Removing the exponential high-degree Hermite variance factor

Determine whether a least-squares estimator based on high-degree Hermite features can remove the exponential-in-degree variance factor present in the projection estimator, thereby permitting the chaos degree to grow faster than logarithmically with the sample size.

Background

For the orthogonal Hermite projection estimator, the variance can grow exponentially with the chaos degree because fourth moments of normalized Hermite polynomials grow geometrically. Consequently, polynomial decay assumptions on the chaos coefficients yield only logarithmic effective degree growth for this estimator.

The authors identify least-squares fitting as a possible alternative, but controlling the smallest eigenvalue of the empirical Gram matrix for high-degree Hermite features and the influence of the truncated expansion remains unresolved.

References

Whether a least-squares estimator removes it is a separate question, since dividing by the empirical Gram matrix would require controlling its smallest eigenvalue for high-degree Hermite features and would also have to absorb the effect of the truncated part of the expansion on the random design; we do not pursue it.

— Sieve Estimation of Optimal Transport Maps from Paired Data in Gaussian Spaces  (2609.09089 - Jin et al., 8 Sep 2026) in Section 5, subsection 5.3, paragraph beginning “The exponential dependence on the degree”