"""The Poincare polydisk.
The Poincare polydisk is defined as a product manifold of the Hyperbolic space
of dimension 2. The Poincare polydisk has a product metric. The metric on each
space is the natural Poincare metric multiplied by a constant.
References
----------
.. [JV2016] B. Jeuris and R. Vandebril. The Kahler mean of Block-Toeplitz
matrices with Toeplitz structured blocks, 2016.
https://epubs.siam.org/doi/pdf/10.1137/15M102112X
"""
import geomstats.backend as gs
from geomstats.geometry.hyperbolic import Hyperbolic
from geomstats.geometry.hyperboloid import Hyperboloid
from geomstats.geometry.hyperboloid import HyperboloidMetric
from geomstats.geometry.product_manifold import ProductManifold
from geomstats.geometry.product_riemannian_metric \
import ProductRiemannianMetric # NOQA
[docs]class PoincarePolydisk(ProductManifold):
r"""Class for the Poincare polydisk.
The Poincare polydisk is a direct product of n Poincare disks,
i.e. hyperbolic spaces of dimension 2.
Parameters
----------
n_disks : int
Number of disks.
coords_type : str, {\'intrinsic\', \'extrinsic\', etc}
Coordinate type.
Optional, default: \'extrinsic\'.
"""
default_coords_type = 'extrinsic'
default_point_type = 'matrix'
def __init__(self, n_disks, coords_type='extrinsic'):
self.n_disks = n_disks
self.coords_type = coords_type
self.point_type = PoincarePolydisk.default_point_type
disk = Hyperboloid(2, coords_type=coords_type)
list_disks = [disk, ] * n_disks
super(PoincarePolydisk, self).__init__(
manifolds=list_disks, default_point_type='matrix')
self.metric = PoincarePolydiskMetric(n_disks=n_disks,
coords_type=coords_type)
[docs] @staticmethod
def intrinsic_to_extrinsic_coords(point_intrinsic):
"""Convert point from intrinsic to extrensic coordinates.
Convert the parameterization of a point in the hyperbolic space
from its intrinsic coordinates, to its extrinsic coordinates
in Minkowski space.
Parameters
----------
point_intrinsic : array-like, shape=[..., n_disk, dim]
Point in intrinsic coordinates.
Returns
-------
point_extrinsic : array-like, shape=[..., n_disks, dim + 1]
Point in extrinsic coordinates.
"""
n_disks = point_intrinsic.shape[1]
point_extrinsic = gs.stack(
[Hyperbolic.change_coordinates_system(
point_intrinsic[:, i_disk, ...], 'intrinsic', 'extrinsic')
for i_disk in range(n_disks)], axis=1)
return point_extrinsic
[docs] def projection_to_tangent_space(self, vector, base_point):
"""Project a vector in the tangent space.
Project a vector in Minkowski space
on the tangent space of the hyperbolic space at a base point.
Parameters
----------
vector : array-like, shape=[..., n_disks, dim + 1]
Vector.
base_point : array-like, shape=[..., n_disks, dim + 1]
Base point.
Returns
-------
tangent_vec : array-like, shape=[..., n_disks, dim + 1]
Tangent vector at base point.
"""
n_disks = base_point.shape[1]
hyperbolic_space = Hyperboloid(2, self.coords_type)
tangent_vec = gs.stack([hyperbolic_space.to_tangent(
vector=vector[:, i_disk, :],
base_point=base_point[:, i_disk, :])
for i_disk in range(n_disks)], axis=1)
return tangent_vec
[docs]class PoincarePolydiskMetric(ProductRiemannianMetric):
r"""Class defining the Poincare polydisk metric.
The Poincare polydisk metric is a product of n Poincare metrics,
each of them being multiplied by a specific constant factor (see
[JV2016]_).
This metric comes from a model used to represent
stationary complex autoregressive Gaussian signals.
Parameters
----------
n_disks : int
Number of disks.
coords_type : str, {\'intrinsic\', \'extrinsic\', etc}
Coordinate type.
Optional, default: \'extrinsic\'.
References
----------
.. [JV2016] B. Jeuris and R. Vandebril. The Kähler mean of Block-Toeplitz
matrices with Toeplitz structured blocks, 2016.
https://epubs.siam.org/doi/pdf/10.1137/15M102112X
"""
default_coords_type = 'extrinsic'
def __init__(self, n_disks, coords_type='extrinsic'):
self.n_disks = n_disks
self.coords_type = coords_type
list_metrics = []
for i_disk in range(n_disks):
scale_i = (n_disks - i_disk) ** 0.5
metric_i = HyperboloidMetric(2, coords_type, scale_i)
list_metrics.append(metric_i)
super(PoincarePolydiskMetric, self).__init__(
metrics=list_metrics, default_point_type='matrix')