"""Affine connections.
Lead author: Nicolas Guigui.
"""
from abc import ABC
import geomstats.backend as gs
import geomstats.errors
def _check_log_solver(connection, raise_=True):
if not hasattr(connection, "log_solver"):
if raise_:
raise ValueError(
"Requires `self.log_solver`. "
"Check `geomstats.numerics.geodesic` for available solvers."
)
return False
return True
def _check_exp_solver(connection, raise_=True):
if not hasattr(connection, "exp_solver"):
if raise_:
raise ValueError(
"Requires `self.exp_solver`. "
"Check `geomstats.numerics.geodesic` for available solvers."
)
return False
return True
[docs]
class Connection(ABC):
r"""Class for affine connections.
Parameters
----------
space : Manifold object
M in the tuple (M, g).
"""
def __init__(self, space):
self._space = space
[docs]
def christoffels(self, base_point=None):
"""Christoffel symbols associated with the connection.
The contravariant index is on the first dimension.
Parameters
----------
base_point : array-like, shape=[..., dim]
Point on the manifold.
Returns
-------
gamma : array-like, shape=[..., dim, dim, dim]
Christoffel symbols, with the contravariant index on
the first dimension.
"""
raise NotImplementedError("The Christoffel symbols are not implemented.")
[docs]
def geodesic_equation(self, state):
"""Compute the geodesic ODE associated with the connection.
Parameters
----------
state : array-like, shape=[..., 2, dim]
Tangent vector at the position.
Returns
-------
geodesic_ode : array-like, shape=[..., 2, dim]
Value of the vector field to be integrated at position.
"""
if self._space.point_ndim != 1:
raise NotImplementedError(
"The geodesic equation is only implemented for points "
"represented by a vector."
)
position = state[..., 0, :]
velocity = state[..., 1, :]
gamma = self.christoffels(position)
equation = gs.einsum("...kij,...i->...kj", gamma, velocity)
equation = -gs.einsum("...kj,...j->...k", equation, velocity)
return gs.stack([velocity, equation], axis=-2)
[docs]
def exp(self, tangent_vec, base_point):
"""Exponential map associated to the affine connection.
Parameters
----------
tangent_vec : array-like, shape=[..., dim]
Tangent vector at the base point.
base_point : array-like, shape=[..., dim]
Point on the manifold.
Returns
-------
exp : array-like, shape=[..., dim]
Point on the manifold.
"""
_check_exp_solver(self)
return self.exp_solver.exp(tangent_vec, base_point)
[docs]
def log(self, point, base_point):
"""Compute logarithm map associated to the affine connection.
Solve the boundary value problem associated to the geodesic equation
using the Christoffel symbols and conjugate gradient descent.
Parameters
----------
point : array-like, shape=[..., dim]
Point on the manifold.
base_point : array-like, shape=[..., dim]
Point on the manifold.
n_steps : int
Number of discrete time steps to take in the integration.
Optional, default: N_STEPS.
step : str, {'euler', 'rk4'}
Numerical scheme to use for integration.
Optional, default: 'euler'.
max_iter
verbose
tol
Returns
-------
tangent_vec : array-like, shape=[..., dim]
Tangent vector at the base point.
"""
_check_log_solver(self)
return self.log_solver.log(point, base_point)
def _pole_ladder_step(
self, base_point, next_point, base_shoot, return_geodesics=False
):
"""Compute one Pole Ladder step.
One step of pole ladder scheme [LP2013a]_ using the geodesic to
transport along as main_geodesic of the parallelogram.
Parameters
----------
base_point : array-like, shape=[..., dim]
Point on the manifold, from which to transport.
next_point : array-like, shape=[..., dim]
Point on the manifold, to transport to.
base_shoot : array-like, shape=[..., dim]
Point on the manifold, end point of the geodesics starting
from the base point with initial speed to be transported.
return_geodesics : bool, optional (defaults to False)
Whether to return the geodesics of the
construction.
