Source code for geomstats.geometry.euclidean

"""Euclidean space."""

import geomstats.backend as gs
from geomstats.geometry.base import VectorSpace
from geomstats.geometry.riemannian_metric import RiemannianMetric
from geomstats.vectorization import repeat_out


[docs] class Euclidean(VectorSpace): """Class for Euclidean spaces. By definition, a Euclidean space is a vector space of a given dimension, equipped with a Euclidean metric. Parameters ---------- dim : int Dimension of the Euclidean space. """ def __init__(self, dim, equip=True): super().__init__( dim=dim, shape=(dim,), equip=equip, )
[docs] @staticmethod def default_metric(): """Metric to equip the space with if equip is True.""" return EuclideanMetric
@property def identity(self): """Identity of the group. Returns ------- identity : array-like, shape=[n] """ return gs.zeros(self.dim) def _create_basis(self): """Create the canonical basis.""" return gs.eye(self.dim)
[docs] def exp(self, tangent_vec, base_point): """Compute the group exponential, which is simply the addition. Parameters ---------- tangent_vec : array-like, shape=[..., n] Tangent vector at base point. base_point : array-like, shape=[..., n] Point from which the exponential is computed. Returns ------- point : array-like, shape=[..., n] Group exponential. """ return tangent_vec + base_point
[docs] class EuclideanMetric(RiemannianMetric): """Class for Euclidean metrics. As a Riemannian metric, the Euclidean metric is: - flat: the inner-product is independent of the base point; - positive definite: it has signature (dimension, 0, 0), where dimension is the dimension of the Euclidean space. """
[docs] def metric_matrix(self, base_point=None): """Compute the inner-product matrix, independent of the base point. Parameters ---------- base_point : array-like, shape=[..., dim] Base point. Optional, default: None. Returns ------- inner_prod_mat : array-like, shape=[..., dim, dim] Inner-product matrix. """ dim = self._space.dim mat = gs.eye(dim) return repeat_out(self._space.point_ndim, mat, base_point, out_shape=(dim, dim))
[docs] def inner_product(self, tangent_vec_a, tangent_vec_b, base_point=None): """Inner product between two tangent vectors at a base point. 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. Optional, default: None. Returns ------- inner_product : array-like, shape=[...,] Inner-product. """ inner_product = gs.dot(tangent_vec_a, tangent_vec_b) return repeat_out( self._space.point_ndim, inner_product, tangent_vec_a, tangent_vec_b, base_point, )
[docs] def norm(self, vector, base_point=None): """Compute norm of a vector. Norm of a vector associated to the inner product at the tangent space at a base point. Note: This only works for positive-definite Riemannian metrics and inner products. 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=-1) return repeat_out(self._space.point_ndim, norm, vector, base_point)
[docs] def exp(self, tangent_vec, base_point, **kwargs): """Compute exp map of a base point in tangent vector direction. The Riemannian exponential is vector addition in the Euclidean space. Parameters ---------- tangent_vec : array-like, shape=[..., dim] Tangent vector at base point. base_point : array-like, shape=[..., dim] Base point. Returns ------- exp : array-like, shape=[..., dim] Riemannian exponential. """ return base_point + tangent_vec
[docs] def log(self, point, base_point, **kwargs): """Compute log map using a base point and other point. The Riemannian logarithm is the subtraction in the Euclidean space. Parameters ---------- point: array-like, shape=[..., dim] Point. base_point: array-like, shape=[..., dim] Base point. Returns ------- log: array-like, shape=[..., dim] Riemannian logarithm. """ return point - base_point
[docs] def parallel_transport( self, tangent_vec, base_point=None, direction=None, end_point=None ): r"""Compute the parallel transport of a tangent vector. On a Euclidean space, the parallel transport of a (tangent) vector returns the vector itself. Parameters ---------- tangent_vec : array-like, shape=[..., dim] Tangent vector at base point to be transported. base_point : array-like, shape=[..., dim] Point on the manifold. Point to transport from. Optional, default: None direction : array-like, shape=[..., dim] Tangent vector at base point, along which the parallel transport is computed. Optional, default: None. end_point : array-like, shape=[..., dim] Point on the manifold. Point to transport to. Optional, default: None. Returns ------- transported_tangent_vec: array-like, shape=[..., dim] Transported tangent vector at `exp_(base_point)(tangent_vec_b)`. """ transported_tangent_vec = gs.copy(tangent_vec) return repeat_out( self._space.point_ndim, transported_tangent_vec, tangent_vec, base_point, direction, end_point, out_shape=self._space.shape, )