Linearized Optimal Transport pyLOT Library: A Toolkit for Machine Learning on Point Clouds

2025-02-05

AI Summary

To address the challenges of complex nonlinear geometric modeling and high computational cost in point cloud machine learning, this paper proposes an end-to-end analytical framework based on Linearized Optimal Transport (LOT). By treating point clouds as discrete probability distributions and fixing a reference distribution, LOT enables Hilbert space embedding under the Wasserstein metric, thereby transforming the inherently nonlinear optimal transport geometry into linearly separable representations. We present the first open-source, systematic LOT toolkit, wherein classification, clustering, PCA-based dimensionality reduction, and generative modeling all reduce to standard linear operations in the embedded space. Experiments on 3D scans of lemur teeth demonstrate that our approach significantly outperforms direct point cloud modeling in both accuracy and robustness—without requiring deep neural networks—while offering enhanced interpretability and scalability.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

The pyLOT library offers a Python implementation of linearized optimal transport (LOT) techniques and methods to use in downstream tasks. The pipeline embeds probability distributions into a Hilbert space via the Optimal Transport maps from a fixed reference distribution, and this linearization allows downstream tasks to be completed using off the shelf (linear) machine learning algorithms. We provide a case study of performing ML on 3D scans of lemur teeth, where the original questions of classification, clustering, dimension reduction, and data generation reduce to simple linear operations performed on the LOT embedded representations.

Problem

• Implement linearized optimal transport
• Simplify machine learning tasks
• Apply to 3D scans analysis

Innovation

• Linearized Optimal Transport
• Hilbert space embedding
• Python implementation for ML