"""The ProductPositiveRealsAndComplexPoincareDisks manifold.
The ProductPositiveRealsAndComplexPoincareDisks manifold is defined as a
product manifold of the positive reals manifold and (n-1) complex Poincaré disks.
The positive reals and complex Poincaré disks product has a product metric.
The product metric on the positive reals and complex Poincaré disks product space is
the positive reals metric and (n - 1) complex Poincaré metrics multiplied by constants.
This product manifold can be used to represent Toeplitz HPD matrices.
The ProductPositiveRealsAndComplexPoincareDisks corresponds to
the one-dimensional case of ProductHPDMatricesAndSiegelDisks.
Indeed, PositiveReals is the one-dimensional case of HPDMatrices and
ComplexPoincareDisk is the one-dimensional case of Siegel.
In these one-dimensional manifolds, many simplifications occur compared with
the multidimensional manifolds since matrices commute in dimension 1.
Lead author: Yann Cabanes.
References
----------
.. [Cabanes_2022] Yann Cabanes. Multidimensional complex stationary
centered Gaussian autoregressive time series machine learning
in Poincaré and Siegel disks: application for audio and radar
clutter classification, PhD thesis, tel-03708515, 2022.
https://theses.hal.science/tel-03708515
.. [Cabanes_CESAR_2019] Yann Cabanes, Frédéric Barbaresco, Marc Arnaudon et
Jérémie Bigot. Unsupervised Machine Learning for Pathological Radar Clutter
Clustering: the P-Mean-Shift Algorithm, IEEE, C&ESAR 2019, Rennes, France, 2019.
https://hal.archives-ouvertes.fr/hal-02875430
.. [Cabanes_RADAR_2019] Yann Cabanes, Frédéric Barbaresco, Marc Arnaudon et
Jérémie Bigot. Non-Supervised High Resolution Doppler Machine Learning for
Pathological Radar Clutter, IEEE, RADAR 2019, Toulon, France, 2019.
https://hal.archives-ouvertes.fr/hal-02875415
.. [Cabanes_GSI_2019] Yann Cabanes, Frédéric Barbaresco, Marc Arnaudon et
Jérémie Bigot. Toeplitz Hermitian Positive Definite Matrix Machine Learning
based on Fisher Metric, IEEE, GSI 2019, Toulouse, France, 2019.
https://hal.archives-ouvertes.fr/hal-02875403
.. [Le_Brigant_2017] Alice Le Brigant. Probability on the spaces of curves and
the associated metric spaces using information geometry; radar applications,
PhD thesis, tel-01635258, 2017.
https://theses.hal.science/tel-01635258
.. [Jeuris_2016] 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
.. [Yang_2013] Marc Arnaudon, Frédéric Barbaresco and Le Yang. Riemannian Medians
and Means With Applications to Radar Signal Processing, IEEE, 2013.
"""
from geomstats.geometry.complex_poincare_disk import ComplexPoincareDisk
from geomstats.geometry.positive_reals import PositiveReals
from geomstats.geometry.product_manifold import ProductManifold, ProductRiemannianMetric
from geomstats.geometry.scalar_product_metric import ScalarProductMetric
[docs]
class ProductPositiveRealsAndComplexPoincareDisks(ProductManifold):
"""Class for the ProductPositiveRealsAndComplexPoincareDisks manifold.
The positive reals and complex Poincaré disks product manifold is a
direct product of the positive reals manifold and (n-1) complex Poincaré disks.
Each manifold of the product is a one-dimensional manifold.
Parameters
----------
n_manifolds : int
Number of manifolds of the product.
"""
def __init__(self, n_manifolds, equip=True):
self.n_manifolds = n_manifolds
factors = [PositiveReals()] + [
ComplexPoincareDisk() for _ in range(n_manifolds - 1)
]
scales = [float(n_manifolds - i_manifold) for i_manifold in range(n_manifolds)]
factors[0].equip_with_metric(ScalarProductMetric(factors[0], scales[0]))
for factor, scale in zip(factors[1:], scales[1:]):
factor.equip_with_metric(ScalarProductMetric(factor, scale))
super().__init__(factors=factors, point_ndim=3, equip=equip)
[docs]
@staticmethod
def default_metric():
"""Metric to equip the space with if equip is True."""
return ProductPositiveRealsAndComplexPoincareDisksMetric
[docs]
class ProductPositiveRealsAndComplexPoincareDisksMetric(ProductRiemannianMetric):
"""Class defining the ProductPositiveRealsAndComplexPoincareDisks metric.
The positive reals and complex Poincaré disks product metric is a product
of the positive reals metric and (n-1) complex Poincaré metrics, each of them
being multiplied by a specific constant factor (see [Cabanes_2022]_,
[Cabanes_CESAR_2019]_, [Cabanes_RADAR_2019]_, [Cabanes_GSI_2019]_,
[Le_Brigant_2017], [Jeuris_2016]_ and [Yang_2013]_).
This metric comes from a model used to represent one-dimensional complex
stationary centered Gaussian autoregressive times series.
Such a times series can be seen as a realization of
a multidimensional complex Gaussian distributions with zero mean,
a Toeplitz HPD covariance matrix and a zero relation matrix.
The ProductPositiveRealsAndComplexPoincareDisks metric is inspired by
information geometry on this specific set of Gaussian distributions.
References
----------
.. [Cabanes_2022] Yann Cabanes. Multidimensional complex stationary
centered Gaussian autoregressive time series machine learning
in Poincaré and Siegel disks: application for audio and radar
clutter classification, PhD thesis, tel-03708515, 2022.
https://theses.hal.science/tel-03708515
.. [Cabanes_CESAR_2019] Yann Cabanes, Frédéric Barbaresco, Marc Arnaudon et
Jérémie Bigot. Unsupervised Machine Learning for Pathological Radar Clutter
Clustering: the P-Mean-Shift Algorithm, IEEE, C&ESAR 2019, Rennes, France, 2019.
https://hal.archives-ouvertes.fr/hal-02875430
.. [Cabanes_RADAR_2019] Yann Cabanes, Frédéric Barbaresco, Marc Arnaudon et
Jérémie Bigot. Non-Supervised High Resolution Doppler Machine Learning for
Pathological Radar Clutter, IEEE, RADAR 2019, Toulon, France, 2019.
https://hal.archives-ouvertes.fr/hal-02875415
.. [Cabanes_GSI_2019] Yann Cabanes, Frédéric Barbaresco, Marc Arnaudon et
Jérémie Bigot. Toeplitz Hermitian Positive Definite Matrix Machine Learning
based on Fisher Metric, IEEE, GSI 2019, Toulouse, France, 2019.
https://hal.archives-ouvertes.fr/hal-02875403
.. [Le_Brigant_2017] Alice Le Brigant. Probability on the spaces of curves and
the associated metric spaces using information geometry; radar applications,
PhD thesis, tel-01635258, 2017.
https://theses.hal.science/tel-01635258
.. [Jeuris_2016] 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
.. [Yang_2013] Marc Arnaudon, Frédéric Barbaresco and Le Yang. Riemannian Medians
and Means With Applications to Radar Signal Processing, IEEE, 2013.
"""