Returns
-------
next_step : dict of array-like and callable with following keys:
next_tangent_vec : array-like, shape=[..., dim]
Tangent vector at end point.
end_point : array-like, shape=[..., dim]
Point on the manifold, closes the geodesic parallelogram of the
construction.
geodesics : list of callable, len=3 (only if
`return_geodesics=True`)
Three geodesics of the construction.
References
----------
.. [LP2013a] Marco Lorenzi, Xavier Pennec. Efficient Parallel Transport
of Deformations in Time Series of Images: from Schild's to
Pole Ladder. Journal of Mathematical Imaging and Vision, Springer
Verlag, 2013,50 (1-2), pp.5-17. ⟨10.1007/s10851-013-0470-3⟩
"""
mid_tangent_vector_to_shoot = (
1.0 / 2.0 * self.log(base_point=base_point, point=next_point)
)
mid_point = self.exp(
base_point=base_point, tangent_vec=mid_tangent_vector_to_shoot
)
tangent_vector_to_shoot = -self.log(base_point=mid_point, point=base_shoot)
end_shoot = self.exp(base_point=mid_point, tangent_vec=tangent_vector_to_shoot)
geodesics = []
if return_geodesics:
main_geodesic = self.geodesic(
initial_point=base_point, end_point=next_point
)
diagonal = self.geodesic(initial_point=mid_point, end_point=base_shoot)
final_geodesic = self.geodesic(
initial_point=next_point, end_point=end_shoot
)
geodesics = [main_geodesic, diagonal, final_geodesic]
return {"geodesics": geodesics, "end_point": end_shoot}
def _schild_ladder_step(
self, base_point, next_point, base_shoot, return_geodesics=False
):
"""Compute one Schild's Ladder step.
One step of the Schild's ladder scheme [LP2013a]_ using the geodesic to
transport along as one side of the parallelogram.
Parameters
----------
base_point : array-like, shape=[..., dim]
Point on the manifold, from which to transport.
next_point : array-like, shape=[..., dim]
Point on the manifold, to transport to.
base_shoot : array-like, shape=[..., dim]
Point on the manifold, end point of the geodesics starting
from the base point with initial speed to be transported.
return_geodesics : bool
Whether to return points computed along each geodesic of the
construction.
Optional, default: False.
Returns
-------
transported_tangent_vector : array-like, shape=[..., dim]
Tangent vector at end point.
end_point : array-like, shape=[..., dim]
Point on the manifold, closes the geodesic parallelogram of the
construction.
References
----------
.. [LP2013a] Marco Lorenzi, Xavier Pennec. Efficient Parallel Transport
of Deformations in Time Series of Images: from Schild's to
Pole Ladder. Journal of Mathematical Imaging and Vision, Springer
Verlag, 2013,50 (1-2), pp.5-17. ⟨10.1007/s10851-013-0470-3⟩
"""
mid_tangent_vector_to_shoot = (
1.0 / 2.0 * self.log(base_point=base_shoot, point=next_point)
)
mid_point = self.exp(
base_point=base_shoot, tangent_vec=mid_tangent_vector_to_shoot
)
tangent_vector_to_shoot = -self.log(base_point=mid_point, point=base_point)
end_shoot = self.exp(base_point=mid_point, tangent_vec=tangent_vector_to_shoot)
geodesics = []
if return_geodesics:
main_geodesic = self.geodesic(
initial_point=base_point, end_point=next_point
)
diagonal = self.geodesic(initial_point=base_point, end_point=end_shoot)
second_diagonal = self.geodesic(
initial_point=base_shoot, end_point=next_point
)
final_geodesic = self.geodesic(
initial_point=next_point, end_point=end_shoot
)
geodesics = [main_geodesic, diagonal, second_diagonal, final_geodesic]
return {"geodesics": geodesics, "end_point": end_shoot}
[docs]
def ladder_parallel_transport(
self,
tangent_vec,
base_point,
direction,
n_rungs=1,
scheme="pole",
alpha=1,
return_geodesics=False,
):
"""Approximate parallel transport using the pole ladder scheme.
