r"""
Manifold of linear subspaces.
The Grassmannian :math:`Gr(n, k)` is the manifold of k-dimensional
subspaces in n-dimensional Euclidean space.
Lead author: Olivier Peltre.
:math:`Gr(n, p)` is represented by
:math:`n \times n` matrices
of rank :math:`p` satisfying :math:`P^2 = P` and :math:`P^T = P`.
Each :math:`P \in Gr(n, p)` is identified with the unique
orthogonal projector onto :math:`{\rm Im}(P)`.
:math:`Gr(n, p)` is a homogoneous space, quotient of the special orthogonal
group by the subgroup of rotations stabilising a :math:`p`-dimensional subspace:
.. math::
Gr(n, p) \simeq \frac {SO(n)} {SO(p) \times SO(n-p)}
It is therefore customary to represent the Grassmannian
by equivalence classes of orthogonal :math:`p`-frames in :math:`{\mathbb R}^n`.
For such a representation, work in the Stiefel manifold instead.
.. math::
Gr(n, p) \simeq St(n, p) / SO(p)
References
----------
.. [Batzies15] Batzies, E., K. Hüper, L. Machado, and F. Silva Leite.
“Geometric Mean and Geodesic Regression on Grassmannians.”
Linear Algebra and Its Applications 466 (February 1, 2015):
83–101. https://doi.org/10.1016/j.laa.2014.10.003.
"""
import geomstats.backend as gs
import geomstats.errors
from geomstats.geometry.base import LevelSet
from geomstats.geometry.general_linear import GeneralLinear
from geomstats.geometry.matrices import Matrices, MatricesMetric
from geomstats.geometry.symmetric_matrices import SymmetricMatrices
from geomstats.vectorization import repeat_out
def _squared_dist_grad_point_a(point_a, point_b, metric):
"""Compute gradient of squared_dist wrt point_a.
Compute the Riemannian gradient of the squared geodesic
distance with respect to the first point point_a.
Parameters
----------
point_a : array-like, shape=[..., dim]
Point.
point_b : array-like, shape=[..., dim]
Point.
metric : SpecialEuclideanMatrixCanonicalLeftMetric
Metric defining the distance.
Returns
-------
_ : array-like, shape=[..., dim]
Riemannian gradient, in the form of a tangent
vector at base point : point_a.
"""
return -2 * metric.log(point_b, point_a)
def _squared_dist_grad_point_b(point_a, point_b, metric):
"""Compute gradient of squared_dist wrt point_b.
Compute the Riemannian gradient of the squared geodesic
distance with respect to the second point point_b.
Parameters
----------
point_a : array-like, shape=[..., dim]
Point.
point_b : array-like, shape=[..., dim]
Point.
metric : SpecialEuclideanMatrixCanonicalLeftMetric
Metric defining the distance.
Returns
-------
_ : array-like, shape=[..., dim]
Riemannian gradient, in the form of a tangent
vector at base point : point_b.
"""
return -2 * metric.log(point_a, point_b)
@gs.autodiff.custom_gradient(_squared_dist_grad_point_a, _squared_dist_grad_point_b)
def _squared_dist(point_a, point_b, metric):
"""Compute geodesic distance between two points.
Compute the squared geodesic distance between point_a
and point_b, as defined by the metric.
This is an auxiliary private function that:
- is called by the method `squared_dist` of the class
SpecialEuclideanMatrixCanonicalLeftMetric,
Parameters
----------
point_a : array-like, shape=[..., dim]
Point.
point_b : array-like, shape=[..., dim]
Point.
metric : SpecialEuclideanMatrixCanonicalLeftMetric
Metric defining the distance.
Returns
-------
_ : array-like, shape=[...,]
Geodesic distance between point_a and point_b.
"""
return metric._squared_dist(point_a, point_b)
[docs]
class Grassmannian(LevelSet):
"""Class for Grassmann manifolds :math:`Gr(n, p)`.
Parameters
----------
n : int
Dimension of the Euclidean space.
p : int
Dimension of the subspaces.
"""
def __init__(self, n, p, equip=True):
geomstats.errors.check_integer(p, "p")
geomstats.errors.check_integer(n, "n")
if p > n:
raise ValueError(
"p < n is required: p-dimensional subspaces in n dimensions."
)
self.n = n
self.p = p
dim = int(p * (n - p))
super().__init__(dim=dim, equip=equip)
def _define_embedding_space(self):
return SymmetricMatrices(self.n)
[docs]
@staticmethod
def default_metric():
"""Metric to equip the space with if equip is True."""
return GrassmannianCanonicalMetric
[docs]
def submersion(self, point):
r"""Submersion that defines the Grassmann manifold.
