MatrixLieGroup: Abstract Class for Matrix Lie Groups
In rgeomstats: Interface to 'Geomstats'
MatrixLieGroup R Documentation
Abstract Class for Matrix Lie Groups
Description
Class for matrix Lie groups.
Super classes
rgeomstats::PythonClass -> rgeomstats::Manifold -> MatrixLieGroup
Public fields
lie_algebraAn object of class MatrixLieAlgebra or NULL
representing the tangent space at the identity.
nThe size of the n \times n matrix elements.
left_canonical_metricAn object of class InvariantMetric
representing the left invariant metric that corresponds to the
Euclidean inner product at the identity.
right_canonical_metricAn object of class InvariantMetric
representing the left invariant metric that corresponds to the
Euclidean inner product at the identity.
Methods
Public methods
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Inherited methods
rgeomstats::PythonClass$get_python_class()rgeomstats::PythonClass$set_python_class()rgeomstats::Manifold$belongs()rgeomstats::Manifold$is_tangent()rgeomstats::Manifold$random_point()rgeomstats::Manifold$random_tangent_vec()rgeomstats::Manifold$regularize()rgeomstats::Manifold$set_metric()rgeomstats::Manifold$to_tangent()
Method new()
The MatrixLieGroup class constructor.
Usage
MatrixLieGroup$new(dim, n, lie_algebra = NULL, ..., py_cls = NULL)
Arguments
dimAn integer value specifying the dimension of the manifold.
nThe size of the n \times n matrix elements.
lie_algebraAn 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_clsA 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$exp(tangent_vec, base_point = NULL)
Arguments
tangent_vecA numeric array of shape [… \times n \times
n] specifying one or more tangent vectors at corresponding base
points.
base_pointA numeric array of shape [… \times n \times n]
specifying one or more base points on the manifold. Defaults to identity if
NULL.
Details
The vector input is not an element of the Lie algebra, but of
the tangent space at base_point: if g denotes base_point,
v the tangent vector, and V = g^{-1} v the associated Lie
algebra vector, then
\exp(v, g) = \mathrm{mul}(g, \exp(V))
.
Therefore, the Lie exponential is obtained when base_point is NULL,
or the identity.
Returns
A numeric array of shape [… \times n \times n] storing
the left multiplication of the Lie exponential of the input tangent
vectors with the corresponding base points.
Examples
if (reticulate::py_module_available("geomstats")) {
so3 <- SpecialOrthogonal(n = 3)
# so3$exp(diag(1, 3)) # TO DO: fix in gs
}
Method log()
Computes a left-invariant vector field bringing base_point
to point.
Usage
MatrixLieGroup$log(point, base_point = NULL)
Arguments
pointA numeric array of shape [… \times n \times n]
specifying one or more points.
base_pointA numeric array of shape [… \times n \times n]
specifying one or more base points on the manifold. Defaults to identity if
NULL.
Details
The output is a vector of the tangent space at base_point, so
not a Lie algebra element if base_point is not the identity.
Furthermore, denoting point by g and base_point by h,
the output satisfies
g = \exp(\log(g, h), h)
.
Returns
A numeric array of shape [… \times n \times n] such
that its Lie exponential at corresponding base points matches
corresponding points.
Examples
if (reticulate::py_module_available("geomstats")) {
so3 <- SpecialOrthogonal(n = 3)
so3$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$lie_bracket(
tangent_vector_a,
tangent_vector_b,
base_point = NULL
)
Arguments
tangent_vector_aA numeric array of shape [… \times n
\times n] specifying one or more tangent vectors at corresponding base
points.
tangent_vector_bA numeric array of shape [… \times n
\times n] specifying one or more tangent vectors at corresponding base
points.
base_pointA numeric array of shape [… \times n \times n]
specifying one or more base points on the manifold. Defaults to identity if
NULL.
Details
For matrix Lie groups with tangent vectors A and B
at the same base point P, this is given by (translate to
identity, compute commutator, go back):
[A,B] = A_P^{-1}B -
B_P^{-1}A
.
Returns
A numeric array of shape [… \times n \times n] storing
the Lie bracket of the two input tangent vectors.
Examples
if (reticulate::py_module_available("geomstats")) {
so3 <- SpecialOrthogonal(n = 3)
so3$lie_bracket(diag(0, 3), diag(1, 3))
}
Method tangent_translation_map()
Computes the push-forward map by the left/right translation.
Usage
MatrixLieGroup$tangent_translation_map(
point,
left_or_right = "left",
inverse = FALSE
)
Arguments
pointA numeric array of shape [… \times \{ \mathrm{dim},
[n \times n] \} ] specifying one or more points at which to compute
the map.
left_or_rightA character string specifying whether to compute the
map for the left or right translation. Choices are "left" or
"right. Defaults to "left".
inverseA boolean specifying whether to inverse the Jacobian
matrix. If set to TRUE, the push forward by the translation by the
inverse of the point is returned. Defaults to FALSE.
Details
Computes the push-forward map of the left/right translation by
the point. It corresponds to the tangent map, or differential of the
group multiplication by the point or its inverse. For groups with a
vector representation, it is only implemented at identity, but it can
be used at other points with inverse = TRUE. This method wraps the
Jacobian translation which actually computes the matrix representation
of the map.