Approximate Parallel transport using either the pole ladder or the
Schild's ladder scheme [LP2013b]_. Pole ladder is exact in symmetric
spaces and of order two in general while Schild's ladder is a first
order approximation [GP2020]_. 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`.
Parameters
----------
tangent_vec : array-like, shape=[..., dim]
Tangent vector at base point to transport.
direction : array-like, shape=[..., dim]
Tangent vector at base point, initial speed of the geodesic along
which to transport.
base_point : array-like, shape=[..., dim]
Point on the manifold, initial position of the geodesic along
which to transport.
n_rungs : int
Number of steps of the ladder.
Optional, default: 1.
scheme : str, {'pole', 'schild'}
The scheme to use for the construction of the ladder at each step.
Optional, default: 'pole'.
alpha : float
Exponent for the scaling of the vector to transport. Must be
greater or equal to 1, 2 is optimal. See [GP2020]_.
Optional, default: 2
Returns
-------
ladder : dict of array-like and callable with following keys
transported_tangent_vector : array-like, shape=[..., dim]
Approximation of the parallel transport of tangent vector a.
trajectory : list of list of callable, len=n_steps
List of lists containing the geodesics of the
construction, only if `return_geodesics=True` in the step
function. The geodesics are methods of the class connection.
References
----------
.. [LP2013b] Lorenzi, Marco, and Xavier Pennec. “Efficient Parallel
Transport of Deformations in Time Series of Images: From Schild to
Pole Ladder.” Journal of Mathematical Imaging and Vision 50, no. 1
(September 1, 2014): 5–17.
https://doi.org/10.1007/s10851-013-0470-3.
.. [GP2020] Guigui, Nicolas, and Xavier Pennec. “Numerical Accuracy
of Ladder Schemes for Parallel Transport on Manifolds.”
Foundations of Computational Mathematics, June 18, 2021.
https://doi.org/10.1007/s10208-021-09515-x.
"""
geomstats.errors.check_integer(n_rungs, "n_rungs")
if alpha < 1:
raise ValueError("alpha must be greater or equal to one")
current_point = base_point
next_tangent_vec = tangent_vec / (n_rungs**alpha)
methods = {"pole": self._pole_ladder_step, "schild": self._schild_ladder_step}
single_step = methods[scheme]
base_shoot = self.exp(base_point=current_point, tangent_vec=next_tangent_vec)
trajectory = []
for i_point in range(n_rungs):
frac_tan_vector_b = (i_point + 1) / n_rungs * direction
next_point = self.exp(base_point=base_point, tangent_vec=frac_tan_vector_b)
next_step = single_step(
base_point=current_point,
next_point=next_point,
base_shoot=base_shoot,
return_geodesics=return_geodesics,
)
current_point = next_point
base_shoot = next_step["end_point"]
trajectory.append(next_step["geodesics"])
transported_tangent_vec = self.log(base_shoot, current_point)
if n_rungs % 2 == 1 and scheme == "pole":
transported_tangent_vec *= -1.0
transported_tangent_vec *= n_rungs**alpha
return {
"transported_tangent_vec": transported_tangent_vec,
"end_point": current_point,
"trajectory": trajectory,
}
[docs]
def riemann_tensor(self, base_point=None):
r"""Compute Riemannian tensor at base_point.
In the literature the Riemannian curvature tensor is noted :math:`R_{ijk}^l`.
Following tensor index convention (ref. Wikipedia), we have:
:math:`R_{ijk}^l = dx^l(R(X_j, X_k)X_i)`
which gives :math:`R_{ijk}^l` as a sum of four terms:
.. math::
\partial_j \Gamma^l_{ki} - \partial_k \Gamma^l_{ji}
+ \Gamma^l_{jm} \Gamma^m_{ki} - \Gamma^l_{km} \Gamma^m_{ji}
Note that geomstats puts the contravariant index on
the first dimension of the Christoffel symbols.
Parameters
----------
base_point : array-like, shape=[..., dim]
Point on the manifold.
Returns
-------
riemann_curvature : array-like, shape=[..., dim, dim, dim, dim]
riemann_tensor[...,i,j,k,l] = R_{ijk}^l
Riemannian tensor curvature,
with the contravariant index on the last dimension.
