Connection: Abstract Class for Connections
In rgeomstats: Interface to 'Geomstats'
Connection R Documentation
Abstract Class for Connections
Description
An R6::R6Class object implementing the base Connection
class for affine connections.
Super class
rgeomstats::PythonClass -> Connection
Public fields
dimAn integer value specifying the dimension of the underlying
manifold.
shapeAn integer vector specifying the shape of one element of the
manifold. Defaults to NULL.
default_coords_typeA string specifying the coordinate type.
Choices are extrensic or intrinsic. Dedaults to intrinsic.
default_point_typeA string specifying the point type. Choices are
vector or matrix. It is automatically determined depending on the
manifold.
Methods
Public methods
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Inherited methods
Method new()
The Connection class constructor.
Usage
Connection$new(
dim,
shape = NULL,
default_coords_type = "intrinsic",
py_cls = NULL
)
Arguments
dimAn integer value specifying the dimension of the manifold.
shapeAn integer vector specifying the shape of one element of the
manifold. Defaults to NULL.
default_coords_typeA string specifying the coordinate type.
Choices are extrensic or intrinsic. Defaults to intrinsic.
py_clsA Python object of class Connection. Defaults to NULL
in which case it is instantiated on the fly using the other input
arguments.
metricA RiemannianMetric object specifying the metric to use
on the manifold. Defaults to NULL.
Returns
An object of class Connection.
Method christoffels()
Christoffel symbols associated with the connection.
Usage
Connection$christoffels(base_point)
Arguments
base_pointA numeric array of shape dim specifying a base point on
the manifold.
Returns
A numeric array of shape c(dim, dim, dim) storing the
Christoffel symbols, with the contravariant index on the first
dimension.
Method geodesic_equation()
Computes the geodesic ODE associated with the connection.
Usage
Connection$geodesic_equation(state, .time)
Arguments
stateA numeric array of shape dim specifying a tangent vector
to the manifold at the position specified by .time.
.timeA numeric array of shape dim specifying a point on the
manifold at which to compute the geodesic ODE.
Returns
A numeric array of shape dim storing the value of the vector
field to be integrated at position.
Method exp()
Exponential map associated to the affine connection.
Usage
Connection$exp(tangent_vec, base_point, n_steps = 100, step = "euler")
Arguments
tangent_vecA numeric array of shape dim specifying a tangent vector
at base_point.
base_pointA numeric array of shape dim specifying a base point on
the manifold.
n_stepsAn integer value specifying the number of discrete time steps
to take in the integration. Defaults to 100L.
stepA string specifying which numerical scheme to use for
integration. Choices are euler or rk4. Defaults to euler.
Details
Exponential map at base_point of tangent_vec computed by
integration of the geodesic equation (initial value problem), using the
christoffel symbols.
Returns
A numeric array of shape dim storing the exponential of the
input tangent vector, which lies on on the manifold.
Method log()
Logarithm map associated to the affine connection.
Usage
Connection$log(
point,
base_point,
n_steps = 100,
step = "euler",
max_iter = 25,
verbose = FALSE,
tol = gs$backend$atol
)
Arguments
pointA numeric array of shape dim specifying a point on the
manifold.
base_pointA numeric array of shape dim specifying a base point on
the manifold.
n_stepsAn integer value specifying the number of discrete time steps
to take in the integration. Defaults to 100L.
stepA string specifying which numerical scheme to use for
integration. Choices are euler or rk4. Defaults to euler.
max_iterAn integer value specifying the number of iterations.
Defaults to 25L.
verboseA boolean specifying whether the optimizer should display
intermediate messages pertaining to its convergence. Defaults to
FALSE.
tolA numeric value specifying the absolute tolerance for
optimization convergence. Defaults to gs$backend$atol.
Details
Solves the boundary value problem associated to the geodesic
equation using the Christoffel symbols and conjugate gradient descent.
Returns
A numeric array of shape dim storing the exponential of the
input tangent vector, which lies on on the manifold.
Method ladder_parallel_transport()
Approximate parallel transport using the pole ladder scheme.
Usage
Connection$ladder_parallel_transport(
tangent_vec,
base_point,
direction,
n_rungs = 1,
scheme = "pole",
alpha = 1,
...
)
Arguments
tangent_vecA numeric array of shape dim specifying a tangent vector
at base_point.
base_pointA numeric array of shape dim specifying a base point on
the manifold.
directionTangent vector at base point specifying the initial
speed of the geodesic along which to transport.
n_rungsA scalar integer specifying the Number of steps of the
ladder. Defaults to 1L.
schemeA string specifying the scheme to use for the construction
of the ladder at each step. Choices are either pole or schild.
