Quantify the condition number χ of the regularized channel

Determine explicit quantitative bounds for the condition number χ of the regularized forward operator K, where χ is defined by χ := sup_{ρ ∈ S_λ} KL(π || ρ) / KL(Kπ || Kρ), under the setting that K is injective, the hypothesis class is parameterized continuously over a compact domain, and the true data distribution π is misspecified (not exactly representable by the model). Provide concrete estimates—beyond mere finiteness—for χ that can be used to sharpen the KL contraction rates established for the self-consistent stochastic interpolant scheme.

Background

In the KL contraction analysis, the authors introduce a condition number χ to relate errors measured in observation space back to data space. Specifically, χ compares KL(π || ρ) to KL(Kπ || Kρ) over a regularized class S_λ induced by constraints on the drift and score models, and it plays a central role in establishing linear convergence rates.

The paper proves that χ is finite when the forward operator K is injective, the parameter space is compact, and the drift/score parametrization is continuous—even in a misspecified setting where π is not exactly representable. However, the authors state they cannot quantify χ under these general assumptions, leaving the development of explicit bounds as an unresolved question that would strengthen the theoretical guarantees (e.g., sharper rates) of the proposed SCSI framework.