const-ae / diffgeo

Working with manifolds in R

diffgeo

The diffgeo R package provides functions to work with six common manifolds:

• Euclidean arrays
• Spheres (in $n$ dimensions)
• Rotation matrices (also called special orthogonal manifolds $\text{SO}(n)$)
• Stiefel manifold (matrices with orthogonal columns)
• Grassmann manifold (matrices with orthogonal columns but they are only considered different if they span separate spaces)
• Symmetric positive definite (SPD) matrices (symmetric matrices with positive eigenvalues)

Installation

You can install diffgeo from Github with:

devtools::install_github("const-ae/diffgeo")

Principles

For each manifold, diffgeo provides:

• Random points
• Random tangents
• Exponential map (a way to go along the manifold from one point to another)
• Logarithm (the inverse exponential map, that is a way calculate the direction between points on the manifold)

Examples

To demonstrate the functionality of the package, I will show the functions for the rotation matrices. All functions work analoguously for the other manifolds.

library(diffgeo)
set.seed(1)

To begin, I create a random rotation matrix in 2D

rotation <- rotation_random_point(2)

We can visualize it’s effect on a random set of points

points <- rbind(rnorm(n = 20), rnorm(20))
rotated_points <- rotation %*% points
plot(t(points), pch = 16, col = "black", asp = 1, xlim = c(-3, 3), ylim = c(-2, 2))
points(t(rotated_points), pch = 16, col = "red")
segments(points[1,], points[2,], rotated_points[1,], rotated_points[2,])

We can also create high-dimensional rotations and check the formal conditions for the rotation manifold

rotation2 <- rotation_random_point(5)
# Determinant is 1
Matrix::det(rotation2)
#> [1] 1
# All columns are orthogonal
round(t(rotation2) %*% rotation2, 3)
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    1    0    0    0    0
#> [2,]    0    1    0    0    0
#> [3,]    0    0    1    0    0
#> [4,]    0    0    0    1    0
#> [5,]    0    0    0    0    1

An important concept in differential geometry are geodesics, i.e. are lines on the manifold. To find the line that connects two rotations we use the rotation_log function. These directions are from the tangent space of the manifold. The tangent space of the rotation manifold is the set of skew-symmetric matrices.

rotation3 <- rotation_random_point(5)
direction <- rotation_log(base_point = rotation2, target_point = rotation3)
round(direction, 3)
#>        [,1]   [,2]   [,3]   [,4]   [,5]
#> [1,]  0.000 -0.483 -0.749  1.063  1.019
#> [2,]  0.483  0.000  1.771  0.734 -0.333
#> [3,]  0.749 -1.771  0.000 -1.685 -0.925
#> [4,] -1.063 -0.734  1.685  0.000  1.083
#> [5,] -1.019  0.333  0.925 -1.083  0.000

We can also reverse the operation and go from a base-point in the direction of a tangent vector (“exponential map”). If we do this using the direction we will land exactly at rotation3 again.

rotation3
#>            [,1]       [,2]       [,3]       [,4]       [,5]
#> [1,]  0.8461783 -0.2831550 -0.2681835  0.3369755  0.1353906
#> [2,] -0.1266562 -0.7590676 -0.3047431 -0.4071623 -0.3861671
#> [3,] -0.4350263 -0.5098987  0.1398897  0.6238921  0.3767557
#> [4,]  0.1644277 -0.2244352  0.3886541 -0.5630449  0.6741816
#> [5,] -0.2272798  0.1824053 -0.8152227 -0.1200224  0.4858793
rotation_map(v = direction, base_point = rotation2)
#>            [,1]       [,2]       [,3]       [,4]       [,5]
#> [1,]  0.8461783 -0.2831550 -0.2681835  0.3369755  0.1353906
#> [2,] -0.1266562 -0.7590676 -0.3047431 -0.4071623 -0.3861671
#> [3,] -0.4350263 -0.5098987  0.1398897  0.6238921  0.3767557
#> [4,]  0.1644277 -0.2244352  0.3886541 -0.5630449  0.6741816
#> [5,] -0.2272798  0.1824053 -0.8152227 -0.1200224  0.4858793

Lastly, it is important to understand that the Sphere, the Grassmann, the Stiefel, and the Rotation manifold have a non-infinite injectivity radius. The injectivity radius is furthest we can go along a manifold and still guarantee that $\log(p, \exp_p(v))$ returns the original tangent vector $v$. The concept is clearest for a 1-Sphere (aka a circle): if we go further than 180° ($\pi$ radians), we end up at a point for which there exist a shorter path to the starting point than the original direction.

# Create a random vector with two elements on a circle
sp <- sphere_random_point(1)
# Make a tangent vector with norm 1
tang <- sphere_random_tangent(sp)
tang <- tang / sqrt(sum(tang^2))

# Plot different step lengths along the direction and 
# highlight the starting point in red and the point at the injectivity radius in green
plot(t(do.call(cbind, lapply(seq(0, 2 * pi, l = 31), \(t) sphere_map(t * tang, sp)))), asp = 1)
points(t(sp), col = "red")
points(t(sphere_map(sphere_injectivity_radius() * tang, sp)), col = "green")

Alternatives

This package is heavily inspired and builds on the Manifolds and Manopt.jl packages developed by Ronny Bergmann et al.. The Manopt.jl package in turn is inspired by the Manopt toolbox for Matlab.