Defaults to pole.
alphaA numeric value specifying the exponent for the scaling of
the vector to transport. Must be greater or equal to 1 and
\insertCiteguigui2022numerical;textualrgeomstats proved that alpha = 2 is optimal. Defaults to 2.
...Extra arguments to be passed to calls to $exp() and $log()
in auxiliary single ladder step functions.
Details
Approximate parallel transport using either the pole ladder or
the Schild's ladder scheme
\insertCitelorenzi2014efficientrgeomstats. Pole ladder is exact in
symmetric spaces and of order two in general while Schild's ladder is a
first order approximation \insertCiteguigui2022numericalrgeomstats.
Both schemes are available on any affine connection manifolds whose
exponential and logarithm maps are implemented. tangent_vec is
transported along the geodesic starting at the base_point with
initial tangent vector direction.
References
\insertCited
Returns
A named list with 3 components:
-
transported_tangent_vector: Approximation of the parallel transport
of the input tangent vector.
-
trajectory : A list of length n_steps storing the geodesics of the
construction, only if return_geodesics = TRUE in the step function. The
geodesics are methods of the class connection.
-
end_point:
Method curvature()
Computes the curvature.
Usage
Connection$curvature(tangent_vec_a, tangent_vec_b, tangent_vec_c, base_point)
Arguments
tangent_vec_aTangent vector at base_point.
tangent_vec_bTangent vector at base_point.
tangent_vec_cTangent vector at base_point.
base_pointA numeric array of shape dim specifying a base point on
the manifold.
Details
For three vector fields X|_P = \mathrm{tangent\_vec\_a},
Y|_P = \mathrm{tangent\_vec\_b}, Z|_P =
\mathrm{tangent\_vec\_c} with tangent vector specified in argument at
the base point P, the curvature is defined by
R(X,Y)Z =
\nabla_{[X,Y]}Z - \nabla_X\nabla_Y Z + \nabla_Y\nabla_X Z.
Returns
Tangent vector at base_point.
Method directional_curvature()
Computes the directional curvature (tidal force operator).
Usage
Connection$directional_curvature(tangent_vec_a, tangent_vec_b, base_point)
Arguments
tangent_vec_aTangent vector at base_point.
tangent_vec_bTangent vector at base_point.
base_pointA numeric array of shape dim specifying a base point on
the manifold.
Details
For two vector fields X|_P = \mathrm{tangent\_vec\_a} and
Y|_P = \mathrm{tangent\_vec\_b} with tangent vector specified in
argument at the base point P, the directional curvature, better
known in relativity as the tidal force operator, is defined by
R_Y(X) = R(Y,X)Y.
Returns
Tangent vector at base_point.
Method curvature_derivative()
Computes the covariant derivative of the curvature.
Usage
Connection$curvature_derivative(
tangent_vec_a,
tangent_vec_b,
tangent_vec_c,
tangent_vec_d,
base_point = NULL
)
Arguments
tangent_vec_aTangent vector at base_point.
tangent_vec_bTangent vector at base_point.
tangent_vec_cTangent vector at base_point.
tangent_vec_dTangent vector at base_point.
base_pointA numeric array of shape dim specifying a base point on
the manifold.
Details
For four vector fields H|_P = \mathrm{tangent\_vec\_a},
X|_P = \mathrm{tangent\_vec\_b}, Y|_P =
\mathrm{tangent\_vec\_c}, Z|_P = \mathrm{tangent\_vec\_d} with
tangent vector value specified in argument at the base point P,
the covariant derivative of the curvature (\nabla_H R)(X, Y) Z
|_P is computed at the base point P.
Returns
Tangent vector at base_point.
Method directional_curvature_derivative()
Computes the covariant derivative of the directional
curvature.
Usage
Connection$directional_curvature_derivative(
tangent_vec_a,
tangent_vec_b,
base_point = NULL
)
Arguments
tangent_vec_aTangent vector at base_point.
tangent_vec_bTangent vector at base_point.
base_pointA numeric array of shape dim specifying a base point on
the manifold.
Details
For two vector fields X|_P = \mathrm{tangent\_vec\_a},
Y|_P = \mathrm{tangent\_vec\_b} with tangent vector value
specified in argument at the base point P, the covariant
derivative (in the direction X) (\nabla_X R_Y)(X) |_P =
(\nabla_X R)(Y, X) Y |_P of the directional curvature (in the
direction Y) R_Y(X) = R(Y, X) Y is a quadratic tensor in
X and Y that plays an important role in the computation of
the moments of the empirical Fréchet mean
\insertCitepennec2019curvaturergeomstats.
