Establish existence of a positive stationary solution for the biased KLXX surrogate

Establish the existence of a positive solution to the fixed-point equation governing stationary densities of the biased KLXX loss under a biased target surrogate, thereby removing the theorem’s assumption that such a positive solution exists.

Background

The accuracy theorem for the biased KLXX loss assumes that the stationary fixed-point equation has a positive solution. The appendix derives a sufficient but conservative domination condition implying positivity, but that condition excludes the default coefficients used in the KLXX experiments. The authors argue heuristically that a solution cannot approach zero where the target density is positive, yet explicitly note that this does not establish existence. A rigorous existence result for a positive stationary solution would therefore strengthen the theoretical analysis and validate the theorem’s applicability under the experimentally used coefficients.

References

The condition is sufficient but conservative, and positivity does not in fact require it. If $\nu_\star(x)$ approaches zero at some $x$ with $\pi(x)>0$, then $w_\star(x)\to\infty$, so $w_\star(x)$ exceeds $w_\star(y)$ for almost every $y$ and both $S{\tilde\pi}_\star(x)$ and $S{\xi}_\star(x)$ tend to $+1$; the right-hand side of eq: surrogate-stationary then tends to $\tilde\pi(x)(1+2\lambda)+2\vartheta\xi(x)>0$. A solution cannot therefore approach zero where $\pi$ is positive, at any $\lambda,\vartheta\geq0$; this indicates that positivity is not restrictive, but it is not an existence proof, and the theorem assumes a positive solution.

— Mode Coverage in Normalizing Flow Boltzmann Generators via Log-Ratio Variation  (2609.09473 - Feng et al., 8 Sep 2026) in Remark, Section 3.3, immediately following the proof of Theorem 3.1 (Appendix: Proof of accuracy under biased target surrogate)