Regularity characterization of SLDB estimators

Characterize the formal regularity properties of the sliced $L^p$ distributional balancing (SLDB) estimator and other distributional balancing methods, including the potential non-regularity induced by sorting and simplex-projection operations.

Background

The paper notes that the SLDB estimator involves non-smooth computational operations, particularly sorting and projection onto the simplex. Although heuristic arguments and simulations suggest that these operations may produce non-regular behavior, the authors do not provide a formal characterization. Establishing such regularity properties is identified as an important theoretical direction, even though the unresolved issue does not invalidate the inferential procedures recommended in the paper.

References

Moving forward, an important direction for future research concerns the fundamental statistical properties of the SLDB estimator, especially its regularity. While heuristic arguments and simulation evidence suggest potential non-regularity arising from non-smooth operations (e.g., sorting and simplex projections), a formal characterization remains an open problem.

— Sliced $L^p$ Distributional Balancing  (2609.09600 - Zhang et al., 9 Sep 2026) in Section 7, Discussion