References
\insertCited
Returns
Tangent vector at base_point.
Method geodesic()
Generates parametrized function for the geodesic curve.
Usage
Connection$geodesic(
initial_point,
end_point = NULL,
initial_tangent_vec = NULL
)
Arguments
initial_pointPoint on the manifold specifying the initial point
of the geodesic.
end_pointPoint on the manifold specifying the end point of the
geodesic. Defaults to NULL, in which case an initial tangent vector
must be given.
initial_tangent_vecTangent vector at base point specifying the
initial speed of the geodesics. Defaults to NULL, in which case an
end point must be given and a logarithm is computed.
Details
Geodesic curve defined by either:
an initial point and an initial tangent vector,
an initial point and an end point.
Returns
A function representing the time-parametrized geodesic curve. If
a list of initial conditions is passed, the output list will contain,
for each time point, a list with the geodesic values each initial
condition.
Method parallel_transport()
Computes the parallel transport of a tangent vector.
Usage
Connection$parallel_transport(
tangent_vec,
base_point,
direction = NULL,
end_point = NULL
)
Arguments
tangent_vecA numeric array of shape dim specifying a tangent vector
at base_point.
base_pointA numeric array of shape dim specifying a base point on
the manifold.
directionTangent vector at base point specifying the point along
which the parallel transport is computed. Defaults to NULL.
end_pointPoint on the manifold specifying the point to transport
to. Defaults to NULL.
Details
Closed-form solution for the parallel transport of a tangent
vector along the geodesic between two points base_point and
end_point or alternatively defined by t \mapsto
\exp_\mathrm{base_point} (t \mathrm{direction}).
Returns
Tangent vector transported at t \mapsto
\exp_\mathrm{base_point} (t \mathrm{direction}).
Method injectivity_radius()
Computes the radius of the injectivity domain.
Usage
Connection$injectivity_radius(base_point)
Arguments
base_pointA numeric array of shape dim specifying a base point on
the manifold.
Details
This is is the supremum of radii r for which the exponential
map is a diffeomorphism from the open ball of radius r centered at the
base point onto its image.
Returns
A numeric value representing the injectivity radius.
Method clone()
The objects of this class are cloneable with this method.
Usage
Connection$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
Author(s)
Nicolas Guigui
rgeomstats documentation built on Nov. 4, 2022, 5:09 p.m.
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| Connection | R Documentation |
Abstract Class for Connections
Description
An R6::R6Class object implementing the base Connection
class for affine connections.
Super class
rgeomstats::PythonClass -> Connection
Public fields
dimAn integer value specifying the dimension of the underlying manifold.
shapeAn integer vector specifying the shape of one element of the manifold. Defaults to
NULL.default_coords_typeA string specifying the coordinate type. Choices are
extrensicorintrinsic. Dedaults tointrinsic.default_point_typeA string specifying the point type. Choices are
vectorormatrix. It is automatically determined depending on the manifold.
Methods
Public methods
Inherited methods
Method new()
The Connection class constructor.
Usage
Connection$new( dim, shape = NULL, default_coords_type = "intrinsic", py_cls = NULL )
Arguments
dimAn integer value specifying the dimension of the manifold.
shapeAn integer vector specifying the shape of one element of the manifold. Defaults to
NULL.default_coords_typeA string specifying the coordinate type. Choices are
extrensicorintrinsic. Defaults tointrinsic.py_clsA Python object of class
Connection. Defaults toNULLin which case it is instantiated on the fly using the other input arguments.metricA
RiemannianMetricobject specifying the metric to use on the manifold. Defaults toNULL.
Returns
An object of class Connection.
Method christoffels()
Christoffel symbols associated with the connection.
Usage
Connection$christoffels(base_point)
Arguments
base_pointA numeric array of shape
dimspecifying a base point on the manifold.
Returns
A numeric array of shape c(dim, dim, dim) storing the
Christoffel symbols, with the contravariant index on the first
dimension.
Method geodesic_equation()
Computes the geodesic ODE associated with the connection.
Usage
Connection$geodesic_equation(state, .time)
Arguments
stateA numeric array of shape
dimspecifying a tangent vector to the manifold at the position specified by.time..timeA numeric array of shape
dimspecifying a point on the manifold at which to compute the geodesic ODE.
Returns
A numeric array of shape dim storing the value of the vector
field to be integrated at position.
Method exp()
Exponential map associated to the affine connection.
