Improving Neural Optimal Transport via Displacement Interpolation
Jaemoo Choi, Yongxin Chen, Jaewoong Choi
TL;DR
This paper introduces DIOTM, a novel neural optimal transport method leveraging displacement interpolation to improve training stability and accuracy in estimating transport maps, especially for image translation tasks.
Contribution
It proposes a new approach using displacement interpolation to enhance stability and performance in neural optimal transport map learning.
Findings
DIOTM outperforms existing models in image-to-image translation.
The method improves training stability and reduces hyperparameter sensitivity.
It effectively utilizes the entire displacement interpolation trajectory for better transport map estimation.
Abstract
Optimal Transport (OT) theory investigates the cost-minimizing transport map that moves a source distribution to a target distribution. Recently, several approaches have emerged for learning the optimal transport map for a given cost function using neural networks. We refer to these approaches as the OT Map. OT Map provides a powerful tool for diverse machine learning tasks, such as generative modeling and unpaired image-to-image translation. However, existing methods that utilize max-min optimization often experience training instability and sensitivity to hyperparameters. In this paper, we propose a novel method to improve stability and achieve a better approximation of the OT Map by exploiting displacement interpolation, dubbed Displacement Interpolation Optimal Transport Model (DIOTM). We derive the dual formulation of displacement interpolation at specific time and prove how…
Peer Reviews
Decision·ICLR 2025 Poster
1) The idea of exploiting the displacement interpolation overall looks interesting and fresh. To my knowledge, it has not been actively studied in the field, so I believe that further developing it may be interesting and fruitful for the community of adversarial/dual-based OT methods. Overall, the contribution of this paper looks as significant for the neural OT field, as WGAN-GP improved WGAN. 2) The HJB based-regularization proposed here seems to be very natural and unbiased in the sense that
1) I believe that there might be a theoretical gap in the proposed DI-OTM approach which lies in the restricted parameterization of the t-dependent transport maps. Specifically, each transport map (for a particular t) should be parameterized the way that it should solve the corresponding inner conjugation (c-transform) minimization for a particular corresponding dual potential (for time t). However, when the authors tighten all the transport maps together via a single function, this may not hold
1. The method has a derivation of the dual problem for displacement interpolation, which opens the possibility of numerical optimal transport computation from the perspective of the Benamou-Brenier dynamic transport formulation. 2. Experiments on toy examples and image-to-image translation problems show that the proposed method achieves good numerical results over competing methods for optimal transport computation and is scalable to image problems. 3. The paper proves numerically that the HJB
1. The method doesn't compare to closely related flow-based optimal transport methods, such as Rectified Flow (Flow straight and fast: Learning to generate and transfer data with rectified flow, ICLR-2023) and Flow Matching (Flow Matching for Generative Modeling, ICLR-2023). I suggest the authors compare with these methods as well. 2. The paper lacks a visual comparison for image-to-image translation problems between different methods and a discussion of why competing methods perform worse. It i
1. The paper presents comprehensive and detailed theoretical derivations, with notable innovations within the OT framework. It leverages the dual formulation of displacement interpolation to derive a new min-max optimization function. 2. In terms of experimental performance, the proposed HJB regularizer is effectively insensitive to the hyperparameter $\lambda$, performing better than other regularizers such as R1 and OTM. And DIOTM outperforms other benchmarks and exhibits more stable training
1. The motivation behind the theoretical innovation is unclear. There is no analysis explaining why decomposing the optimization of $T_\theta$ in OTM into optimizations for forward $\overrightarrow{T_\theta}$ and backward $\overleftarrow{T_\theta}$ improves training stability. 2. The experimental results in Table 2 appear unusual. I couldn't find related experimental setups for the benchmarks, and some references don’t report similar experiments or use different resolution datasets. Since the F
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TopicsNeural Networks and Applications · Muscle activation and electromyography studies