"""Classes for fiber bundles and quotient metrics.
Lead author: Nicolas Guigui.
"""
import geomstats.backend as gs
from geomstats.geometry.riemannian_metric import RiemannianMetric
[docs]
class QuotientMetric(RiemannianMetric):
"""Quotient metric.
Given a (principal) fiber bundle, or more generally a manifold with a
Lie group acting on it by the right, the quotient space is the space of
orbits under this action. The quotient metric is defined such that the
canonical projection is a Riemannian submersion, i.e. it is isometric to
the restriction of the metric of the total space to horizontal subspaces.
Parameters
----------
space : Manifold
Base.
total_space : Manifold
Total space equipped with a fiber bundle structure.
"""
def __init__(self, space, total_space, signature=None):
self._total_space = total_space
super().__init__(space=space, signature=signature)
@property
def _fiber_bundle(self):
"""Fiber bundle associated to total space."""
return self._total_space.fiber_bundle
[docs]
def inner_product(
self, tangent_vec_a, tangent_vec_b, base_point=None, fiber_point=None
):
"""Compute the inner-product of two tangent vectors at a base point.
Parameters
----------
tangent_vec_a : array-like, shape=[..., {dim, [n, n]}]
Tangent vector to the quotient manifold.
tangent_vec_b : array-like, shape=[..., {dim, [n, n]}]
Tangent vector to the quotient manifold.
base_point : array-like, shape=[..., {dim, [n, n]}]
Point on the quotient manifold.
Optional, default: None.
fiber_point : array-like, shape=[..., {dim, [n, n]}]
Point on the total space, lift of `base_point`, i.e. such that
`submersion` applied to `point` results in `base_point`.
Optional, default: None. In this case, it is computed using the
method lift.
Returns
-------
inner_product : float, shape=[...]
Inner products
"""
if base_point is None and fiber_point is None:
raise ValueError(
"Either a point (of the total space) or a "
"base point (of the base manifold) must be "
"given."
)
if fiber_point is None:
fiber_point = self._fiber_bundle.lift(base_point)
horizontal_a = self._fiber_bundle.horizontal_lift(
tangent_vec_a, base_point=base_point, fiber_point=fiber_point
)
horizontal_b = self._fiber_bundle.horizontal_lift(
tangent_vec_b, base_point=base_point, fiber_point=fiber_point
)
return self._total_space.metric.inner_product(
horizontal_a, horizontal_b, fiber_point
)
[docs]
def exp(self, tangent_vec, base_point):
"""Compute the Riemannian exponential of a tangent vector.
Parameters
----------
tangent_vec : array-like, shape=[..., {dim, [n, n]}]
Tangent vector to the quotient manifold.
base_point : array-like, shape=[..., {dim, [n, n]}]
Point on the quotient manifold.
Optional, default: None.
Returns
-------
exp : array-like, shape=[..., {dim, [n, n]}]
Point on the quotient manifold.
"""
fiber_point = self._fiber_bundle.lift(base_point)
horizontal_vec = self._fiber_bundle.horizontal_lift(
tangent_vec, fiber_point=fiber_point, base_point=base_point
)
return self._fiber_bundle.riemannian_submersion(
self._total_space.metric.exp(horizontal_vec, fiber_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.
"""
fiber_point = self._fiber_bundle.lift(initial_point)
horizontal_vec = self._fiber_bundle.horizontal_lift(
initial_tangent_vec, fiber_point=fiber_point, base_point=initial_point
)
total_path = self._total_space.metric._geodesic_from_exp(
fiber_point, horizontal_vec
)
def path(t):
total_geod_points = total_path(t)
out = self._fiber_bundle.riemannian_submersion(total_geod_points)
return out
return path
[docs]
def log(self, point, base_point):
"""Compute the Riemannian logarithm of a point.
Parameters
----------
point : array-like, shape=[..., {dim, [n, n]}]
Point on the quotient manifold.
base_point : array-like, shape=[..., {dim, [n, n]}]
Point on the quotient manifold.
Returns
-------
log : array-like, shape=[..., {dim, [n, n]}]
Tangent vector at the base point equal to the Riemannian logarithm
of point at the base point.
"""
fiber_point = self._fiber_bundle.lift(point)
fiber_base_point = self._fiber_bundle.lift(base_point)
aligned = self._fiber_bundle.align(fiber_point, fiber_base_point)
return self._fiber_bundle.tangent_riemannian_submersion(
self._total_space.metric.log(aligned, fiber_base_point),
fiber_base_point,
)
[docs]
def squared_dist(self, point_a, point_b):
"""Squared geodesic distance between two points.
Parameters
----------
point_a : array-like, shape=[..., {dim, [n, n]}]
Point.
point_b : array-like, shape=[..., {dim, [n, n]}]
Point.
