Sharper concentration and coercivity for nonparametric estimation

Develop sharper concentration bounds (e.g., Bernstein-type) for the empirical normal matrix and vector in the self-test loss and establish coercivity in infinite-dimensional function spaces to enable a nonparametric statistical theory for learning Φ and V from unlabeled data.

Background

The parametric error analysis uses Chebyshev-type concentration with fixed finite-dimensional bases and a coercivity condition ensuring identifiability. Extending the analysis to nonparametric settings would benefit from sharper (e.g., Bernstein) concentration to control empirical fluctuations more tightly.

A central theoretical obstacle is verifying a suitable coercivity (or identifiability) condition in infinite-dimensional hypothesis spaces, which is necessary for well-posedness of the inverse problem with unlabeled data.

References

Sharper concentration bounds can be obtained by replacing the Chebyshev bounds in Lemma~\ref{lem:concentration} with Bernstein-type bounds under stronger tail assumptions. This is particularly useful in nonparametric estimation settings, where finer control over the concentration of empirical quantities is needed. The main challenge for nonparametric estimation is the coercivity condition in the infinite-dimensional function space. We leave this as a direction for future work.

— Learning interacting particle systems from unlabeled data  (2604.02581 - Wei et al., 2 Apr 2026) in Remark (Sharper concentration and nonparametric estimation) following Lemma ‘Concentration of normal matrix and vector’ in Appendix: Proofs for the error bound of the parametric estimator