WLbarycenter: Barycenter of vMF Distributions Under a Wasserstein-Like...

In maotai: Tools for Matrix Algebra, Optimization and Inference

Barycenter of vMF Distributions Under a Wasserstein-Like Geometry

Description

Given a collection of von Mises-Fisher (vMF) distributions, each characterized by a mean direction \mathbf{\mu} and a concentration parameter \kappa, this function solves the geometric mean problem to compute the barycentric vMF distribution under an approximate Wasserstein geometry.

Usage

WLbarycenter(means, concentrations, weights = NULL)

Arguments

means	An (n \times p) matrix where each row represents the mean direction of one of the n vMF distributions.
concentrations	A length-n vector of nonnegative concentration parameters.
weights	A weight vector of length n. If NULL, equal weights (rep(1/n, n)) are used.

Value

A named list containing:

mean

A length-p vector representing the barycenter direction.

concentration

A scalar representing the barycenter concentration.

Examples

# Set seed for reproducibility
set.seed(123)

# Number of vMF distributions
n <- 5   

# Generate mean directions concentrated around a specific angle (e.g., 45 degrees)
base_angle <- pi / 4  # 45 degrees in radians
angles <- rnorm(n, mean = base_angle, sd = pi / 20)  # Small deviation from base_angle
means <- cbind(cos(angles), sin(angles))  # Convert angles to unit vectors

# Generate concentration parameters with large magnitudes (tight distributions)
concentrations <- rnorm(n, mean = 50, sd = 5)  # Large values around 50

# Compute the barycenter under the Wasserstein-like geometry
barycenter <- WLbarycenter(means, concentrations)

# Convert barycenter mean direction to an angle
bary_angle <- atan2(barycenter$mean[2], barycenter$mean[1])

## Visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")

# Plot the unit circle
plot(cos(seq(0, 2 * pi, length.out = 200)), sin(seq(0, 2 * pi, length.out = 200)), 
     type = "l", col = "gray", lwd = 2, xlab = "x", ylab = "y", 
     main = "Barycenter of vMF Distributions on S^1")

# Add input mean directions
points(means[,1], means[,2], col = "blue", pch = 19, cex = 1.5)

# Add the computed barycenter
points(cos(bary_angle), sin(bary_angle), col = "red", pch = 17, cex = 2)

# Add legend
legend("bottomleft", legend = c("vMF Means", "Barycenter"), col = c("blue", "red"), 
       pch = c(19, 17), cex = 1)

# Plot the concentration parameters
hist(concentrations, main = "Concentration Parameters", xlab = "Concentration")
abline(v=barycenter$concentration, col="red", lwd=2)
par(opar)

maotai documentation built on April 12, 2025, 2:10 a.m.