Debiasing optimal transport: classical and entropic
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| Format: | Preprint |
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2026
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| _version_ | 1866911613910515712 |
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| author | Aubin-Frankowski, Pierre-Cyril Ehrlacher, Virginie Todeschi, Gabriele |
| author_facet | Aubin-Frankowski, Pierre-Cyril Ehrlacher, Virginie Todeschi, Gabriele |
| contents | We study the notion of debiasability for cost functions arising in optimal transport. We call a symmetric cost function $c:\mathscr{X}\times\mathscr{X}\to\mathbb{R}\cup\{+\infty\}$ debiasable if it satisfies $c(x,y)\ge \tfrac{1}{2}c(x,x)+\tfrac{1}{2}c(y,y)$ for all $x,y\in\mathscr{X}$. Building on an equivalent characterization by an inf-representation $c(x,y)=\inf_{z\in\mathscr{Z}}ψ(x,z)+ψ(y,z)$ for some set $\mathscr{Z}$ and some function $ψ: \mathscr{X}\times \mathscr{Z} \to \mathbb{R} \cup \{+\infty\}$, interpreted as a generalization of the midpoint identity for squared geodesic distances, we investigate the debiasability of costs defined on spaces of probability measures. Our primary focus is the entropic regularization of optimal transport across different regimes of the regularization parameter $\varepsilon \in [0,+\infty]$, encompassing classical optimal transport ($\varepsilon=0$), entropic optimal transport ($\varepsilon>0$), and the Maximum Mean Discrepancy ($\varepsilon=+\infty$). For $\varepsilon \in (0,+\infty]$, we investigate sufficient conditions, such as negative definiteness of the ground cost or continuity and positive definiteness of the induced kernel, handled then via a convex-nonconcave minimax argument. All our results extend naturally to unbalanced optimal transport settings and we generalize in this way the findings of \cite{feydy2019interpolating} and \cite{sejourne2019sinkhorn}. As a byproduct, we derive novel decomposition formulas for entropic optimal transport, which may be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_19981 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Debiasing optimal transport: classical and entropic Aubin-Frankowski, Pierre-Cyril Ehrlacher, Virginie Todeschi, Gabriele Optimization and Control Functional Analysis 49Q22, 46E22 We study the notion of debiasability for cost functions arising in optimal transport. We call a symmetric cost function $c:\mathscr{X}\times\mathscr{X}\to\mathbb{R}\cup\{+\infty\}$ debiasable if it satisfies $c(x,y)\ge \tfrac{1}{2}c(x,x)+\tfrac{1}{2}c(y,y)$ for all $x,y\in\mathscr{X}$. Building on an equivalent characterization by an inf-representation $c(x,y)=\inf_{z\in\mathscr{Z}}ψ(x,z)+ψ(y,z)$ for some set $\mathscr{Z}$ and some function $ψ: \mathscr{X}\times \mathscr{Z} \to \mathbb{R} \cup \{+\infty\}$, interpreted as a generalization of the midpoint identity for squared geodesic distances, we investigate the debiasability of costs defined on spaces of probability measures. Our primary focus is the entropic regularization of optimal transport across different regimes of the regularization parameter $\varepsilon \in [0,+\infty]$, encompassing classical optimal transport ($\varepsilon=0$), entropic optimal transport ($\varepsilon>0$), and the Maximum Mean Discrepancy ($\varepsilon=+\infty$). For $\varepsilon \in (0,+\infty]$, we investigate sufficient conditions, such as negative definiteness of the ground cost or continuity and positive definiteness of the induced kernel, handled then via a convex-nonconcave minimax argument. All our results extend naturally to unbalanced optimal transport settings and we generalize in this way the findings of \cite{feydy2019interpolating} and \cite{sejourne2019sinkhorn}. As a byproduct, we derive novel decomposition formulas for entropic optimal transport, which may be of independent interest. |
| title | Debiasing optimal transport: classical and entropic |
| topic | Optimization and Control Functional Analysis 49Q22, 46E22 |
| url | https://arxiv.org/abs/2604.19981 |