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Generative Modeling with Optimal Transport Maps
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Generative Modeling with Optimal Transport Maps
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With the discovery of Wasserstein GANs, Optimal Transport (OT) has become a powerful tool for large-scale generative modeling tasks. In these tasks, OT cost is typically used as the loss for training GANs. In contrast to this approach, we show that the OT map itself can be used as a generative model, providing comparable performance. Previous analogous approaches consider OT maps as generative models only in the latent spaces due to their poor performance in the original high-dimensional ambient space. In contrast, we apply OT maps directly in the ambient space, e.g., a space of high-dimensional images. First, we derive a min-max optimization algorithm to efficiently compute OT maps for the quadratic cost (Wasserstein-2 distance). Next, we extend the approach to the case when the input and output distributions are located in the spaces of different dimensions and derive error bounds for the computed OT map. We evaluate the algorithm on image generation and unpaired image restoration tasks. In particular, we consider denoising, colorization, and inpainting, where the optimality of the restoration map is a desired attribute, since the output (restored) image is expected to be close to the input (degraded) one.
Forward citations
Cited by 6 Pith papers
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Rectified flow learns straight-path neural ODEs for distribution transport, yielding efficient generative models and domain transfers that work well even with a single simulation step.
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Implicit Neural Optimal Transport via Fixed-Point Optimization
A single-network fixed-point formulation for neural optimal transport eliminates adversarial min-max optimization and implicit differentiation while enforcing dual feasibility exactly.
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Implicit Neural Optimal Transport via Fixed-Point Optimization
A single-network implicit neural optimal transport method that solves the c-transform via proximal fixed-point iteration for stable, non-adversarial training.
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Stability of the Monge Map in Semi-Dual Optimal Transport
Semi-dual optimal transport has a degenerate saddle-point structure equivalent to constrained optimization, with necessary and sufficient conditions derived for Monge map convergence independent of dual potential optimality.
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Stability of the Monge Map in Semi-Dual Optimal Transport
Semi-dual optimal transport has a degenerate saddle-point structure whose solution is a constrained optimization problem, giving necessary and sufficient conditions for Monge map convergence independent of dual optimality.
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Stability of the Monge Map in Semi-Dual Optimal Transport
Semi-dual OT formulation has degenerate saddle-point structure; necessary and sufficient conditions for Monge map convergence are derived without requiring dual potential optimality.
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