# Source code for geomstats.geometry.grassmannian

r"""
Manifold of linear subspaces.

The Grassmannian :math:Gr(n, k) is the manifold of k-dimensional
subspaces in n-dimensional Euclidean space.

:math:Gr(n, k) is represented by
:math:n \times n matrices
of rank :math:k  satisfying :math:P^2 = P and :math:P^T = P.
Each :math:P \in Gr(n, k) is identified with the unique
orthogonal projector onto :math:{\rm Im}(P).

:math:Gr(n, k) is a homogoneous space, quotient of the special orthogonal
group by the subgroup of rotations stabilising a k-dimensional subspace:

.. math::

Gr(n, k) \simeq \frac {SO(n)} {SO(k) \times SO(n-k)}

It is therefore customary to represent the Grassmannian
by equivalence classes of orthogonal :math:k-frames in :math:{\mathbb R}^n.
For such a representation, work in the Stiefel manifold instead.

.. math::

Gr(n, k) \simeq St(n, k) / SO(k)

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.euclidean import EuclideanMetric
from geomstats.geometry.general_linear import GeneralLinear
from geomstats.geometry.matrices import Matrices, MatricesMetric
from geomstats.geometry.riemannian_metric import RiemannianMetric
from geomstats.geometry.symmetric_matrices import SymmetricMatrices

[docs]def submersion(point, k):
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 k columns of the identity
matrix by (see Exercise E.25 in [Pau07]_).
.. math:

\begin{pmatrix} I_k + A & B^T \\ B & D \end{pmatrix} \mapsto
(D - B(I_k + A)^{-1}B^T, A + A^2 + B^TB

This map is a submersion and its zero space is the set of orthogonal
rank-k projectors.

References
----------
.. [Pau07]   Paulin, Frédéric. “Géométrie diﬀérentielle élémentaire,” 2007.
https://www.imo.universite-paris-saclay.fr/~paulin
/notescours/cours_geodiff.pdf.
"""
_, eigvecs = gs.linalg.eigh(point)
eigvecs = gs.flip(eigvecs, -1)
flipped_point = Matrices.mul(Matrices.transpose(eigvecs), point, eigvecs)
b = flipped_point[..., k:, :k]
d = flipped_point[..., k:, k:]
a = flipped_point[..., :k, :k] - gs.eye(k)
first = d - Matrices.mul(
b, GeneralLinear.inverse(a + gs.eye(k)), 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)

"""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 : SpecialEuclideanMatrixCannonicalLeftMetric
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)

"""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 : SpecialEuclideanMatrixCannonicalLeftMetric
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)

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
SpecialEuclideanMatrixCannonicalLeftMetric,
- has been created to support the implementation

Parameters
----------
point_a : array-like, shape=[..., dim]
Point.
point_b : array-like, shape=[..., dim]
Point.
metric : SpecialEuclideanMatrixCannonicalLeftMetric
Metric defining the distance.

Returns
-------
_ : array-like, shape=[...,]
Geodesic distance between point_a and point_b.
"""
return metric.private_squared_dist(point_a, point_b)

[docs]class Grassmannian(LevelSet):
"""Class for Grassmann manifolds Gr(n, k).

Parameters
----------
n : int
Dimension of the Euclidean space.
k : int
Dimension of the subspaces.
"""

def __init__(self, n, k, **kwargs):
geomstats.errors.check_integer(k, "k")
geomstats.errors.check_integer(n, "n")
if k > n:
raise ValueError(
"k < n is required: k-dimensional subspaces in n dimensions."
)

self.n = n
self.k = k

kwargs.setdefault("metric", GrassmannianCanonicalMetric(n, k))
dim = int(k * (n - k))
super(Grassmannian, self).__init__(
dim=dim,
embedding_space=SymmetricMatrices(n),
submersion=lambda x: submersion(x, k),
value=gs.zeros((n, n)),
tangent_submersion=lambda v, x: 2
* Matrices.to_symmetric(Matrices.mul(x, v))
- v,
**kwargs
)

[docs]    def random_uniform(self, n_samples=1):
"""Sample random points from a uniform distribution.

Following [Chikuse03]_, :math: n_samples * n * k scalars are sampled
from a standard normal distribution and reshaped to matrices,
the projectors on their first k 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
"""
points = gs.random.normal(size=(n_samples, self.n, self.k))
full_rank = Matrices.mul(Matrices.transpose(points), points)
projector = Matrices.mul(
points, GeneralLinear.inverse(full_rank), Matrices.transpose(points)
)
return projector if n_samples == 1 else projector

[docs]    def random_point(self, n_samples=1, bound=1.0):
"""Sample random points from a uniform distribution.

Following [Chikuse03]_, :math: n_samples * n * k scalars are sampled
from a standard normal distribution and reshaped to matrices,
the projectors on their first k 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.k) + [1.0] * self.k)
p_d = gs.einsum("...ij,...j->...ij", eigvecs, diagonal)
return Matrices.mul(p_d, Matrices.transpose(eigvecs))

[docs]class GrassmannianCanonicalMetric(MatricesMetric, RiemannianMetric):
"""Canonical metric of the Grassmann manifold.

Coincides with the Frobenius metric.

Parameters
----------
n : int
Dimension of the Euclidean space.
k : int
Dimension of the subspaces.
"""

def __init__(self, n, p):
geomstats.errors.check_integer(p, "p")
geomstats.errors.check_integer(n, "n")
if p > n:
raise ValueError("p <= n is required.")

dim = int(p * (n - p))
super(GrassmannianCanonicalMetric, self).__init__(
m=n, n=n, dim=dim, signature=(dim, 0, 0)
)

self.n = n
self.p = p
self.embedding_metric = EuclideanMetric(n * p)

[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 Gr(n, k) 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.
"""
GLn = GeneralLinear(self.n)
id_n = GLn.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 = GLn.compose(sym2, sym1)
return Matrices.bracket(GLn.log(rot) / 2, base_point)

[docs]    def parallel_transport(
self, tangent_vec, base_point, tangent_vec_b=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.
tangent_vec_b : 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 tangent_vec_b
is given.
Optional, default: None

Returns
-------
transported_tangent_vec: array-like, shape=[..., n, n]
Transported tangent vector at exp_(base_point)(tangent_vec_b).

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 tangent_vec_b is None:
if end_point is not None:
tangent_vec_b = self.log(end_point, base_point)
else:
raise ValueError(
"Either an end_point or a tangent_vec_b must be given to define the"
" geodesic along which to transport."
)
expm = gs.linalg.expm
mul = Matrices.mul
rot = -Matrices.bracket(base_point, tangent_vec_b)
return mul(expm(rot), tangent_vec, expm(-rot))

[docs]    def private_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,
- has been created to support the implementation

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.
"""
dist = super().squared_dist(point_a, point_b)
return dist

[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.
"""
dist = _squared_dist(point_a, point_b, metric=self)
return dist

"""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
-------