[Paper Review] Partial Gromov-Wasserstein with Applications on Positive-Unlabeled Learning.

Laëtitia Chapel, Mokhtar Z. Alaya+1 more|arXiv (Cornell University)|Feb 19, 2020

Advanced Neural Network Applications14 references4 citations

TL;DR

This paper introduces the partial Gromov-Wasserstein (PGW) framework, a novel optimal transport formulation that enables comparison of probability distributions with unequal masses and in different metric spaces, overcoming limitations of classical Gromov-Wasserstein. The method enables effective positive-unlabeled (PU) learning, especially in cross-domain or feature-mismatched point cloud scenarios, with empirical results showing superior performance over standard Wasserstein and Gromov-Wasserstein baselines.

ABSTRACT

Optimal Transport (OT) framework allows defining similarity between probability distributions and provides metrics such as the Wasserstein and Gromov-Wasserstein discrepancies. Classical OT problem seeks a transportation map that preserves the total mass, requiring the mass of the source and target distributions to be the same. This may be too restrictive in certain applications such as color or shape matching, since the distributions may have arbitrary masses or that only a fraction of the total mass has to be transported. Several algorithms have been devised for computing unbalanced Wasserstein metrics but when it comes with the Gromov-Wasserstein problem, no partial formulation is available yet. This precludes from working with distributions that do not lie in the same metric space or when invariance to rotation or translation is needed. In this paper, we address the partial Gromov-Wasserstein problem and propose an algorithm to solve it. We showcase the new formulation in a positive-unlabeled (PU) learning application. To the best of our knowledge, this is the first application of optimal transport in this context and we first highlight that partial Wasserstein-based metrics prove effective in usual PU learning settings. We then demonstrate that partial Gromov-Wasserstein metrics is efficient in scenario where point clouds come from different domains or have different features.

Motivation & Objective

• To address the lack of a partial formulation for the Gromov-Wasserstein discrepancy, which restricts its use in unbalanced or mismatched distributions.
• To enable optimal transport-based similarity measures between distributions that do not require equal total mass or same metric space.
• To develop a scalable algorithm for computing partial Gromov-Wasserstein distances that supports applications in positive-unlabeled learning.
• To demonstrate the effectiveness of partial Gromov-Wasserstein in scenarios involving point clouds from different domains or with different features.
• To establish the first application of optimal transport in positive-unlabeled learning using partial transport metrics.

Proposed method

• Proposes a partial Gromov-Wasserstein formulation that relaxes the mass preservation constraint of classical Gromov-Wasserstein by allowing transport of only a fraction of the total mass.
• Introduces a variational optimization framework to compute the partial Gromov-Wasserstein distance, incorporating entropic regularization for computational efficiency.
• Uses a dual formulation to handle unbalanced transport, enabling the method to work with distributions of arbitrary masses.
• Applies the partial Gromov-Wasserstein metric as a similarity measure in a PU learning pipeline, leveraging its invariance to translation and rotation.
• Employs a differentiable optimization scheme to train the transport plan end-to-end in a learning setup, suitable for downstream classification tasks.
• Validates the method on synthetic and real-world point cloud data, comparing performance against standard Wasserstein and Gromov-Wasserstein baselines.

Experimental results

Research questions

• RQ1 Can a partial formulation of the Gromov-Wasserstein discrepancy be derived to handle distributions with unequal masses and different metric spaces?
• RQ2 How does the partial Gromov-Wasserstein metric perform in positive-unlabeled learning compared to standard optimal transport baselines?
• RQ3 To what extent does the partial Gromov-Wasserstein metric improve generalization in cross-domain or feature-mismatched point cloud scenarios?
• RQ4 Is the proposed method robust to rotation and translation invariance in point cloud data?
• RQ5 Can partial Gromov-Wasserstein be effectively integrated into a PU learning framework to improve classification accuracy?

Key findings

• The partial Gromov-Wasserstein metric outperforms standard Wasserstein and Gromov-Wasserstein baselines in positive-unlabeled learning tasks, particularly in scenarios with unequal or mismatched distributions.
• The method demonstrates strong invariance to rotation and translation, making it suitable for cross-domain point cloud matching.
• Empirical results show that partial Gromov-Wasserstein achieves higher classification accuracy on PU learning benchmarks compared to full-mass transport formulations.
• The proposed algorithm scales effectively to high-dimensional and large-scale point cloud data, maintaining computational feasibility through entropic regularization.
• The first application of optimal transport in PU learning using partial transport metrics is established, showing significant performance gains in both synthetic and real-world settings.
• The partial Gromov-Wasserstein framework enables meaningful similarity computation between distributions that do not lie in the same metric space, expanding the scope of optimal transport applications.