KSD on Lie Groups

This repository contains the official implementation of the experiments presented in the paper:

Kernel Stein Discrepancy on Lie Groups: Theory and Applications

This code provides methods to compute the Minimum Kernel Stein Discrepancy Estimator (MKSDE) and Maximum Likelihood Estimator (MLE) of the parameter $F$ of a von-Mises Fisher (vMF) distribution defined on the Special Orthogonal Group $SO(N)$. The experiments demonstrate the performance of MKSDE in both parameter estimation and a goodness-of-fit test on Lie Groups.

• Synthetic Data: The code generates synthetic data on $SO(N)$ for analysis.
• Parameter Settings: The initial parameter values and settings are included in the scipts.

• Estimates (Figure 1 in the paper): The computed MKSDE and MLE for the parameter $F$ and their respective distances to the ground truth $F_0$.
• Geodesic Distances (Table I in the paper) The geodesic distances between the mode of estimated $F$ and the ground truth orientation of the samples in Figure 2.
• Goodness-of-Fit Results (Table II in the paper): The test outputs the computed statistic $n(wKSD^2_n(\hat{\theta}))$ and the $(1-\beta)$-quantile.

• Number of Samples (n_samples): 100 for goodness-of-fit, varies (100-200) for estimation.
• Lie Group Dimension ($N$): 3
• Kernel Bandwidth ($\tau$): 1.0
• Initial Parameter ($F$):
  • $F$ for estimation: [8.5, 1.1, 4.1, 7.8, 3.9, 6, 4.3, 6.4, 4.8]
  • $F$ for goodness-of-fit: Identity matrix [1, 0, 0, 0, 1, 0, 0, 0, 1]
• Cayley Kappa : 2.0
• Bootstrap Samples (n_bootstrap): 1000
• Significance Level ($\beta$): 0.1

These values can be modified within the provided scripts to suit different experimental settings.

Execution: The scripts are written in R. When you execute the script, all results, including intermediate steps and computed values, will be displayed directly.

• MKSDE.R:

  • Functionality: Computes the Minimum Kernel Stein Discrepancy Estimator (MKSDE) and Maximum Likelihood Estimator (MLE) for the parameter $F$ of a von-Mises Fisher (vMF) distribution defined on $SO(N)$.
  • Details: This script estimates the parameters $F$ using synthetic data generated from the vMF distribution, and compares the performance of MKSDE against MLE by computing distances and kernel values.

• rotations/rotations*.R:

  • Functionality: Computes the geodesic distances between the mode of estimated $F$ and the ground truth orientations.

  • Details: These scripts reproduce the results in Table II in the paper, computing the geodesic distances between the modes of MKSDE and MLE with the ground truth orientations.

• gof.R:

  • Functionality: Performs the MKSDE goodness-of-fit test to evaluate the fit of the model distribution to the data.

  • Details: This script generates synthetic data using the Cayley distribution and computes the goodness-of-fit statistics, including the $n(wKSD^2_n(\hat{\theta}))$ statistic. A bootstrap procedure is used to obtain the $(1-\beta)$-quantile.

If you use this codebase, or otherwise found our work valuable, please cite:

X. Qu, X. Fan and B. C. Vemuri, "Kernel Stein Discrepancy on Lie Groups: Theory and Applications," in IEEE Transactions on Information Theory,
vol. 70, no. 12, pp. 8961-8974, Dec. 2024, doi: 10.1109/TIT.2024.3468212.