Variational Wasserstein Clustering.

Area of Science:

• Computational geometry
• Machine learning
• Data analysis

Background:

• Clustering algorithms group data points.
• Optimal transportation theory provides a framework for comparing probability distributions.
• Existing methods like k-means have limitations in handling complex data structures.

Purpose of the Study:

• To develop a new clustering method leveraging optimal transportation principles.
• To establish a connection between optimal transportation and k-means clustering.
• To apply the method to domain adaptation, remeshing, and representation learning.

Main Methods:

• Formulating clustering as an optimal transportation problem.
• Solving optimal transportation using a variational principle.
• Employing power diagrams as transportation plans for domain aggregation.
• Adjusting power diagrams to minimize clustering energy and Wasserstein distance simultaneously.
• Achieving a measure-preserving mapping between data domains.

Main Results:

• Demonstrated a novel clustering approach based on optimal transportation.
• Established a theoretical link between optimal transportation and k-means.
• Successfully applied the method to domain adaptation, remeshing, and representation learning tasks.
• Validated the method on both synthetic and real-world datasets.

Conclusions:

• The proposed optimal transportation-based clustering method offers a powerful alternative to traditional techniques.
• The method provides a measure-preserving mapping, beneficial for various data transformation tasks.
• This approach shows promise for advancing unsupervised learning and data analysis.