"""
if len(self._space.shape) > 1:
raise NotImplementedError(
"Riemann tensor not implemented for manifolds with points of ndim > 1."
)
christoffels = self.christoffels(base_point)
jacobian_christoffels = gs.autodiff.jacobian_vec(self.christoffels)(base_point)
prod_christoffels = gs.einsum(
"...ijk,...klm->...ijlm", christoffels, christoffels
)
riemann_curvature = (
gs.einsum("...ijlm->...lmji", jacobian_christoffels)
- gs.einsum("...ijlm->...ljmi", jacobian_christoffels)
+ gs.einsum("...ijlm->...mjli", prod_christoffels)
- gs.einsum("...ijlm->...lmji", prod_christoffels)
)
return riemann_curvature
[docs]
def curvature(self, tangent_vec_a, tangent_vec_b, tangent_vec_c, base_point=None):
r"""Compute the Riemann curvature map R.
For three tangent vectors at base point :math:`P`:
- :math:`X|_P = tangent\_vec\_a`,
- :math:`Y|_P = tangent\_vec\_b`,
- :math:`Z|_P = tangent\_vec\_c`,
the curvature :math:`(X, Y, Z, P)` is defined by
:math:`R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]}Z`.
The output is the tangent vector:
:math:`dx^l(R(X, Y)Z) = R_{ijk}^l X_j Y_k Z_i`
written with Einstein notation.
Parameters
----------
tangent_vec_a : array-like, shape=[..., dim]
Tangent vector at `base_point`.
tangent_vec_b : array-like, shape=[..., dim]
Tangent vector at `base_point`.
tangent_vec_c : array-like, shape=[..., dim]
Tangent vector at `base_point`.
base_point : array-like, shape=[..., dim]
Point on the manifold.
Returns
-------
curvature : array-like, shape=[..., dim]
curvature(X, Y, Z, P)[..., l] = dx^l(R(X, Y)Z)
Tangent vector at `base_point`.
"""
riemann = self.riemann_tensor(base_point)
curvature = gs.einsum(
"...ijkl, ...j, ...k, ...i -> ...l",
riemann,
tangent_vec_a,
tangent_vec_b,
tangent_vec_c,
)
return curvature
[docs]
def ricci_tensor(self, base_point=None):
r"""Compute Ricci curvature tensor at base_point.
The Ricci curvature tensor :math:`\mathrm{Ric}_{ij}` is defined as:
:math:`\mathrm{Ric}_{ij} = R_{ikj}^k`
with Einstein notation.
Parameters
----------
base_point : array-like, shape=[..., dim]
Point on the manifold.
Returns
-------
ricci_tensor : array-like, shape=[..., dim, dim]
ricci_tensor[...,i,j] = Ric_{ij}
Ricci tensor curvature.
"""
riemann_tensor = self.riemann_tensor(base_point)
ricci_tensor = gs.einsum("...ijkj -> ...ik", riemann_tensor)
return ricci_tensor
[docs]
def directional_curvature(self, tangent_vec_a, tangent_vec_b, base_point=None):
r"""Compute the directional curvature (tidal force operator).
For two tangent vectors at `base_point` :math:`P`:
- :math:`X|_P = tangent\_vec\_a`,
- :math:`Y|_P = tangent\_vec\_b`,
the directional curvature, better known
in relativity as the tidal force operator,
is defined by
:math:`R_Y(X) = R(Y,X)Y`.
Parameters
----------
tangent_vec_a : array-like, shape=[..., dim]
Tangent vector at `base_point`.
tangent_vec_b : array-like, shape=[..., dim]
Tangent vector at `base_point`.
base_point : array-like, shape=[..., dim]
Base-point on the manifold.
Returns
-------
directional_curvature : array-like, shape=[..., dim]
Tangent vector at `base_point`.
"""
return self.curvature(tangent_vec_b, tangent_vec_a, tangent_vec_b, base_point)
[docs]
def curvature_derivative(
self,
tangent_vec_a,
tangent_vec_b,
tangent_vec_c,
tangent_vec_d,
base_point=None,
):
r"""Compute the covariant derivative of the curvature.
