"""Manifold module.
In other words, a topological space that locally resembles
Euclidean space near each point.
"""
import geomstats.backend as gs
import geomstats.errors
ATOL = gs.atol
[docs]class Manifold:
r"""Class for manifolds.
Parameters
----------
dim : int
Dimension of the manifold.
default_point_type : str, {\'vector\', \'matrix\'}
Point type.
Optional, default: 'vector'.
default_coords_type : str, {\'intrinsic\', \'extrinsic\', etc}
Coordinate type.
Optional, default: 'intrinsic'.
"""
def __init__(
self, dim, default_point_type='vector',
default_coords_type='intrinsic', **kwargs):
super(Manifold, self).__init__(**kwargs)
geomstats.errors.check_integer(dim, 'dim')
geomstats.errors.check_parameter_accepted_values(
default_point_type, 'default_point_type', ['vector', 'matrix'])
self.dim = dim
self.default_point_type = default_point_type
self.default_coords_type = default_coords_type
[docs] def belongs(self, point, atol=ATOL):
"""Evaluate if a point belongs to the manifold.
Parameters
----------
point : array-like, shape=[..., dim]
Point to evaluate.
atol : float
Absolute tolerance.
Optional, default: 1e-6.
Returns
-------
belongs : array-like, shape=[...,]
Boolean evaluating if point belongs to the manifold.
"""
raise NotImplementedError('belongs is not implemented.')
[docs] def is_tangent(self, vector, base_point=None, atol=ATOL):
"""Check whether the vector is tangent at base_point.
Parameters
----------
vector : array-like, shape=[..., dim]
Vector.
base_point : array-like, shape=[..., dim]
Point on the manifold.
Optional, default: none.
atol : float
Absolute tolerance.
Optional, default: 1e-6.
Returns
-------
is_tangent : bool
Boolean denoting if vector is a tangent vector at the base point.
"""
raise NotImplementedError(
'is_tangent is not implemented.')
[docs] def to_tangent(self, vector, base_point=None):
"""Project a vector to a tangent space of the manifold.
Parameters
----------
vector : array-like, shape=[..., dim]
Vector.
base_point : array-like, shape=[..., dim]
Point on the manifold.
Returns
-------
tangent_vec : array-like, shape=[..., dim]
Tangent vector at base point.
"""
raise NotImplementedError(
'to_tangent is not implemented.')
[docs] def random_point(self, n_samples=1, bound=1.):
"""Sample random points on the manifold.
If the manifold is compact, a uniform distribution is used.
Parameters
----------
n_samples : int
Number of samples.
Optional, default: 1.
bound : float
Bound of the interval in which to sample for non compact manifolds.
Optional, default: 1.
Returns
-------
samples : array-like, shape=[..., {dim, [n, n]}]
Points sampled on the hypersphere.
"""
raise NotImplementedError('random_point is not implemented')
[docs] def regularize(self, point):
"""Regularize a point to the canonical representation for the manifold.
Parameters
----------
point : array-like, shape=[..., dim]
Point.
Returns
-------
regularized_point : array-like, shape=[..., dim]
Regularized point.
"""
regularized_point = point
return regularized_point