Structured wavelet-based sparse discriminant analysis in high dimensions

High-dimensional classification presents substantial challenges when the number of variables is large relative to the sample size, particularly when the underlying signals exhibit multiscale or structured patterns. Classical linear discriminant analysis and its sparse extensions often treat predictors as unstructured, which may limit their ability to capture such complexity and reduce robustness under heavy-tailed noise. In this paper, we propose a wavelet-based sparse discriminant framework that constructs the discriminant function directly in the wavelet domain. The method incorporates an l1-type regularization with scale-dependent penalization, allowing for adaptive shrinkage across resolution levels and explicitly exploiting multiscale structure in the data. Theoretical properties of the proposed estimator are established, including an oracle-type bound on the excess risk and sparsistency under standard high-dimensional assumptions. The empirical performance is evaluated through extensive simulation studies as well as a real data application based on the ionosphere dataset. The results demonstrate consistent improvements over classical linear discriminant analysis and standard sparse classifiers, particularly in terms of AUC and robustness to variability across samples. Overall, the proposed approach provides a flexible and theoretically grounded methodology for high-dimensional classification problems involving structured or multiscale predictors, with strong empirical support from both simulated and real-world data.