Returns
-------
sq_dist : array-like, shape=[...,]
Squared distance.
"""
fiber_point_a = self._fiber_bundle.lift(point_a)
fiber_point_b = self._fiber_bundle.lift(point_b)
aligned = self._fiber_bundle.align(fiber_point_b, fiber_point_a)
return self._total_space.metric.squared_dist(fiber_point_a, aligned)
[docs]
def curvature(self, tangent_vec_a, tangent_vec_b, tangent_vec_c, base_point):
r"""Compute the curvature.
For three vectors fields :math:`X|_P = tangent\_vec\_a,
Y|_P = tangent\_vec\_b, Z|_P = tangent\_vec\_c` with tangent vector
specified in argument at the base point :math:`P`,
the curvature is defined by :math:`R(X,Y)Z = \nabla_{[X,Y]}Z
- \nabla_X\nabla_Y Z + \nabla_Y\nabla_X Z`.
In the case of quotient metrics, the fundamental equations of a
Riemannian submersion allow to compute the curvature of the base
manifold from the one of the total space and a correction term that
uses the integrability tensor A [ONeill]_.
In more details, let :math:`X, Y, Z` be the horizontal lift of
vector fields extending the tangent vectors given in argument in a
neighborhood of the base-point P in the base-space. Then the
curvature of the base-space at the base-points is
:math:`R(X,Y) Z = hor( R^T(X,Y) Z) - 2 A_Z A_X Y + A_X A_Y Z - A_Y
A_X Z`, where :math:`R^T(X,Y)Z` is the curvature tensor of the
total space.
Parameters
----------
tangent_vec_a : array-like, shape=[..., {dim, [n, n]}]
Tangent vector at `base_point`.
tangent_vec_b : array-like, shape=[..., {dim, [n, n]}]
Tangent vector at `base_point`.
tangent_vec_c : array-like, shape=[..., {dim, [n, n]}]
Tangent vector at `base_point`.
base_point : array-like, shape=[..., {dim, [n, n]}]
Point on the base manifold.
Returns
-------
curvature : array-like, shape=[..., {dim, [n, n]}]
Tangent vector at `base_point`.
References
----------
.. [ONeill] O’Neill, Barrett. The Fundamental Equations of a
Submersion, Michigan Mathematical Journal 13, no. 4
(December 1966): 459–69. https://doi.org/10.1307/mmj/1028999604.
"""
bundle = self._fiber_bundle
fiber_point = bundle.lift(base_point)
horizontal_a = bundle.horizontal_lift(tangent_vec_a, base_point)
horizontal_b = bundle.horizontal_lift(tangent_vec_b, base_point)
horizontal_c = bundle.horizontal_lift(tangent_vec_c, base_point)
top_curvature = self._total_space.metric.curvature(
horizontal_a, horizontal_b, horizontal_c, fiber_point
)
projected_top_curvature = bundle.tangent_riemannian_submersion(
top_curvature, fiber_point
)
f_ab = bundle.integrability_tensor(horizontal_a, horizontal_b, fiber_point)
f_c_f_ab = bundle.integrability_tensor(horizontal_c, f_ab, fiber_point)
f_c_f_ab = bundle.tangent_riemannian_submersion(f_c_f_ab, fiber_point)
f_ac = bundle.integrability_tensor(horizontal_a, horizontal_c, fiber_point)
f_b_f_ac = bundle.integrability_tensor(horizontal_b, f_ac, fiber_point)
f_b_f_ac = bundle.tangent_riemannian_submersion(f_b_f_ac, fiber_point)
f_bc = bundle.integrability_tensor(horizontal_b, horizontal_c, fiber_point)
f_a_f_bc = bundle.integrability_tensor(horizontal_a, f_bc, fiber_point)
f_a_f_bc = bundle.tangent_riemannian_submersion(f_a_f_bc, fiber_point)
return projected_top_curvature - 2 * f_c_f_ab + f_a_f_bc - f_b_f_ac
[docs]
def curvature_derivative(
self,
tangent_vec_a,
tangent_vec_b,
tangent_vec_c,
tangent_vec_d,
base_point,
):
r"""Compute the covariant derivative of the curvature.
For four vectors fields :math:`H|_P =` `tangent_vec_a`,
:math:`X|_P =` `tangent_vec_b`, :math:`Y|_P =` `tangent_vec_c`,
:math:`Z|_P =` `tangent_vec_d` with
tangent vector value specified in argument at the `base_point` :math:`P`,
the covariant derivative of the curvature
:math:`(\nabla_H R)(X, Y)Z |_P` is computed at the `base_point` :math:`P`.
In the case of quotient metrics, the fundamental equations of a
Riemannian submersion allow to compute this tensor on the base manifold
from the one of the total space T and its covariant derivative with
additional correction terms involving the integrability tensor A and
its covariant derivatives [Pennec]_.