The Grassmann manifold is defined here as embedded in the set of
symmetric matrices, as the pre-image of the function defined around the
projector on the space spanned by the first :math:`p` columns of the identity
matrix by (see Exercise E.25 in [Pau07]_).
.. math::
\begin{pmatrix} I_p + A & B^T \\ B & D \end{pmatrix} \mapsto
(D - B(I_p + A)^{-1}B^T, A + A^2 + B^TB)
This map is a submersion and its zero space is the set of orthogonal
rank-:math:`p` projectors.
Parameters
----------
point : array-like, shape=[..., n, n]
Returns
-------
submersed_point : array-like, shape=[..., n, n]
References
----------
.. [Pau07] Paulin, Frédéric. “Géométrie différentielle élémentaire,” 2007.
https://www.imo.universite-paris-saclay.fr/~paulin/notescours/
cours_geodiff.pdf.
"""
p = self.p
_, eigvecs = gs.linalg.eigh(point)
eigvecs = gs.flip(eigvecs, -1)
flipped_point = Matrices.mul(Matrices.transpose(eigvecs), point, eigvecs)
b = flipped_point[..., p:, :p]
d = flipped_point[..., p:, p:]
a = flipped_point[..., :p, :p] - gs.eye(p)
first = d - Matrices.mul(
b, GeneralLinear.inverse(a + gs.eye(p)), Matrices.transpose(b)
)
second = a + Matrices.mul(a, a) + Matrices.mul(Matrices.transpose(b), b)
row_1 = gs.concatenate([first, gs.zeros_like(b)], axis=-1)
row_2 = gs.concatenate([Matrices.transpose(gs.zeros_like(b)), second], axis=-1)
return gs.concatenate([row_1, row_2], axis=-2)
[docs]
def tangent_submersion(self, vector, point):
"""Tangent submersion.
Parameters
----------
vector : array-like, shape=[..., n, n]
point : array-like, shape=[..., n, n]
Returns
-------
submersed_vector : array-like, shape=[..., n, n]
"""
return 2 * Matrices.to_symmetric(Matrices.mul(point, vector)) - vector
[docs]
def random_point(self, n_samples=1, bound=1.0):
r"""Sample random points from a uniform distribution.
Following [Chikuse03]_, :math:`n\_samples * n * p` scalars are sampled
from a standard normal distribution and reshaped to matrices,
the projectors on their first :math:`p` columns follow a uniform
distribution.
Parameters
----------
n_samples : int
The number of points to sample
Optional. default: 1.
Returns
-------
projectors : array-like, shape=[..., n, n]
Points following a uniform distribution.
References
----------
.. [Chikuse03] Yasuko Chikuse, Statistics on special manifolds,
New York: Springer-Verlag. 2003, 10.1007/978-0-387-21540-2
"""
return self.random_uniform(n_samples)
[docs]
def to_tangent(self, vector, base_point):
"""Project a vector to a tangent space of the manifold.
Compute the bracket (commutator) of the base_point with
the skew-symmetric part of vector.
Parameters
----------
vector : array-like, shape=[..., n, n]
Vector.
base_point : array-like, shape=[..., n, n]
Point on the manifold.
Returns
-------
tangent_vec : array-like, shape=[..., n, n]
Tangent vector at base point.
"""
sym = Matrices.to_symmetric(vector)
return Matrices.bracket(base_point, Matrices.bracket(base_point, sym))
[docs]
def projection(self, point):
"""Project a matrix to the Grassmann manifold.
An eigenvalue decomposition of (the symmetric part of) point is used.
Parameters
----------
point : array-like, shape=[..., n, n]
Point in embedding manifold.
Returns
-------
projected : array-like, shape=[..., n, n]
Projected point.
"""
mat = Matrices.to_symmetric(point)
_, eigvecs = gs.linalg.eigh(mat)
diagonal = gs.array([0.0] * (self.n - self.p) + [1.0] * self.p)
p_d = gs.einsum("...ij,...j->...ij", eigvecs, diagonal)
return Matrices.mul(p_d, Matrices.transpose(eigvecs))
[docs]
class GrassmannianCanonicalMetric(MatricesMetric):
"""Canonical metric of the Grassmann manifold.
Coincides with the Frobenius metric.
"""
def __init__(self, space):
super().__init__(space=space, signature=(space.dim, 0, 0))
self._general_linear = GeneralLinear(space.n, equip=False)
[docs]
def exp(self, tangent_vec, base_point, **kwargs):
"""Exponentiate the invariant vector field v from base point p.
Parameters
----------
tangent_vec : array-like, shape=[..., n, n]
Tangent vector at base point.
