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.

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
from geomstats.geometry.riemannian_metric import RiemannianMetric
from geomstats.geometry.symmetric_matrices import SymmetricMatrices
from geomstats.vectorization import repeat_out


[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_uniform(self, n_samples=1): 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 """ points = gs.random.normal(size=(n_samples, self.n, self.p)) full_rank = Matrices.mul(Matrices.transpose(points), points) projector = Matrices.mul( points, GeneralLinear.inverse(full_rank), Matrices.transpose(points) ) return projector[0] if n_samples == 1 else projector
[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(RiemannianMetric): """Canonical metric of the Grassmann manifold.""" 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): """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): 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))
[docs] def squared_dist(self, point_a, point_b): """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. """ sdist_func = super().squared_dist def _squared_dist_grad_point_a(point_a, point_b): """Compute gradient of squared_dist wrt point_a.""" return -2 * self.log(point_b, point_a) def _squared_dist_grad_point_b(point_a, point_b): """Compute gradient of squared_dist wrt point_b.""" return -2 * self.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): """Compute geodesic distance between two points. 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 sdist_func(point_a, point_b) return _squared_dist(point_a, point_b)
[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. 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)
[docs] def inner_product(self, tangent_vec_a, tangent_vec_b, base_point=None): """Compute Frobenius inner-product of two tangent vectors. Coincides with the Frobenius metric. Parameters ---------- tangent_vec_a : array-like, shape=[..., m, n] Tangent vector. tangent_vec_b : array-like, shape=[..., m, n] Tangent vector. base_point : array-like, shape=[..., m, n] Base point. Optional, default: None. Returns ------- inner_prod : array-like, shape=[...,] Frobenius inner-product of tangent_vec_a and tangent_vec_b. """ inner_prod = Matrices.frobenius_product(tangent_vec_a, tangent_vec_b) return repeat_out( self._space.point_ndim, inner_prod, tangent_vec_a, tangent_vec_b, base_point )
[docs] def squared_norm(self, vector, base_point=None): """Compute the square of the norm of a vector. Squared norm of a vector associated to the inner product at the tangent space at a base point. Parameters ---------- vector : array-like, shape=[..., dim] Vector. base_point : array-like, shape=[..., dim] Base point. Optional, default: None. Returns ------- sq_norm : array-like, shape=[...,] Squared norm. """ sq_norm = gs.linalg.norm(vector, axis=(-2, -1)) ** 2 return repeat_out(self._space.point_ndim, sq_norm, vector, base_point)
[docs] def norm(self, vector, base_point=None): """Compute norm of a matrix. Norm of a matrix associated to the Frobenius inner product. Parameters ---------- vector : array-like, shape=[..., dim] Vector. base_point : array-like, shape=[..., dim] Base point. Optional, default: None. Returns ------- norm : array-like, shape=[...,] Norm. """ norm = gs.linalg.norm(vector, axis=(-2, -1)) return repeat_out(self._space.point_ndim, norm, vector, base_point)