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Abstract Class for Matrix Lie Groups

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

Class for matrix Lie groups.

Author

Nina Miolane

Super classes

rgeomstats::PythonClass -> rgeomstats::Manifold -> MatrixLieGroup

Public fields

lie_algebra

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

n

The size of the \(n \times n\) matrix elements.

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.

Methods

Public methods

Inherited methods

Method new()

The MatrixLieGroup class constructor.

Usage

MatrixLieGroup$new(dim, n, lie_algebra = NULL, ..., py_cls = NULL)

Arguments

dim

An integer value specifying the dimension of the manifold.

n

The size of the \(n \times n\) matrix elements.

lie_algebra

An object of class MatrixLieAlgebra or NULL representing 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 MatrixLieGroup. Defaults to NULL in which case it is instantiated on the fly using the other input arguments.

Returns

An object of class MatrixLieGroup.

Method exp()

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

Usage

MatrixLieGroup$exp(tangent_vec, base_point = NULL)

Arguments

tangent_vec

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 \([\dots \times n \times n]\) 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 \([\dots \times n \times n]\) storing the left multiplication of the Lie exponential of the input tangent vectors with the corresponding base points.

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3)
  # so3$exp(diag(1, 3)) # TO DO: fix in gs
}

Method log()

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

Usage

MatrixLieGroup$log(point, base_point = NULL)

Arguments

point

A numeric array of shape \([\dots \times n \times n]\) specifying one or more points.

base_point

A numeric array of shape \([\dots \times n \times n]\) 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 \([\dots \times n \times n]\) such that its Lie exponential at corresponding base points matches corresponding points.

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3)
  so3$log(diag(1, 3))
}

Method get_identity()

Gets the identity of the group.

Usage

MatrixLieGroup$get_identity()

Returns

A numeric array of shape \(n \times n\) storing the identity of the Lie group.

Examples

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

Method lie_bracket()

Computes the lie bracket of two tangent vectors.

Usage

MatrixLieGroup$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 \([\dots \times n \times n]\) 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)
  so3$lie_bracket(diag(0, 3), diag(1, 3))
}

Method tangent_translation_map()

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

Usage

MatrixLieGroup$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 at which to compute the map.

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 taking as argument a numeric array tangent_vec of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \} ]\) specifying one or more tangent vectors and returning a numeric array of shape \([\dots \times \{ \mathrm{dim}, [n \times n] \} ]\) storing the result of the tangent mapping of the left/right translation of input tangent points by corresponding base points.

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3)
  tangent_map <- so3$tangent_translation_map(diag(1, 3))
  tangent_map(diag(1, 3))
}

Method compose()

Performs function composition corresponding to the Lie group.

Usage

MatrixLieGroup$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)
  so3$compose(diag(1, 3), diag(1, 3))
}

Method inverse()

Computes the inverse law of the Lie group.

Usage

MatrixLieGroup$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 the same shape storing the inverted points.

Examples

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3)
  so3$inverse(diag(1, 3))
}

Method clone()

The objects of this class are cloneable with this method.

Usage

MatrixLieGroup$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

Examples


## ------------------------------------------------
## Method `MatrixLieGroup$exp`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3)
  # so3$exp(diag(1, 3)) # TO DO: fix in gs
}

## ------------------------------------------------
## Method `MatrixLieGroup$log`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3)
  so3$log(diag(1, 3))
}

## ------------------------------------------------
## Method `MatrixLieGroup$get_identity`
## ------------------------------------------------

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

## ------------------------------------------------
## Method `MatrixLieGroup$lie_bracket`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3)
  so3$lie_bracket(diag(0, 3), diag(1, 3))
}

## ------------------------------------------------
## Method `MatrixLieGroup$tangent_translation_map`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3)
  tangent_map <- so3$tangent_translation_map(diag(1, 3))
  tangent_map(diag(1, 3))
}

## ------------------------------------------------
## Method `MatrixLieGroup$compose`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3)
  so3$compose(diag(1, 3), diag(1, 3))
}

## ------------------------------------------------
## Method `MatrixLieGroup$inverse`
## ------------------------------------------------

if (reticulate::py_module_available("geomstats")) {
  so3 <- SpecialOrthogonal(n = 3)
  so3$inverse(diag(1, 3))
}