Compactness of the entropic transport-map class

Prove that the class of entropic transport maps \(\mathbb{T}_\rho^\epsilon(\mathcal{P}(\Omega))=\{T^\epsilon_{\rho\rightarrow\nu}:\nu\in\mathcal{P}(\Omega)\}\) is compact in \(L^2(\rho)\).

Background

The paper derives generalization bounds for linearized Wasserstein dictionary learning with entropic transport-map atoms by controlling covering numbers of the candidate atom class. For optimal transport maps, compactness is established under bounded-domain and density assumptions. For entropic maps, the authors instead take the closure of the candidate class and use a covering-number argument, avoiding the need to prove compactness of the original class directly.

The unresolved issue is whether the full class of entropic transport maps generated from all target measures in P(Ω)\mathcal{P}(\Omega) is itself compact in L2(ρ)L^2(\rho). Establishing this would provide a direct analogue of the compactness result proved for ordinary optimal transport maps and would clarify the functional-analytic structure underlying the entropic-map dictionary class.

References

We conjecture that $\mathbb{T}_\rho\epsilon(\mathcal{P}(\Omega))$ is indeed compact in $L2(\rho)$, but for simplicity will use an alternate approach using the following lemma:

— Sparse Dictionary Learning for Measure-Valued Data via Linearized Optimal Transport: Efficient Algorithms and Sample Complexity  (2610.12924 - Mallery et al., 9 Oct 2026) in Section "Sample Complexity of LWDL-EOT" in Appendix "Proofs for the Generalization Error Bounds"