Returns
A function taking as argument a numeric array tangent_vec of
shape [… \times \{ \mathrm{dim}, [n \times n] \} ] specifying
one or more tangent vectors and returning a numeric array of shape
[… \times \{ \mathrm{dim}, [n \times n] \} ] storing the
result of the tangent mapping of the left/right translation of input
tangent points by corresponding base points.
Examples
if (reticulate::py_module_available("geomstats")) {
so3 <- SpecialOrthogonal(n = 3)
tangent_map <- so3$tangent_translation_map(diag(1, 3))
tangent_map(diag(1, 3))
}
Method compose()
Performs function composition corresponding to the Lie
group.
Usage
MatrixLieGroup$compose(point_a, point_b)
Arguments
point_aA numeric array of shape [… \times \{
\mathrm{dim}, n \times n \}] specifying one or more left factors in
the product.
point_bA numeric array of shape [… \times \{
\mathrm{dim}, n \times n \}] specifying one or more right factors in
the product.
Returns
A numeric array of shape [… \times \{ \mathrm{dim}, n
\times n \}] storing the product of point_a and point_b along the
first dimension.
Examples
if (reticulate::py_module_available("geomstats")) {
so3 <- SpecialOrthogonal(n = 3)
so3$compose(diag(1, 3), diag(1, 3))
}
Method inverse()
Computes the inverse law of the Lie group.
Usage
MatrixLieGroup$inverse(point)
Arguments
pointA numeric array of shape [… \times \{ \mathrm{dim},
n \times n \}] specifying one or more points to be inverted.
Returns
A numeric array of the same shape storing the inverted points.
Examples
if (reticulate::py_module_available("geomstats")) {
so3 <- SpecialOrthogonal(n = 3)
so3$inverse(diag(1, 3))
}
Method clone()
The objects of this class are cloneable with this method.
Usage
MatrixLieGroup$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
Author(s)
Nina Miolane
Examples
## ------------------------------------------------
## Method `MatrixLieGroup$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$inverse(diag(1, 3))
}
rgeomstats documentation built on Nov. 4, 2022, 5:09 p.m.
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| MatrixLieGroup | R Documentation |
Abstract Class for Matrix Lie Groups
Description
Class for matrix Lie groups.
Super classes
rgeomstats::PythonClass -> rgeomstats::Manifold -> MatrixLieGroup
Public fields
lie_algebraAn object of class
MatrixLieAlgebraorNULLrepresenting the tangent space at the identity.nThe size of the n \times n matrix elements.
left_canonical_metricAn object of class
InvariantMetricrepresenting the left invariant metric that corresponds to the Euclidean inner product at the identity.right_canonical_metricAn object of class
InvariantMetricrepresenting the left invariant metric that corresponds to the Euclidean inner product at the identity.
Methods
Public methods
Inherited methods
rgeomstats::PythonClass$get_python_class()rgeomstats::PythonClass$set_python_class()rgeomstats::Manifold$belongs()rgeomstats::Manifold$is_tangent()rgeomstats::Manifold$random_point()rgeomstats::Manifold$random_tangent_vec()rgeomstats::Manifold$regularize()rgeomstats::Manifold$set_metric()rgeomstats::Manifold$to_tangent()
Method new()
The MatrixLieGroup class constructor.
Usage
MatrixLieGroup$new(dim, n, lie_algebra = NULL, ..., py_cls = NULL)
Arguments
dimAn integer value specifying the dimension of the manifold.
nThe size of the n \times n matrix elements.
lie_algebraAn object of class
MatrixLieAlgebraorNULLrepresenting the tangent space at the identity....Extra arguments to be passed to parent class constructors. See
Manifoldclass.py_clsA Python object of class
MatrixLieGroup. Defaults toNULLin which case it is instantiated on the fly using the other input arguments.
Returns
An object of class MatrixLieGroup.
Method exp()
Exponentiates a left-invariant vector field from a base point.
Usage
MatrixLieGroup$exp(tangent_vec, base_point = NULL)
Arguments
tangent_vecA numeric array of shape [… \times n \times n] specifying one or more tangent vectors at corresponding base points.
base_pointA numeric array of shape [… \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 [… \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
pointA numeric array of shape [… \times n \times n] specifying one or more points.
base_pointA numeric array of shape [… \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 [… \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_aA numeric array of shape [… \times n \times n] specifying one or more tangent vectors at corresponding base points.
tangent_vector_bA numeric array of shape [… \times n \times n] specifying one or more tangent vectors at corresponding base points.
base_pointA numeric array of shape [… \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 [… \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
pointA numeric array of shape [… \times \{ \mathrm{dim}, [n \times n] \} ] specifying one or more points at which to compute the map.
left_or_rightA character string specifying whether to compute the map for the left or right translation. Choices are
"left"or"right. Defaults to"left".inverseA 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 toFALSE.
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 [… \times \{ \mathrm{dim}, [n \times n] \} ] specifying
one or more tangent vectors and returning a numeric array of shape
[… \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_aA numeric array of shape [… \times \{ \mathrm{dim}, n \times n \}] specifying one or more left factors in the product.
point_bA numeric array of shape [… \times \{ \mathrm{dim}, n \times n \}] specifying one or more right factors in the product.
Returns
A numeric array of shape [… \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
pointA numeric array of shape [… \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
deepWhether to make a deep clone.
Author(s)
Nina Miolane
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))
}
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