Usage
Connection$exp(tangent_vec, base_point, n_steps = 100, step = "euler")
Arguments
tangent_vecA numeric array of shape
dimspecifying a tangent vector atbase_point.base_pointA numeric array of shape
dimspecifying a base point on the manifold.n_stepsAn integer value specifying the number of discrete time steps to take in the integration. Defaults to
100L.stepA string specifying which numerical scheme to use for integration. Choices are
eulerorrk4. Defaults toeuler.
Details
Exponential map at base_point of tangent_vec computed by integration of the geodesic equation (initial value problem), using the christoffel symbols.
Returns
A numeric array of shape dim storing the exponential of the
input tangent vector, which lies on on the manifold.
Method log()
Logarithm map associated to the affine connection.
Usage
Connection$log( point, base_point, n_steps = 100, step = "euler", max_iter = 25, verbose = FALSE, tol = gs$backend$atol )
Arguments
pointA numeric array of shape
dimspecifying a point on the manifold.base_pointA numeric array of shape
dimspecifying a base point on the manifold.n_stepsAn integer value specifying the number of discrete time steps to take in the integration. Defaults to
100L.stepA string specifying which numerical scheme to use for integration. Choices are
eulerorrk4. Defaults toeuler.max_iterAn integer value specifying the number of iterations. Defaults to
25L.verboseA boolean specifying whether the optimizer should display intermediate messages pertaining to its convergence. Defaults to
FALSE.tolA numeric value specifying the absolute tolerance for optimization convergence. Defaults to
gs$backend$atol.
Details
Solves the boundary value problem associated to the geodesic equation using the Christoffel symbols and conjugate gradient descent.
Returns
A numeric array of shape dim storing the exponential of the
input tangent vector, which lies on on the manifold.
Method ladder_parallel_transport()
Approximate parallel transport using the pole ladder scheme.
Usage
Connection$ladder_parallel_transport( tangent_vec, base_point, direction, n_rungs = 1, scheme = "pole", alpha = 1, ... )
Arguments
tangent_vecA numeric array of shape
dimspecifying a tangent vector atbase_point.base_pointA numeric array of shape
dimspecifying a base point on the manifold.directionTangent vector at base point specifying the initial speed of the geodesic along which to transport.
n_rungsA scalar integer specifying the Number of steps of the ladder. Defaults to
1L.schemeA string specifying the scheme to use for the construction of the ladder at each step. Choices are either
poleorschild. Defaults topole.alphaA numeric value specifying the exponent for the scaling of the vector to transport. Must be greater or equal to 1 and \insertCiteguigui2022numerical;textualrgeomstats proved that
alpha = 2is optimal. Defaults to2....Extra arguments to be passed to calls to
$exp()and$log()in auxiliary single ladder step functions.
Details
Approximate parallel transport using either the pole ladder or
the Schild's ladder scheme
\insertCitelorenzi2014efficientrgeomstats. Pole ladder is exact in
symmetric spaces and of order two in general while Schild's ladder is a
first order approximation \insertCiteguigui2022numericalrgeomstats.
Both schemes are available on any affine connection manifolds whose
exponential and logarithm maps are implemented. tangent_vec is
transported along the geodesic starting at the base_point with
initial tangent vector direction.
References
\insertCitedReturns
A named list with 3 components:
-
transported_tangent_vector: Approximation of the parallel transport of the input tangent vector. -
trajectory: A list of lengthn_stepsstoring the geodesics of the construction, only ifreturn_geodesics = TRUEin the step function. The geodesics are methods of the class connection. -
end_point:
Method curvature()
Computes the curvature.
Usage
Connection$curvature(tangent_vec_a, tangent_vec_b, tangent_vec_c, base_point)
Arguments
tangent_vec_aTangent vector at
base_point.tangent_vec_bTangent vector at
base_point.tangent_vec_cTangent vector at
base_point.base_pointA numeric array of shape
dimspecifying a base point on the manifold.
Details
For three vector fields X|_P = \mathrm{tangent\_vec\_a}, Y|_P = \mathrm{tangent\_vec\_b}, Z|_P = \mathrm{tangent\_vec\_c} with tangent vector specified in argument at the base point P, the curvature is defined by
R(X,Y)Z = \nabla_{[X,Y]}Z - \nabla_X\nabla_Y Z + \nabla_Y\nabla_X Z.
Returns
Tangent vector at base_point.
Method directional_curvature()
Computes the directional curvature (tidal force operator).
Usage
Connection$directional_curvature(tangent_vec_a, tangent_vec_b, base_point)
Arguments
tangent_vec_aTangent vector at
base_point.tangent_vec_bTangent vector at
base_point.base_pointA numeric array of shape
dimspecifying a base point on the manifold.
