Ground-penetrating radar (GPR) combines depth resolution, non-destructive operation, and broad material sensitivity, yet it has seen limited use in diagnosing building envelopes. The compact geometry of wall assemblies, where reflections from closely spaced studs, sheathing, and cladding strongly overlap, has made systematic inversion difficult. Recent advances in data-driven interpretation provide an opportunity to revisit this challenge and assess whether machine learning can reliably extract structural information from such complex signals. Here, we develop a GPR-based inversion framework that decomposes wall diagnostics into classification tasks addressing vertical (stud presence) and lateral (wall-type) variations. Alongside model development, we implement multiple feature minimization strategies - including recursive elimination, agglomerative clustering, and L0-based sparsity - to promote fidelity and interpretability. Among these approaches, the L0-based sparse neural network (SparseNN) emerges as particularly effective: it exceeds Random Forest accuracy while relying on only a fraction of the input features, each linked to identifiable dielectric interfaces. SHAP analysis further confirms that the SparseNN learns reflection patterns consistent with physical layer boundaries. In summary, this framework establishes a foundation for physically interpretable and data-efficient inversion of wall assemblies using GPR radargrams. Although defect detection is not addressed here, the ability to reconstruct intact envelope structure and isolate features tied to key elements provides a necessary baseline for future inversion and anomaly-analysis tasks.