In more details, let :math:`H, X, Y, Z` be the horizontal lift of
parallel vector fields extending the tangent vectors given in argument
by parallel transport in a neighborhood of the `base_point` :math:`P` in the
base-space. Such vector fields verify :math:`\nabla^T_H H=0`,
:math:`\nabla^T_H X = A_H X` (and similarly for Y and Z) using the
connection :math:`\nabla^T` of the total space. Then the covariant
derivative of the curvature tensor is given by
.. math::
\nabla_H (R(X, Y) Z) =
hor\nabla_H^T(R^T(X,Y)Z) - A_H(ver R^T(X,Y)Z )
+ (2 A_H A_Z A_X Y - A_H A_X A_Y Z + A_H A_Y A_X Z)
- (2 \nabla_H^T A_Z A_X Y - \nabla_H^T A_X A_Y Z + \nabla_H^T A_Y A_X Z)
where :math:`R^T(X,Y)Z` is the curvature tensor of the total space.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
Tangent vector at `base_point` (derivative direction).
tangent_vec_b : array-like, shape=[..., n, n]
Tangent vector at `base_point`.
tangent_vec_c : array-like, shape=[..., n, n]
Tangent vector at `base_point`.
tangent_vec_d : array-like, shape=[..., n, n]
Tangent vector at `base_point`.
base_point : array-like, shape=[..., n, n]
Point on the base manifold.
Returns
-------
curvature_derivative : array-like, shape=[..., n, n]
Tangent vector at `base_point`.
References
----------
.. [Pennec] Pennec, Xavier. Computing the curvature and its gradient
in Kendall shape spaces. Unpublished.
"""
bundle = self._fiber_bundle
point_fiber = bundle.lift(base_point)
hor_h = bundle.horizontal_lift(tangent_vec_a, point_fiber)
hor_x = bundle.horizontal_lift(tangent_vec_b, point_fiber)
hor_y = bundle.horizontal_lift(tangent_vec_c, point_fiber)
hor_z = bundle.horizontal_lift(tangent_vec_d, point_fiber)
nabla_h_x = bundle.integrability_tensor(hor_h, hor_x, point_fiber)
nabla_h_y = bundle.integrability_tensor(hor_h, hor_y, point_fiber)
nabla_h_z = bundle.integrability_tensor(hor_h, hor_z, point_fiber)
nabla_curvature_top = self._total_space.metric.curvature_derivative(
hor_h, hor_x, hor_y, hor_z, point_fiber
)
hor_nabla_curvature_top = bundle.horizontal_projection(
nabla_curvature_top, point_fiber
)
ver_nabla_curvature_top = nabla_curvature_top - hor_nabla_curvature_top
a_h_ver_nabla_curvature_top = bundle.integrability_tensor(
hor_h, ver_nabla_curvature_top, point_fiber
)
# A_H A_Z A_X Y and \nabla_H A_Z A_X Y
nabla_h_a_x_y, a_x_y = bundle.integrability_tensor_derivative(
hor_h, hor_x, nabla_h_x, hor_y, nabla_h_y, point_fiber
)
nabla_h_a_z_a_x_y, a_z_a_x_y = bundle.integrability_tensor_derivative(
hor_h, hor_z, nabla_h_z, a_x_y, nabla_h_a_x_y, point_fiber
)
a_h_a_z_a_x_y = bundle.integrability_tensor(hor_h, a_z_a_x_y, point_fiber)
# A_H A_X A_Y Z and \nabla_H A_X A_Y Z
nabla_h_a_y_z, a_y_z = bundle.integrability_tensor_derivative(
hor_h, hor_y, nabla_h_y, hor_z, nabla_h_z, point_fiber
)
nabla_h_a_x_a_y_z, a_x_a_y_z = bundle.integrability_tensor_derivative(
hor_h, hor_x, nabla_h_x, a_y_z, nabla_h_a_y_z, point_fiber
)
a_h_a_x_a_y_z = bundle.integrability_tensor(hor_h, a_x_a_y_z, point_fiber)
# A_H A_Y A_X Z and \nabla_H A_Y A_X Z
nabla_h_a_x_z, a_x_z = bundle.integrability_tensor_derivative(
hor_h, hor_x, nabla_h_x, hor_z, nabla_h_z, point_fiber
)
nabla_h_a_y_a_x_z, a_y_a_x_z = bundle.integrability_tensor_derivative(
hor_h, hor_y, nabla_h_y, a_x_z, nabla_h_a_x_z, point_fiber
)
a_h_a_y_a_x_z = bundle.integrability_tensor(hor_h, a_y_a_x_z, point_fiber)
return (
hor_nabla_curvature_top
- a_h_ver_nabla_curvature_top
- 2.0 * (nabla_h_a_z_a_x_y - a_h_a_z_a_x_y)
+ (nabla_h_a_x_a_y_z - a_h_a_x_a_y_z)
- (nabla_h_a_y_a_x_z - a_h_a_y_a_x_z)
)
[docs]
def directional_curvature_derivative(
self, tangent_vec_a, tangent_vec_b, base_point
):
r"""Compute the covariant derivative of the directional curvature.
