Building the Bridge of Schr\"odinger: A Continuous Entropic Optimal Transport Benchmark

• URL: http://arxiv.org/abs/2306.10161v2
• Date: Wed, 1 Nov 2023 10:26:59 GMT
• Title: Building the Bridge of Schr\"odinger: A Continuous Entropic Optimal Transport Benchmark
• Authors: Nikita Gushchin, Alexander Kolesov, Petr Mokrov, Polina Karpikova, Andrey Spiridonov, Evgeny Burnaev, Alexander Korotin
• Abstract summary: We propose a novel way to create pairs of probability distributions for which the ground truth OT solution is known by the construction. We use these benchmark pairs to test how well existing neural EOT/SB solvers actually compute the EOT solution.
• Score: 96.06787302688595
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Over the last several years, there has been significant progress in developing neural solvers for the Schr\"odinger Bridge (SB) problem and applying them to generative modelling. This new research field is justifiably fruitful as it is interconnected with the practically well-performing diffusion models and theoretically grounded entropic optimal transport (EOT). Still, the area lacks non-trivial tests allowing a researcher to understand how well the methods solve SB or its equivalent continuous EOT problem. We fill this gap and propose a novel way to create pairs of probability distributions for which the ground truth OT solution is known by the construction. Our methodology is generic and works for a wide range of OT formulations, in particular, it covers the EOT which is equivalent to SB (the main interest of our study). This development allows us to create continuous benchmark distributions with the known EOT and SB solutions on high-dimensional spaces such as spaces of images. As an illustration, we use these benchmark pairs to test how well existing neural EOT/SB solvers actually compute the EOT solution. Our code for constructing benchmark pairs under different setups is available at: https://github.com/ngushchin/EntropicOTBenchmark.

Related papers

• Light and Optimal Schrödinger Bridge Matching [67.93806073192938]
  We propose a novel procedure to learn Schr"odinger Bridges (SB) which we call the textbf Schr"odinger bridge matching. We show that the optimal bridge matching objective coincides with the recently discovered energy-based modeling (EBM) objectives to learn EOT/SB. We develop a light solver (which we call LightSB-M) to implement optimal matching in practice using the mixture parameterization of the Schr"odinger potential.
  arXiv  Detail & Related papers  (2024-02-05T17:17:57Z)
• Double-Bounded Optimal Transport for Advanced Clustering and Classification [58.237576976486544]
  We propose Doubly Bounded Optimal Transport (DB-OT), which assumes that the target distribution is restricted within two boundaries instead of a fixed one. We show that our method can achieve good results with our improved inference scheme in the testing stage.
  arXiv  Detail & Related papers  (2024-01-21T07:43:01Z)
• Generalized Schrödinger Bridge Matching [54.171931505066]
  Generalized Schr"odinger Bridge (GSB) problem setup is prevalent in many scientific areas both within and without machine learning. We propose Generalized Schr"odinger Bridge Matching (GSBM), a new matching algorithm inspired by recent advances. We show that such a generalization can be cast as solving conditional optimal control, for which variational approximations can be used.
  arXiv  Detail & Related papers  (2023-10-03T17:42:11Z)
• Energy-Guided Continuous Entropic Barycenter Estimation for General Costs [95.33926437521046]
  We propose a novel algorithm for approximating the continuous Entropic OT (EOT) barycenter for arbitrary OT cost functions. Our approach is built upon the dual reformulation of the EOT problem based on weak OT.
  arXiv  Detail & Related papers  (2023-10-02T11:24:36Z)
• Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport [9.980822222343921]
  We propose a novel generative model based on the semi-dual formulation of Unbalanced Optimal Transport (UOT) Unlike OT, UOT relaxes the hard constraint on distribution matching. This approach provides better robustness against outliers, stability during training, and faster convergence. Our model outperforms existing OT-based generative models, achieving FID scores of 2.97 on CIFAR-10 and 6.36 on CelebA-HQ-256.
  arXiv  Detail & Related papers  (2023-05-24T06:31:05Z)
• Entropic Neural Optimal Transport via Diffusion Processes [105.34822201378763]
  We propose a novel neural algorithm for the fundamental problem of computing the entropic optimal transport (EOT) plan between continuous probability distributions. Our algorithm is based on the saddle point reformulation of the dynamic version of EOT which is known as the Schr"odinger Bridge problem. In contrast to the prior methods for large-scale EOT, our algorithm is end-to-end and consists of a single learning step.
  arXiv  Detail & Related papers  (2022-11-02T14:35:13Z)
• Likelihood Training of Schr\"odinger Bridge using Forward-Backward SDEs Theory [29.82841891919951]
  It remains unclear whether the optimization principle of SB relates to the modern training of deep generative models. We present a novel computational framework for likelihood training of SB models grounded on Forward-Backward Theory. We show that the resulting training achieves comparable results on generating realistic images on MNIST, CelebA, and CIFAR10.
  arXiv  Detail & Related papers  (2021-10-21T17:18:59Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.