The G -invariant graph Laplacian Part I: Convergence rate and eigendecomposition

Area of Science:

• Manifold Learning
• Geometric Deep Learning
• Harmonic Analysis

Background:

• Graph Laplacian algorithms are effective for manifold data analysis.
• Existing methods lack efficiency for data with group symmetries.

Purpose of the Study:

• To develop a novel graph Laplacian construction for manifold data with group symmetries.
• To improve convergence rates and computational efficiency in data analysis tasks.

Main Methods:

• Constructing a G-invariant Graph Laplacian (G-GL) by incorporating distances from group actions.
• Analyzing the convergence properties of G-GL to the Laplace-Beltrami operator.
• Deriving eigenfunctions of G-GL using FFT-type algorithms.

Main Results:

• G-GL shows improved convergence rates compared to standard graph Laplacians.
• G-GL eigenfunctions are efficiently computable.
• Demonstrated effectiveness on filtering noisy data on SU(2) manifolds.

Conclusions:

• The G-invariant Graph Laplacian offers a powerful tool for analyzing symmetric manifold data.
• This approach enhances existing manifold learning techniques.