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Abstract Class for Lie Groups

Source: R/lie-group.R
LieGroup.Rd

Class for Lie groups. In this class, point_type ('vector' or 'matrix') will be used to describe the format of the points on the Lie group. If point_type is 'vector', the format of the inputs is dimension, where dimension is the dimension of the Lie group. If point_type is 'matrix', the format of the inputs is c(n, n) where n is the parameter of \(\mathrm{GL}(n)\) e.g. the amount of rows and columns of the matrix.

Author

Nina Miolane

Super classes

rgeomstats::PythonClass -> rgeomstats::Manifold -> LieGroup

Public fields

lie_algebra

An object of class MatrixLieAlgebra or NULL representing the tangent space at the identity.

left_canonical_metric

An object of class InvariantMetric representing the left invariant metric that corresponds to the Euclidean inner product at the identity.

right_canonical_metric

An object of class InvariantMetric representing the left invariant metric that corresponds to the Euclidean inner product at the identity.

metrics

A list of objects of class RiemannianMetric.

Methods

Public methods

Inherited methods

Method new()

The LieGroup class constructor.

Usage

LieGroup$new(dim, shape, lie_algebra = NULL, ..., py_cls = NULL)

Arguments

dim

An integer value specifying the dimension of the manifold.

shape

An integer vector specifying the shape of one element of the Lie group.

lie_algebra

An object of class MatrixLieAlgebra or NULL specifying the tangent space at the identity.

...

Extra arguments to be passed to parent class constructors. See Manifold class.

py_cls

A Python object of class LieGroup. Defaults to NULL in which case it is instantiated on the fly using the other input arguments.

Returns

An object of class LieGroup.

Method exp()

Exponentiates a left-invariant vector field from a base point.

Usage

LieGroup$exp(tangent_vec, base_point = NULL)

Arguments

tangent_vec

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more tangent vectors at corresponding base points.

base_point

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more base points on the manifold. Defaults to identity if NULL.

Details

The vector input is not an element of the Lie algebra, but of the tangent space at base_point: if \(g\) denotes base_point, \(v\) the tangent vector, and \(V = g^{-1} v\) the associated Lie algebra vector, then $$\exp(v, g) = \mathrm{mul}(g, \exp(V))$$. Therefore, the Lie exponential is obtained when base_point is NULL, or the identity.

Returns

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) storing the group exponential of the input tangent vector(s).

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$exp(rep(0, 3))
}

Method exp_from_identity()

Compute the group exponential of tangent vector from the identity.

Usage

LieGroup$exp_from_identity(tangent_vec)

Arguments

tangent_vec

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more tangent vectors at corresponding base points.

Returns

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) storing the group exponential of the input tangent vector(s).

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$exp_from_identity(rep(0, 3))
}

Method exp_not_from_identity()

Calculate the group exponential at base_point.

Usage

LieGroup$exp_not_from_identity(tangent_vec, base_point)

Arguments

tangent_vec

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more tangent vectors at corresponding base points.

base_point

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more base points on the manifold. Defaults to identity if NULL.

Returns

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) storing the group exponential of the input tangent vector(s).

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$exp_not_from_identity(rep(0, 3), rep(0, 3))
}

Method log()

Computes a left-invariant vector field bringing base_point to point.

Usage

LieGroup$log(point, base_point = NULL)

Arguments

point

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more points on the manifold.

base_point

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more base points on the manifold. Defaults to identity if NULL.

Details

The output is a vector of the tangent space at base_point, so not a Lie algebra element if base_point is not the identity. Furthermore, denoting point by \(g\) and base_point by \(h\), the output satisfies $$g = \exp(\log(g, h), h)$$.

Returns

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) storing the group logarithm of the input point(s).

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$log(rep(0, 3))
}

Method log_from_identity()

Computes the group logarithm of point from the identity.

Usage

LieGroup$log_from_identity(point)

Arguments

point

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more points on the manifold.

Returns

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) storing the group logarithm of the input point(s).

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$log_from_identity(rep(0, 3))
}

Method log_not_from_identity()

Computes the group logarithm at base_point.

Usage

LieGroup$log_not_from_identity(point, base_point)

Arguments

point

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more points on the manifold.

base_point

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more base points on the manifold. Defaults to identity if NULL.

Returns

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) storing the group logarithm of the input point(s).

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$log_not_from_identity(rep(0, 3), rep(0, 3))
}

Method get_identity()

Gets the identity of the group.

Usage

LieGroup$get_identity()

Returns

A numeric array of shape \(\{ \mathrm{dim}, [n \times n] \}\) storing the identity of the Lie group.

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$get_identity()
}

Method lie_bracket()

Computes the lie bracket of two tangent vectors.

