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https://github.com/const-ae/diffgeo

Working with manifolds in R
https://github.com/const-ae/diffgeo

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Working with manifolds in R

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README 

        
---

output: github_document

---



```{r, include = FALSE}

knitr::opts_chunk$set(

  collapse = TRUE,

  comment = "#>",

  fig.path = "man/figures/README-",

  out.width = "100%"

)

```



# 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](https://github.com/const-ae/diffgeo) with:



``` r

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.



```{r}

library(diffgeo)

set.seed(1)

```



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



```{r}

rotation <- rotation_random_point(2)

```



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

```{r rotation_visualization_2d}

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



```{r}

rotation2 <- rotation_random_point(5)

# Determinant is 1

Matrix::det(rotation2)

# All columns are orthogonal

round(t(rotation2) %*% rotation2, 3)

```



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.



```{r}

rotation3 <- rotation_random_point(5)

direction <- rotation_log(base_point = rotation2, target_point = rotation3)

round(direction, 3)

```



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.



```{r}

rotation3

rotation_map(v = direction, base_point = rotation2)

```



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.



```{r circle_injectivity_radius}

# 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](https://juliamanifolds.github.io/Manifolds.jl/latest/index.html) and [Manopt.jl](https://manoptjl.org/stable/) packages developed by Ronny Bergmann et al.. The Manopt.jl package in turn is inspired by the [Manopt](https://www.manopt.org/) toolbox for Matlab.