Eliminate the sqrt(log) factor in low-dimensional GW estimation rates

Determine whether the extra sqrt(log(n ∧ m)) factor in the upper bound for E[|D(ĥμ_n, ĥν_m) − D(μ, ν)|] established for dimensions d_x ∧ d_y < 4 can be removed; specifically, ascertain whether the empirical estimator D(ĥμ_n, ĥν_m) achieves the rate (n ∧ m)^{-1/2} without a logarithmic factor under the finite-moment conditions assumed in Theorem 3.1 (i.e., μ, ν ∈ P_{4q} with q ≥ d_x d_y + 2).

Background

The paper proves sample complexity upper bounds for estimating the squared (2,2)-Gromov–Wasserstein distance D(μ, ν) in unbounded settings under finite moment assumptions. For d_x ∧ d_y ≥ 4, the authors recover the (n ∧ m)^{-2/(d_x ∧ d_y)} rate up to a potential logarithmic factor in the boundary case, extending prior compact-support results.

In contrast, for d_x ∧ d_y < 4, their general upper bound reduces to (n ∧ m)^{-1/2} multiplied by an extra sqrt(log(n ∧ m)) factor when q ≥ d_x d_y + 2. The authors suspect this logarithmic factor may be an artifact of their proof (stemming from discretizing the domain of matrices A in the variational representation for S_2), and explicitly state uncertainty about whether it can be removed, raising a concrete open question about achieving the (n ∧ m)^{-1/2} rate without a log factor under their moment conditions.