Variational Entropic Optimal Transport

TL;DR

This work addresses the practical inefficiency of entropic OT with quadratic cost in continuous spaces by proposing Variational Entropic Optimal Transport (VarEOT), a simulation-free solver built on a variational reformulation of the log-partition term $\log Z(f,x_0)$. By introducing an auxiliary normalizer and jointly learning a dual potential $f$ and a variational function $\xi$, VarEOT yields a differentiable objective that can be optimized with standard neural networks and stochastic gradients, avoiding MCMC during training. The authors establish finite-sample generalization bounds and approximation results under universal function approximation, and demonstrate competitive performance on synthetic 2D transports and unpaired image-to-image translation in the ALAE latent space, with favorable comparisons to existing weak-dual solvers. Overall, VarEOT delivers a practical, flexible, and theoretically grounded framework for continuous EOT that obviates costly simulation while preserving expressive transport plans and enabling out-of-sample generation.

Abstract

Entropic optimal transport (EOT) in continuous spaces with quadratic cost is a classical tool for solving the domain translation problem. In practice, recent approaches optimize a weak dual EOT objective depending on a single potential, but doing so is computationally not efficient due to the intractable log-partition term. Existing methods typically resolve this obstacle in one of two ways: by significantly restricting the transport family to obtain closed-form normalization (via Gaussian-mixture parameterizations), or by using general neural parameterizations that require simulation-based training procedures. We propose Variational Entropic Optimal Transport (VarEOT), based on an exact variational reformulation of the log-partition $\log \mathbb{E}[\exp(\cdot)]$ as a tractable minimization over an auxiliary positive normalizer. This yields a differentiable learning objective optimized with stochastic gradients and avoids the necessity of MCMC simulations during the training. We provide theoretical guarantees, including finite-sample generalization bounds and approximation results under universal function approximation. Experiments on synthetic data and unpaired image-to-image translation demonstrate competitive or improved translation quality, while comparisons within the solvers that use the same weak dual EOT objective support the benefit of the proposed optimization principle.

Figures (7)

• Figure 1: Optimal plan learned with VarEOT (ours) in Gaussian$\!\rightarrow\!$Swiss roll example.
• Figure 2: Qualitative comparison for Man$\rightarrow$Woman translation with $\varepsilon=1.0$. From top to bottom: input samples, VarEOT (ours), LightSB, and EgNOT. Input images are selected from the test set: we take the first 300 samples and rank them by encoder-decoder reconstruction quality (LPIPS), displaying the top-ranked examples.
• Figure 3: Qualitative comparison for Female$\rightarrow$Male translation with $\varepsilon=1.0$. From top to bottom: input samples, VarEOT (ours), LightSB, and EgNOT. Input images are selected from the test set: we take the first 300 samples and rank them by encoder-decoder reconstruction quality (LPIPS), displaying the top-ranked examples.
• Figure 4: Qualitative comparison for Adult$\rightarrow$Child translation with $\varepsilon=1.0$. From top to bottom: input samples, VarEOT (ours), LightSB, and EgNOT. Input images are selected from the test set: we take the first 300 samples and rank them by encoder-decoder reconstruction quality (LPIPS), displaying the top-ranked examples.
• Figure 5: Qualitative comparison for Child$\rightarrow$Adult translation with $\varepsilon=1.0$. From top to bottom: input samples, VarEOT (ours), LightSB, and EgNOT. Input images are selected from the test set: we take the first 300 samples and rank them by encoder-decoder reconstruction quality (LPIPS), displaying the top-ranked examples.

Theorems & Definitions (20)

• Proposition 3.1: Variational bound for the partition function
• Theorem 3.2: Variational dual form of EOT
• Theorem 3.3
• Proposition 3.4
• Theorem 3.5: Bound on estimation error
• Remark 3.6
• Theorem 3.7: Vanishing of approximation error
• Definition 1.1
• Definition 1.2: Rademacher complexity
• Lemma 1.3: Representativeness estimation