Linear Spherical Sliced Optimal Transport: A Fast Metric for Comparing Spherical Data.

Area of Science:

• Computational geometry
• Probability theory
• Machine learning

Background:

• Comparing spherical probability distributions is crucial in computer vision, geosciences, and medicine.
• Existing methods like spherical sliced Wasserstein distances reduce computational cost but may not fully preserve geometry.
• Linear optimal transport embeds distributions into L^2 spaces for simpler comparisons.

Purpose of the Study:

• Introduce the Linear Spherical Sliced Optimal Transport (LSSOT) framework.
• Develop a computationally efficient metric for spherical probability measures that preserves intrinsic geometry.
• Demonstrate LSSOT's effectiveness in various applications.

Main Methods:

• Utilizing slicing techniques to embed spherical distributions into L^2 spaces.
• Leveraging linear optimal transport principles.
• Establishing the metricity of the proposed LSSOT framework.

Main Results:

• LSSOT provides a computationally efficient metric for spherical probability measures.
• The framework preserves the intrinsic geometry of spherical distributions.
• Demonstrated superior computational efficiency and high accuracy in applications like cortical surface registration and 3D point cloud interpolation.

Conclusions:

• LSSOT offers significant computational benefits and high accuracy for comparing spherical distributions.
• The framework effectively preserves geometric properties, making it suitable for complex applications.
• LSSOT represents a promising advancement in the field of optimal transport for spherical data.