For four tangent vectors at `base_point` :math:`P`:
- :math:`H|_P = tangent\_vec\_a`,
- :math:`X|_P = tangent\_vec\_b`,
- :math:`Y|_P = tangent\_vec\_c`,
- :math:`Z|_P = tangent\_vec\_d`,
the covariant derivative of the curvature is defined as:
:math:`(\nabla_H R)(X, Y) Z |_P`.
Parameters
----------
tangent_vec_a : array-like, shape=[..., dim]
Tangent vector at `base_point`.
tangent_vec_b : array-like, shape=[..., dim]
Tangent vector at `base_point`.
tangent_vec_c : array-like, shape=[..., dim]
Tangent vector at `base_point`.
tangent_vec_d : array-like, shape=[..., dim]
Tangent vector at `base_point`.
base_point : array-like, shape=[..., dim]
Point on the manifold.
Returns
-------
curvature_derivative : array-like, shape=[..., dim]
Tangent vector at base-point.
"""
raise NotImplementedError(
"The covariant derivative of the curvature is not implemented."
)
[docs]
def directional_curvature_derivative(
self, tangent_vec_a, tangent_vec_b, base_point=None
):
r"""Compute the covariant derivative of the directional curvature.
For tangent vector fields at `base_point` :math:`P`:
- :math:`X|_P = tangent\_vec\_a`,
- :math:`Y|_P = tangent\_vec\_b`,
the covariant derivative (in the direction `X`)
:math:`(\nabla_X R_Y)(X) |_P = (\nabla_X R)(Y, X) Y |_P` of the
directional curvature (in the direction `Y`)
:math:`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.
References
----------
.. [Pennec] Pennec, Xavier. Curvature effects on the empirical mean in
Riemannian and affine Manifolds: a non-asymptotic high
concentration expansion in the small-sample regime. Preprint. 2019.
https://arxiv.org/abs/1906.07418
"""
return self.curvature_derivative(
tangent_vec_a, tangent_vec_b, tangent_vec_a, tangent_vec_b, base_point
)
def _geodesic_from_exp(self, initial_point, initial_tangent_vec):
"""Generate parameterized function for the geodesic curve.
Parameters
----------
initial_point : array-like, shape=[..., dim]
Point on the manifold, initial point of the geodesic.
initial_tangent_vec : array-like, shape=[..., dim],
Tangent vector at base point, the initial speed of the geodesics.
Optional, default: None.
If None, an end point must be given and a logarithm is computed.
Returns
-------
path : callable
Time parameterized geodesic curve. If a batch of initial
conditions is passed, the output array's first dimension
represents the different initial conditions, and the second
corresponds to time.
"""
ndim = self._space.point_ndim
ijk = "ijk"[: self._space.point_ndim]
multiple_tangent = initial_tangent_vec.ndim > ndim
multiple_point = initial_point.ndim > ndim
is_batch = multiple_tangent or multiple_point
def path(t):
"""Generate parameterized function for geodesic curve.
Parameters
----------
t : array-like, shape=[n_points,]
Times at which to compute points of the geodesics.
"""
t = gs.to_ndarray(t, to_ndim=1, dtype=initial_tangent_vec.dtype)
tangent_vecs = gs.einsum(f"n,...{ijk}->...n{ijk}", t, initial_tangent_vec)
if is_batch:
if multiple_point and multiple_tangent:
points_at_time_t = [
self.exp(tv, pt) for tv, pt in zip(tangent_vecs, initial_point)
]
elif multiple_point:
points_at_time_t = [
self.exp(tangent_vecs, pt) for pt in initial_point
]
elif multiple_tangent:
points_at_time_t = [
self.exp(tv, initial_point) for tv in tangent_vecs
]
return gs.stack(points_at_time_t, axis=0)
return self.exp(tangent_vecs, initial_point)
return path
def _geodesic_ivp(self, initial_point, initial_tangent_vec):
"""Solve geodesic initial value problem.
