Neural Entropic Optimal Transport and Gromov-Wasserstein Alignment
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912869143019520 |
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| author | Wang, Tao Goldfeld, Ziv |
| author_facet | Wang, Tao Goldfeld, Ziv |
| contents | Optimal transport (OT) and Gromov-Wasserstein (GW) alignment are powerful frameworks for geometrically driven matching of probability distributions, yet their large-scale usage is hampered by high statistical and computational costs. Entropic regularization has emerged as a promising solution, allowing parametric convergence rates via the plug-in estimator, which can be computed using the Sinkhorn algorithm (or its iterations in the GW case). However, Sinkhorn's $O(n^2)$ time complexity for an $n$-sized dataset becomes prohibitive for modern, massive datasets. In this work, we propose a new computational framework for the entropic OT and GW problems that replaces the Sinkhorn step with a neural network trained via backpropagation on mini-batches. By shifting the computational load from the entire dataset to the mini-batch, our approach enables reliable estimation of both the optimal transport/alignment cost and plan at dataset sizes and dimensions far exceeding those tractable with standard Sinkhorn methods. We derive non-asymptotic error bounds for these estimates, showing they achieve minimax-optimal parametric convergence rates for compactly supported distributions. Numerical experiments confirm the accuracy of our method in high-dimensional, large-sample regimes where Sinkhorn is infeasible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_07397 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Neural Entropic Optimal Transport and Gromov-Wasserstein Alignment Wang, Tao Goldfeld, Ziv Statistics Theory Optimal transport (OT) and Gromov-Wasserstein (GW) alignment are powerful frameworks for geometrically driven matching of probability distributions, yet their large-scale usage is hampered by high statistical and computational costs. Entropic regularization has emerged as a promising solution, allowing parametric convergence rates via the plug-in estimator, which can be computed using the Sinkhorn algorithm (or its iterations in the GW case). However, Sinkhorn's $O(n^2)$ time complexity for an $n$-sized dataset becomes prohibitive for modern, massive datasets. In this work, we propose a new computational framework for the entropic OT and GW problems that replaces the Sinkhorn step with a neural network trained via backpropagation on mini-batches. By shifting the computational load from the entire dataset to the mini-batch, our approach enables reliable estimation of both the optimal transport/alignment cost and plan at dataset sizes and dimensions far exceeding those tractable with standard Sinkhorn methods. We derive non-asymptotic error bounds for these estimates, showing they achieve minimax-optimal parametric convergence rates for compactly supported distributions. Numerical experiments confirm the accuracy of our method in high-dimensional, large-sample regimes where Sinkhorn is infeasible. |
| title | Neural Entropic Optimal Transport and Gromov-Wasserstein Alignment |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2312.07397 |