"""Minkowski space."""
import geomstats.backend as gs
from geomstats.geometry.manifold import Manifold
from geomstats.geometry.riemannian_metric import RiemannianMetric
[docs]class Minkowski(Manifold):
"""Class for Minkowski space.
Parameters
----------
dim : int
Dimension of Minkowski space.
"""
def __init__(self, dim):
super(Minkowski, self).__init__(dim=dim)
self.metric = MinkowskiMetric(dim)
[docs] def belongs(self, point):
"""Evaluate if a point belongs to the Minkowski space.
Parameters
----------
point : array-like, shape=[..., dim]
Point to evaluate.
Returns
-------
belongs : array-like, shape=[...,]
Boolean evaluating if point belongs to the Minkowski space.
"""
point_dim = point.shape[-1]
belongs = point_dim == self.dim
if gs.ndim(point) == 2:
belongs = gs.tile([belongs], (point.shape[0],))
return belongs
[docs] def random_point(self, n_samples=1, bound=1.):
"""Sample in the Minkowski space from the uniform distribution.
Parameters
----------
n_samples: int
Number of samples.
Optional, default: 1
bound : float
Side of hypercube support of the uniform distribution.
Optional, default: 1
Returns
-------
points : array-like, shape=[..., dim]
Sample.
"""
size = (self.dim,)
if n_samples != 1:
size = (n_samples, self.dim)
point = bound * gs.random.rand(*size) * 2 - 1
return point
[docs]class MinkowskiMetric(RiemannianMetric):
"""Class for the pseudo-Riemannian Minkowski metric.
The metric is flat: the inner product is independent of the base point.
Parameters
----------
dim : int
Dimension of the Minkowski space.
"""
def __init__(self, dim):
super(MinkowskiMetric, self).__init__(
dim=dim,
signature=(dim - 1, 1, 0))
[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.
Returns
-------
inner_prod_mat : array-like, shape=[..., dim, dim]
Inner-product matrix.
"""
inner_prod_mat = gs.eye(self.dim - 1, self.dim - 1)
first_row = gs.array([0.] * (self.dim - 1))
first_row = gs.to_ndarray(first_row, to_ndim=2, axis=1)
inner_prod_mat = gs.vstack(
[gs.transpose(first_row), inner_prod_mat])
first_column = gs.array([-1.] + [0.] * (self.dim - 1))
first_column = gs.to_ndarray(first_column, to_ndim=2, axis=1)
inner_prod_mat = gs.hstack([first_column, inner_prod_mat])
return inner_prod_mat
[docs] def exp(self, tangent_vec, base_point):
"""Compute the Riemannian exponential of `tangent_vec` at `base_point`.
The Riemannian exponential is the addition in the Minkowski 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.
"""
exp = base_point + tangent_vec
return exp
[docs] def log(self, point, base_point):
"""Compute the Riemannian logarithm of `point` at `base_point`.
The Riemannian logarithm is the subtraction in the Minkowski space.
Parameters
----------
point : array-like, shape=[..., dim]
Point.
base_point : array-like, shape=[..., dim]
Base point.
Returns
-------
log : array-like, shape=[..., dim]
Riemannian logarithm.
"""
log = point - base_point
return log