Usage

LieGroup$lie_bracket(tangent_vector_a, tangent_vector_b, base_point = NULL)

Arguments

tangent_vector_a

A numeric array of shape \([\dots \times n \times n]\) specifying one or more tangent vectors at corresponding base points.

tangent_vector_b

A numeric array of shape \([\dots \times n \times n]\) specifying one or more tangent vectors at corresponding base points.

base_point

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more base points on the manifold. Defaults to identity if NULL.

Details

For matrix Lie groups with tangent vectors \(A\) and \(B\) at the same base point \(P\), this is given by (translate to identity, compute commutator, go back): $$[A,B] = A_P^{-1}B - B_P^{-1}A$$.

Returns

A numeric array of shape \([\dots \times n \times n]\) storing the Lie bracket of the two input tangent vectors.

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$lie_bracket(diag(0, 3), diag(0, 3))
}

Method tangent_translation_map()

Computes the push-forward map by the left/right translation.

Usage

LieGroup$tangent_translation_map(
  point,
  left_or_right = "left",
  inverse = FALSE
)

Arguments

point

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more points on the manifold.

left_or_right

A character string specifying whether to compute the map for the left or right translation. Choices are "left" or "right. Defaults to "left".

inverse

A boolean specifying whether to inverse the Jacobian matrix. If set to TRUE, the push forward by the translation by the inverse of the point is returned. Defaults to FALSE.

Details

Computes the push-forward map of the left/right translation by the point. It corresponds to the tangent map, or differential of the group multiplication by the point or its inverse. For groups with a vector representation, it is only implemented at identity, but it can be used at other points with inverse = TRUE. This method wraps the Jacobian translation which actually computes the matrix representation of the map.

Returns

A function computing the tangent map of the left/right translation by point. It can be applied to tangent vectors.

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$tangent_translation_map(rep(0, 3))
}

Method compose()

Performs function composition corresponding to the Lie group.

Usage

LieGroup$compose(point_a, point_b)

Arguments

point_a

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more left factors in the product.

point_b

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more right factors in the product.

Returns

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) storing the product of point_a and point_b along the first dimension.

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$compose(rep(0, 3), rep(0, 3))
}

Method jacobian_translation()

Computes the Jacobian of left/right translation by a point.

Usage

LieGroup$jacobian_translation(point, left_or_right = "left")

Arguments

point

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more points on the manifold.

left_or_right

A character string specifying whether to compute the map for the left or right translation. Choices are "left" or "right. Defaults to "left".

Returns

A numeric array of shape \([\dots \times \mathrm{dim} \times \mathrm{dim}]\) storing the Jacobian of the left/right translation by point.

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$jacobian_translation(rep(0, 3))
}

Method inverse()

Computes the inverse law of the Lie group.

Usage

LieGroup$inverse(point)

Arguments

point

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) specifying one or more points to be inverted.

Returns

A numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \}]\) storing the inverted points.

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$inverse(rep(0, 3))
}

Method add_metric()

Adds a metric to the class $metrics attribute.

Usage

LieGroup$add_metric(metric)

Arguments

metric

An object of class RiemannianMetric.

Returns

The class itself invisibly.

Method clone()

The objects of this class are cloneable with this method.

Usage

LieGroup$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

Examples


## ------------------------------------------------
## Method `LieGroup$exp`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$exp(rep(0, 3))
}

## ------------------------------------------------
## Method `LieGroup$exp_from_identity`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$exp_from_identity(rep(0, 3))
}

## ------------------------------------------------
## Method `LieGroup$exp_not_from_identity`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$exp_not_from_identity(rep(0, 3), rep(0, 3))
}

## ------------------------------------------------
## Method `LieGroup$log`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$log(rep(0, 3))
}

## ------------------------------------------------
## Method `LieGroup$log_from_identity`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$log_from_identity(rep(0, 3))
}

## ------------------------------------------------
## Method `LieGroup$log_not_from_identity`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$log_not_from_identity(rep(0, 3), rep(0, 3))
}

## ------------------------------------------------
## Method `LieGroup$get_identity`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$get_identity()
}

## ------------------------------------------------
## Method `LieGroup$lie_bracket`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$lie_bracket(diag(0, 3), diag(0, 3))
}

## ------------------------------------------------
## Method `LieGroup$tangent_translation_map`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$tangent_translation_map(rep(0, 3))
}

## ------------------------------------------------
## Method `LieGroup$compose`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$compose(rep(0, 3), rep(0, 3))
}

## ------------------------------------------------
## Method `LieGroup$jacobian_translation`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$jacobian_translation(rep(0, 3))
}

## ------------------------------------------------
## Method `LieGroup$inverse`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3, point_type = "vector")
  so3$inverse(rep(0, 3))
}