Selection-aware inference after preliminary feature screening
Develop sample-splitting or other selection-aware procedures that account theoretically for preliminary feature screening in high-dimensional sparse support vector machine applications.
References
Several directions remain open. First, the present theory is restricted to balanced, linearly separable data; extending it to class-imbalanced designs \citep{Loffredo2024RestoringBalance} or to the nonseparable perceptron regime \citep{Franz2019CriticalJammed} is a natural next step. Second, the theoretical framework is based on a replica-symmetric characterization and does not provide a rigorous proof of the RS ansatz or its stability; establishing such results, or identifying conditions under which the RS solution can be justified rigorously, is an important theoretical problem. Third, the framework assumes a Gaussian-mixture model for the class-conditional distributions, and extending the theory to broader distributional classes would increase its applicability. Fourth, the current analysis is restricted to the $L_1$ penalty; extending the proportional-asymptotic theory to nonconvex penalties such as SCAD \citep{fan2001variable} or MCP could combine reduced regularization bias with formal inferential guarantees. Fifth, a theoretically justified, data-driven procedure for selecting the tuning parameter $\lambda$ remains an important problem. Finally, developing sample-splitting or other selection-aware procedures for the preliminary screening step in high-dimensional applications would provide a cleaner end-to-end inferential framework.