Cross-Predictive Sparse Bayesian Learning with Application to XL-MIMO Channel Estimation

Accurate channel estimation is a key requirement in extremely large-scale multiple-input multiple-output (XL-MIMO) systems. Sparse Bayesian learning (SBL) is a well-established framework for exploiting channel sparsity, but its performance depends on parametric prior assumptions and hyperparameter optimization based on marginal likelihood, which may be sensitive to noise, limited pilot observations, and model mismatch. In this work, we propose \textit{cross-predictive SBL (CP-SBL)}, a data-driven variant of SBL in which the sparsity-inducing weights are learned by minimizing a randomized cross-predictive objective rather than through likelihood maximization. The proposed method preserves the hierarchical Bayesian structure of SBL while replacing parametric prior learning with a predictive consistency criterion derived from random data splitting. Numerical results for near-field XL-MIMO channel estimation show that CP-SBL consistently achieves lower normalized mean squared error than the baseline SBL across a wide range of signal-to-noise ratios, pilot lengths, numbers of antennas, and numbers of propagation paths, with comparable complexity and without requiring manual hyperparameter tuning.