Details
For two vector fields X|_P = \mathrm{tangent\_vec\_a} and Y|_P = \mathrm{tangent\_vec\_b} with tangent vector specified in argument at the base point P, the directional curvature, better known in relativity as the tidal force operator, is defined by
R_Y(X) = R(Y,X)Y.
Returns
Tangent vector at base_point.
Method curvature_derivative()
Computes the covariant derivative of the curvature.
Usage
Connection$curvature_derivative( tangent_vec_a, tangent_vec_b, tangent_vec_c, tangent_vec_d, base_point = NULL )
Arguments
tangent_vec_aTangent vector at
base_point.tangent_vec_bTangent vector at
base_point.tangent_vec_cTangent vector at
base_point.tangent_vec_dTangent vector at
base_point.base_pointA numeric array of shape
dimspecifying a base point on the manifold.
Details
For four vector fields H|_P = \mathrm{tangent\_vec\_a}, X|_P = \mathrm{tangent\_vec\_b}, Y|_P = \mathrm{tangent\_vec\_c}, Z|_P = \mathrm{tangent\_vec\_d} with tangent vector value specified in argument at the base point P, the covariant derivative of the curvature (\nabla_H R)(X, Y) Z |_P is computed at the base point P.
Returns
Tangent vector at base_point.
Method directional_curvature_derivative()
Computes the covariant derivative of the directional curvature.
Usage
Connection$directional_curvature_derivative( tangent_vec_a, tangent_vec_b, base_point = NULL )
Arguments
tangent_vec_aTangent vector at
base_point.tangent_vec_bTangent vector at
base_point.base_pointA numeric array of shape
dimspecifying a base point on the manifold.
Details
For two vector fields X|_P = \mathrm{tangent\_vec\_a}, Y|_P = \mathrm{tangent\_vec\_b} with tangent vector value specified in argument at the base point P, the covariant derivative (in the direction X) (\nabla_X R_Y)(X) |_P = (\nabla_X R)(Y, X) Y |_P of the directional curvature (in the direction Y) R_Y(X) = R(Y, X) Y is a quadratic tensor in X and Y that plays an important role in the computation of the moments of the empirical Fréchet mean \insertCitepennec2019curvaturergeomstats.
References
\insertCitedReturns
Tangent vector at base_point.
Method geodesic()
Generates parametrized function for the geodesic curve.
Usage
Connection$geodesic( initial_point, end_point = NULL, initial_tangent_vec = NULL )
Arguments
initial_pointPoint on the manifold specifying the initial point of the geodesic.
end_pointPoint on the manifold specifying the end point of the geodesic. Defaults to
NULL, in which case an initial tangent vector must be given.initial_tangent_vecTangent vector at base point specifying the initial speed of the geodesics. Defaults to
NULL, in which case an end point must be given and a logarithm is computed.
Details
Geodesic curve defined by either:
an initial point and an initial tangent vector,
an initial point and an end point.
Returns
A function representing the time-parametrized geodesic curve. If a list of initial conditions is passed, the output list will contain, for each time point, a list with the geodesic values each initial condition.
Method parallel_transport()
Computes the parallel transport of a tangent vector.
Usage
Connection$parallel_transport( tangent_vec, base_point, direction = NULL, end_point = NULL )
Arguments
tangent_vecA numeric array of shape
dimspecifying a tangent vector atbase_point.base_pointA numeric array of shape
dimspecifying a base point on the manifold.directionTangent vector at base point specifying the point along which the parallel transport is computed. Defaults to
NULL.end_pointPoint on the manifold specifying the point to transport to. Defaults to
NULL.
Details
Closed-form solution for the parallel transport of a tangent
vector along the geodesic between two points base_point and
end_point or alternatively defined by t \mapsto
\exp_\mathrm{base_point} (t \mathrm{direction}).
Returns
Tangent vector transported at t \mapsto \exp_\mathrm{base_point} (t \mathrm{direction}).
Method injectivity_radius()
Computes the radius of the injectivity domain.
Usage
Connection$injectivity_radius(base_point)
Arguments
base_pointA numeric array of shape
dimspecifying a base point on the manifold.
Details
This is is the supremum of radii r for which the exponential map is a diffeomorphism from the open ball of radius r centered at the base point onto its image.
Returns
A numeric value representing the injectivity radius.
Method clone()
The objects of this class are cloneable with this method.
Usage
Connection$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
Author(s)
Nicolas Guigui
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