KL-based TV control via Girsanov: need a bound on score differences

Develop a dimension-free bound on \(\sum_{i=1}^n (S_i(\rho_t V_t) - S_i(\rho_t W_t))^2\) for the scores along the coupled reverse jump processes V and W, enabling the use of Girsanov’s theorem to bound total variation via Kullback–Leibler divergence.

Background

The authors consider an alternative route to bound total variation distance between the coupled processes via KL divergence and Pinsker’s inequality, invoking Girsanov’s theorem for jump processes.

This approach requires controlling the squared difference of coordinate-wise scores along the coupled reverse heat processes. The authors explicitly state their inability to obtain a suitable bound, identifying this as a main obstacle in the discrete cube setting.

References

However, to bound KL divergence this way requires a bound on \sum_{i=1}n {S_i(\rho_t V_t) - S_i(\rho_t W_t)}2, and the author was unable to obtain a good bound of this type.

— Talagrand's convolution conjecture up to loglog via perturbed reverse heat (2511.19374 - Chen, 24 Nov 2025) in Remark after Lemma 3.2, Section 3.2 (Proof of Lemma on total variation control)