Compute the parameterized function for the geodesic starting at
initial_point with initial velocity given by initial_tangent_vec.
Parameters
----------
initial_point : array-like, shape=[..., dim]
Initial point.
initial_tangent_vec : array-like, shape=[..., dim]
Tangent vector at initial point.
Returns
-------
path : function
Parameterized function for the geodesic curve starting at
initial_point with velocity initial_tangent_vec.
"""
if _check_exp_solver(self, raise_=False) and self.exp_solver.solves_ivp:
return self.exp_solver.geodesic_ivp(initial_tangent_vec, initial_point)
return self._geodesic_from_exp(initial_point, initial_tangent_vec)
def _geodesic_bvp(self, initial_point, end_point):
"""Solve geodesic boundary problem.
Compute the parameterized function for the geodesic starting at
initial_point and ending at end_point.
Parameters
----------
initial_point : array-like, shape=[..., dim]
Initial point.
end_point : array-like, shape=[..., dim]
End point.
Returns
-------
path : function
Parameterized function for the geodesic curve starting at
initial_point and ending at end_point.
"""
if _check_log_solver(self, raise_=False) and self.log_solver.solves_bvp:
return self.log_solver.geodesic_bvp(end_point, initial_point)
return NotImplemented
[docs]
def geodesic(self, initial_point, end_point=None, initial_tangent_vec=None):
"""Generate parameterized function for the geodesic curve.
Geodesic curve defined by either:
- an initial point and an initial tangent vector,
- an initial point and an end point.
Parameters
----------
initial_point : array-like, shape=[..., dim]
Point on the manifold, initial point of the geodesic.
end_point : array-like, shape=[..., dim], optional
Point on the manifold, end point of the geodesic. If None,
an initial tangent vector must be given.
initial_tangent_vec : array-like, shape=[..., dim],
Tangent vector at base point, the initial speed of the geodesics.
Optional, default: None.
If None, an end point must be given and a logarithm is computed.
Returns
-------
path : callable
Time parameterized geodesic curve. If a batch of initial
conditions is passed, the output array's first dimension
represents the different initial conditions, and the second
corresponds to time.
"""
if end_point is None and initial_tangent_vec is None:
raise ValueError(
"Specify an end point or an initial tangent "
"vector to define the geodesic."
)
if end_point is not None:
if initial_tangent_vec is not None:
raise ValueError(
"Cannot specify both an end point and an initial tangent vector."
)
out = self._geodesic_bvp(initial_point, end_point)
if out is not NotImplemented:
return out
initial_tangent_vec = self.log(end_point, initial_point)
return self._geodesic_ivp(initial_point, initial_tangent_vec)
[docs]
def parallel_transport(
self, tangent_vec, base_point, direction=None, end_point=None
):
r"""Compute the parallel transport of a tangent vector.
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 :math:`t \mapsto exp_{(base\_point)}(
t*direction)`.
Parameters
----------
tangent_vec : array-like, shape=[..., {dim, [n, m]}]
Tangent vector at base point to be transported.
base_point : array-like, shape=[..., {dim, [n, m]}]
Point on the manifold. Point to transport from.
direction : array-like, shape=[..., {dim, [n, m]}]
Tangent vector at base point, along which the parallel transport
is computed.
Optional, default: None.
end_point : array-like, shape=[..., {dim, [n, m]}]
Point on the manifold. Point to transport to.
Optional, default: None.
Returns
-------
transported_tangent_vec: array-like, shape=[..., {dim, [n, m]}]
Transported tangent vector at `exp_(base_point)(tangent_vec_b)`.
"""
raise NotImplementedError(
"The closed-form solution of parallel transport is not known, "
"use the ladder_parallel_transport instead."
)
[docs]
def injectivity_radius(self, base_point=None):
"""Compute the radius of the injectivity domain.
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.
Parameters
----------
base_point : array-like, shape=[..., {dim, [n, m]}]
Point on the manifold.
Returns
-------
radius : array-like, shape=[...,]
Injectivity radius.
"""
raise NotImplementedError("The injectivity range is not implemented yet.")