"""Complex Hermitian space.
Lead author: Yann Cabanes.
"""
import math
import geomstats.backend as gs
from geomstats.geometry.base import ComplexVectorSpace
from geomstats.geometry.complex_riemannian_metric import ComplexRiemannianMetric
from geomstats.vectorization import repeat_out
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class Hermitian(ComplexVectorSpace):
"""Class for Hermitian spaces.
By definition, a Hermitian space is a complex vector space
of a given dimension, equipped with a Hermitian metric.
Parameters
----------
dim : int
Dimension of the Hermitian space.
"""
def __init__(self, dim, equip=True):
super().__init__(shape=(dim,), equip=equip)
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@staticmethod
def default_metric():
"""Metric to equip the space with if equip is True."""
return HermitianMetric
@property
def identity(self):
"""Identity of the group.
Returns
-------
identity : array-like, shape=[n]
"""
return gs.zeros(self.dim, dtype=gs.get_default_cdtype())
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def exp(self, tangent_vec, base_point=None):
"""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
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class HermitianMetric(ComplexRiemannianMetric):
"""Class for Hermitian metrics.
As a Riemannian metric, the Hermitian 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 Hermitian space.
"""
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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.
"""
mat = gs.eye(self._space.dim, dtype=gs.get_default_cdtype())
if base_point is not None and base_point.ndim > 1:
return gs.broadcast_to(mat, base_point.shape + (self._space.dim,))
return mat
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@staticmethod
def inner_product(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.
"""
return gs.dot(gs.conj(tangent_vec_a), tangent_vec_b)
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@staticmethod
def norm(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.
"""
return gs.linalg.norm(vector, axis=-1)
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@staticmethod
def exp(tangent_vec, base_point):
"""Compute exp map of a base point in tangent vector direction.
The Riemannian exponential is vector addition in the Hermitian 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
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@staticmethod
def log(point, base_point):
"""Compute log map using a base point and other point.
The Riemannian logarithm is the subtraction in the Hermitian 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
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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 Hermitian 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.
direction : array-like, shape=[..., dim]
Tangent vector at base point, along which the parallel transport
is computed.
end_point : array-like, shape=[..., dim]
Point on the manifold. Point to transport to.
Returns
-------
transported_tangent_vec : array-like, shape=[..., dim]
Transported tangent vector at `exp_(base_point)(direction)`.
"""
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,
)
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def injectivity_radius(self, base_point=None):
"""Compute the radius of the injectivity domain.
In a Hermitian space, it is infinite everywhere.
Parameters
----------
base_point : array-like, shape=[..., dim]
Point on the manifold.
Returns
-------
radius : array-like, shape=[...,]
Injectivity radius.
"""
radius = gs.array(math.inf)
return repeat_out(self._space.point_ndim, radius, base_point)