Intrinsic nature of the logarithmic factor for q=1

Determine whether the logarithmic factor at q=1 in the partial optimal transport rate is intrinsic.

Background

For the partial optimal transport estimator, the stochastic rate contains an additional logarithmic factor when the transport exponent is q=1. The paper derives this factor from a multiscale argument but does not determine whether it reflects a genuine minimax or methodological limitation. Resolving this question would clarify the sharpness of the q=1 rate.

References

Several directions for future work therefore arise naturally. On the theoretical side, it remains to establish minimax lower bounds for direct estimation in partial optimal transport and to determine whether the logarithmic factor at $q=1$ in \Cref{thm:pot-rate} is intrinsic.

— Weighted persistence intensity regression  (2609.01387 - Pegoraro et al., 1 Sep 2026) in Section Discussion