For two vectors fields :math:`X|_P =` `tangent_vec_a`,
:math:`Y|_P =` `tangent_vec_b` with tangent vector value specified in
argument at the `base_point` :math:`P`,
the covariant derivative (in the direction :math:`X`)
:math:`(\nabla_X R_Y)(X) |_P = (\nabla_X R)(Y, X) Y |_P` of the
directional curvature (in the direction :math:`Y`)
:math:`R_Y(X) = R(Y, X) Y` is a quadratic tensor in :math:`X` and :math:`Y`
that plays an important role in the computation of the moments of the
empirical Fréchet mean.
This tensor can be computed from the covariant derivative of the
curvature tensor as in done generically the Connection class.
However, in the case of quotient metrics, a simplified expression can
be implemented based on the directional curvature of the total space T
and its covariant derivative with additional correction terms
involving the integrability tensor A and its covariant derivatives
[Pennec]_.
In more details, let :math:`X, Y` be the horizontal lift of parallel
vector fields extending the tangent vectors given in argument by
parallel transport in a neighborhood of the `base_point` :math:`P` in the
base-space. Such vector fields verify :math:`\nabla^T_X X=0` and
:math:`\nabla^T_X^Y = A_X Y` using the connection :math:`\nabla^T`
of the total space. Then the covariant derivative of the
directional curvature tensor is given by
.. math::
\nabla_X (R_Y(X)) = hor \nabla^T_X (R^T_Y(X)) - A_X( ver R^T_Y(X))
+ 3 A_X A_Y A_X Y - 3 \nabla_X^T A_Y A_X Y
where :math:`R^T_Y(X)` is the directional curvature tensor of the total space.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
Tangent vector at `base_point`.
tangent_vec_b : array-like, shape=[..., n, n]
Tangent vector at `base_point`.
base_point : array-like, shape=[..., n, n]
Point on the base manifold.
Returns
-------
curvature_derivative : array-like, shape=[..., n, n]
Tangent vector at `base_point`.
References
----------
.. [Pennec] Pennec, Xavier. Computing the curvature and its gradient
in Kendall shape spaces. Unpublished.
"""
bundle = self._fiber_bundle
point_fiber = bundle.lift(base_point)
hor_x = bundle.horizontal_lift(tangent_vec_a, point_fiber)
hor_y = bundle.horizontal_lift(tangent_vec_b, point_fiber)
nabla_x_x = gs.zeros_like(hor_x)
nabla_x_y = bundle.integrability_tensor(hor_x, hor_y, point_fiber)
nabla_curvature_top = self._total_space.metric.curvature_derivative(
hor_x, hor_x, hor_y, hor_y, point_fiber
)
hor_nabla_curvature_top = bundle.horizontal_projection(
nabla_curvature_top, point_fiber
)
ver_nabla_curvature_top = nabla_curvature_top - hor_nabla_curvature_top
a_x_ver_nabla_curvature_top = bundle.integrability_tensor(
hor_x, ver_nabla_curvature_top, point_fiber
)
# A_X A_Y A_X Y and \nabla_X A_Y A_X Y
nabla_x_a_x_y, a_x_y = bundle.integrability_tensor_derivative(
hor_x, hor_x, nabla_x_x, hor_y, nabla_x_y, point_fiber
)
nabla_x_a_y_a_x_y, a_y_a_x_y = bundle.integrability_tensor_derivative(
hor_x, hor_y, nabla_x_y, a_x_y, nabla_x_a_x_y, point_fiber
)
a_x_a_y_a_x_y = bundle.integrability_tensor(hor_x, a_y_a_x_y, point_fiber)
return (
hor_nabla_curvature_top
- a_x_ver_nabla_curvature_top
+ 3.0 * (nabla_x_a_y_a_x_y - a_x_a_y_a_x_y)
)
[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."
)
initial_fiber_point = self._fiber_bundle.lift(initial_point)
end_fiber_point = self._fiber_bundle.lift(end_point)
aligned_end_fiber_point = self._fiber_bundle.align(
end_fiber_point, initial_fiber_point
)
geodesic = self._total_space.metric.geodesic(
initial_point=initial_fiber_point, end_point=aligned_end_fiber_point
)
def projected_geodesic(t):
return self._fiber_bundle.riemannian_submersion(geodesic(t))
return projected_geodesic
return self._geodesic_ivp(initial_point, initial_tangent_vec)