) a margin amplification result showing that GEO-trained quantum embeddings achieve improved separation bounds over Euclidean-trained counterparts due to the spectral regularization provided by the Wasserstein component of the Fisher-Wasserstein metric; and (iii) a distributional stability result proving that Sinkhorn- projected quantum kernel matrices preserve a doubly stochas- tic spectral structure that mitigates kernel collapse in imbalanced settings. We validate the framework on the UCI Online Retail II dataset (N = 5,942 customers, d = 11 RFM-extended features, churn rate ≈37%), a publicly available transactional benchmark. Under nested cross-validation, Q-GEO achieves 0.8614 accuracy, 0.8103 pre- cision, 0.7891 recall, 0.7996 F1, and 0.9047 ROC AUC, outperform- ing both classical baselines (including logistic regression, random forest, XGBoost, and CatBoost) and standard variational quantum kernel methods. We complement the accuracy analysis with SHAP- based explainability, computation time comparisons, and a detailed feature engineering appendix to support interpretability and repro- ducibility. We interpret these results as evidence that geometric optimization principles can materially enhance quantum machine learning for management science applications.