`tangent_vec` is the bracket of a skew-symmetric matrix with the
base_point.
base_point : array-like, shape=[..., n, n]
Base point.
Returns
-------
exp : array-like, shape=[..., n, n]
Riemannian exponential.
"""
expm = gs.linalg.expm
mul = Matrices.mul
rot = Matrices.bracket(base_point, -tangent_vec)
return mul(expm(rot), base_point, expm(-rot))
[docs]
def log(self, point, base_point, **kwargs):
r"""Compute the Riemannian logarithm of point w.r.t. base_point.
Given :math:`P, P'` in :math:`Gr(n, p)` the logarithm from :math:`P`
to :math:`P` is induced by the infinitesimal rotation [Batzies2015]_:
.. math::
Y = \frac 1 2 \log \big((2 P' - 1)(2 P - 1)\big)
The tangent vector :math:`X` at :math:`P`
is then recovered by :math:`X = [Y, P]`.
Parameters
----------
point : array-like, shape=[..., n, n]
Point.
base_point : array-like, shape=[..., n, n]
Base point.
Returns
-------
tangent_vec : array-like, shape=[..., n, n]
Riemannian logarithm, a tangent vector at `base_point`.
References
----------
.. [Batzies2015] Batzies, Hüper, Machado, Leite.
"Geometric Mean and Geodesic Regression on Grassmannians"
Linear Algebra and its Applications, 466, 83-101, 2015.
"""
id_n = self._general_linear.identity
id_n, point, base_point = gs.convert_to_wider_dtype([id_n, point, base_point])
sym2 = 2 * point - id_n
sym1 = 2 * base_point - id_n
rot = self._general_linear.compose(sym2, sym1)
return Matrices.bracket(self._general_linear.log(rot) / 2, base_point)
[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=[..., n, n]
Tangent vector at base point to be transported.
base_point : array-like, shape=[..., n, n]
Point on the Grassmann manifold. Point to transport from.
direction : array-like, shape=[..., n, n]
Tangent vector at base point, along which the parallel transport
is computed.
Optional, default: None
end_point : array-like, shape=[..., n, n]
Point on the Grassmann manifold to transport to. Unused if
`direction` is given.
Optional, default: None
Returns
-------
transported_tangent_vec: array-like, shape=[..., n, n]
Transported tangent vector at `exp_(base_point)(direction)`.
References
----------
.. [BZA20] Bendokat, Thomas, Ralf Zimmermann, and P.-A. Absil.
“A Grassmann Manifold Handbook: Basic Geometry and Computational
Aspects.” ArXiv:2011.13699 [Cs, Math], November 27, 2020.
https://arxiv.org/abs/2011.13699.
"""
if direction is None:
if end_point is not None:
direction = self.log(end_point, base_point)
else:
raise ValueError(
"Either an end_point or a direction must be given to define the"
" geodesic along which to transport."
)
expm = gs.linalg.expm
mul = Matrices.mul
rot = -Matrices.bracket(base_point, direction)
return mul(expm(rot), tangent_vec, expm(-rot))
def _squared_dist(self, point_a, point_b):
"""Compute geodesic distance between two points.
Compute the squared geodesic distance between point_a
and point_b, as defined by the metric.
This is an auxiliary private function that:
- is called by the method `squared_dist` of the class
GrassmannianCanonicalMetric,
Parameters
----------
point_a : array-like, shape=[..., dim]
Point.
point_b : array-like, shape=[..., dim]
Point.
Returns
-------
_ : array-like, shape=[...,]
Geodesic distance between point_a and point_b.
"""
return super().squared_dist(point_a, point_b)
[docs]
def squared_dist(self, point_a, point_b, **kwargs):
"""Squared geodesic distance between two points.
Parameters
----------
point_a : array-like, shape=[..., dim]
Point.
point_b : array-like, shape=[..., dim]
Point.
Returns
-------
sq_dist : array-like, shape=[...,]
Squared distance.
"""
return _squared_dist(point_a, point_b, metric=self)
[docs]
def injectivity_radius(self, base_point):
"""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.
In this case it is Pi / 2 everywhere.
Parameters
----------
base_point : array-like, shape=[..., n, n]
Point on the manifold.
Returns
-------
radius : array-like, shape=[...,]
Injectivity radius.
References
----------
.. [BZA20] Bendokat, Thomas, Ralf Zimmermann, and P.-A. Absil.
“A Grassmann Manifold Handbook: Basic Geometry and
Computational Aspects.”
ArXiv:2011.13699 [Cs, Math], November 27, 2020.
https://arxiv.org/abs/2011.13699.
"""
radius = gs.array(gs.pi / 2)
return repeat_out(self._space.point_ndim, radius, base_point)