# -*- coding: utf-8 -*-
# The Procrustes library provides a set of functions for transforming
# a matrix to make it as similar as possible to a target matrix.
#
# Copyright (C) 2017-2022 The QC-Devs Community
#
# This file is part of Procrustes.
#
# Procrustes is free software; you can redistribute it and/or
# modify it under the terms of the GNU General Public License
# as published by the Free Software Foundation; either version 3
# of the License, or (at your option) any later version.
#
# Procrustes is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program; if not, see <http://www.gnu.org/licenses/>
#
# --
"""Orthogonal Procrustes Module."""
# import warnings
from typing import Optional
import numpy as np
from procrustes.utils import compute_error, ProcrustesResult, setup_input_arrays
import scipy
__all__ = [
"orthogonal",
"orthogonal_2sided",
]
[docs]def orthogonal(
a: np.ndarray,
b: np.ndarray,
pad: bool = True,
translate: bool = False,
scale: bool = False,
unpad_col: bool = False,
unpad_row: bool = False,
check_finite: bool = True,
weight: Optional[np.ndarray] = None,
lapack_driver: str = "gesvd",
) -> ProcrustesResult:
r"""Perform orthogonal Procrustes.
Given a matrix :math:`\mathbf{A}_{m \times n}` and a reference matrix :math:`\mathbf{B}_{m
\times n}`, find the orthogonal transformation matrix :math:`\mathbf{Q}_{n
\times n}` that makes :math:`\mathbf{AQ}` as close as possible to :math:`\mathbf{B}`.
In other words,
.. math::
\underbrace{\min}_{\left\{\mathbf{Q} | \mathbf{Q}^{-1} = {\mathbf{Q}}^\dagger \right\}}
\|\mathbf{A}\mathbf{Q} - \mathbf{B}\|_{F}^2
This Procrustes method requires the :math:`\mathbf{A}` and :math:`\mathbf{B}` matrices to
have the same shape, which is gauranteed with the default ``pad`` argument for any given
:math:`\mathbf{A}` and :math:`\mathbf{B}` matrices. In preparing the :math:`\mathbf{A}` and
:math:`\mathbf{B}` matrices, the (optional) order of operations is: **1)** unpad zero
rows/columns, **2)** translate the matrices to the origin, **3)** weight entries of
:math:`\mathbf{A}`, **4)** scale the matrices to have unit norm, **5)** pad matrices with zero
rows/columns so they have the same shape.
Parameters
----------
a : ndarray
The 2D-array :math:`\mathbf{A}` which is going to be transformed.
b : ndarray
The 2D-array :math:`\mathbf{B}` representing the reference matrix.
pad : bool, optional
Add zero rows (at the bottom) and/or columns (to the right-hand side) of matrices
:math:`\mathbf{A}` and :math:`\mathbf{B}` so that they have the same shape.
translate : bool, optional
If True, both arrays are centered at origin (columns of the arrays will have mean zero).
scale : bool, optional
If True, both arrays are normalized with respect to the Frobenius norm, i.e.,
:math:`\text{Tr}\left[\mathbf{A}^\dagger\mathbf{A}\right] = 1` and
:math:`\text{Tr}\left[\mathbf{B}^\dagger\mathbf{B}\right] = 1`.
unpad_col : bool, optional
If True, zero columns (with values less than 1.0e-8) on the right-hand side of the intial
:math:`\mathbf{A}` and :math:`\mathbf{B}` matrices are removed.
unpad_row : bool, optional
If True, zero rows (with values less than 1.0e-8) at the bottom of the intial
:math:`\mathbf{A}` and :math:`\mathbf{B}` matrices are removed.
check_finite : bool, optional
If True, convert the input to an array, checking for NaNs or Infs.
weight : ndarray, optional
The 1D-array representing the weights of each row of :math:`\mathbf{A}`. This defines the
elements of the diagonal matrix :math:`\mathbf{W}` that is multiplied by :math:`\mathbf{A}`
matrix, i.e., :math:`\mathbf{A} \rightarrow \mathbf{WA}`.
lapack_driver : {'gesvd', 'gesdd'}, optional
Whether to use the more efficient divide-and-conquer approach ('gesdd') or the more robust
general rectangular approach ('gesvd') to compute the singular-value decomposition with
`scipy.linalg.svd`.
Returns
-------
res : ProcrustesResult
The Procrustes result represented as a class:`utils.ProcrustesResult` object.
Notes
-----
The optimal orthogonal matrix is obtained by,
.. math::
\mathbf{Q}^{\text{opt}} =
\arg \underbrace{\min}_{\left\{\mathbf{Q} \left| {\mathbf{Q}^{-1} = {\mathbf{Q}}^\dagger}
\right. \right\}} \|\mathbf{A}\mathbf{Q} - \mathbf{B}\|_{F}^2 =
\arg \underbrace{\max}_{\left\{\mathbf{Q} \left| {\mathbf{Q}^{-1} = {\mathbf{Q}}^\dagger}
\right. \right\}} \text{Tr}\left[\mathbf{Q^\dagger}\mathbf{A^\dagger}\mathbf{B}\right]
The solution is obtained using the singular value decomposition (SVD) of the
:math:`\mathbf{A}^\dagger \mathbf{B}` matrix,
.. math::
\mathbf{A}^\dagger \mathbf{B} &= \tilde{\mathbf{U}} \tilde{\mathbf{\Sigma}}
\tilde{\mathbf{V}}^{\dagger} \\
\mathbf{Q}^{\text{opt}} &= \tilde{\mathbf{U}} \tilde{\mathbf{V}}^{\dagger}
The singular values are always listed in decreasing order, with the smallest singular
value in the bottom-right-hand corner of :math:`\tilde{\mathbf{\Sigma}}`.
Examples
--------
>>> import numpy as np
>>> from scipy.stats import ortho_group
>>> from procrustes import orthogonal
>>> a = np.random.rand(5, 3) # random input matrix
>>> q = ortho_group.rvs(3) # random orthogonal transformation
>>> b = np.dot(a, q) + np.random.rand(1, 3) # random target matrix
>>> result = orthogonal(a, b, translate=True, scale=False)
>>> print(result.error) # error (should be zero)
>>> print(result.t) # transformation matrix (same as q)
>>> print(result.new_a) # translated array a
>>> print(result.new_b) # translated array b
"""
# check inputs
new_a, new_b = setup_input_arrays(
a,
b,
unpad_col,
unpad_row,
pad,
translate,
scale,
check_finite,
weight,
)
if new_a.shape != new_b.shape:
raise ValueError(
f"Shape of A and B does not match: {new_a.shape} != {new_b.shape} "
"Check pad, unpad_col, and unpad_row arguments."
)
# calculate SVD of A.T * B
u, _, vt = scipy.linalg.svd(np.dot(new_a.T, new_b), lapack_driver=lapack_driver)
# compute optimal orthogonal transformation
u_opt = np.dot(u, vt)
# compute one-sided error
error = compute_error(new_a, new_b, u_opt)
return ProcrustesResult(error=error, new_a=new_a, new_b=new_b, t=u_opt, s=None)
[docs]def orthogonal_2sided(
a: np.ndarray,
b: np.ndarray,
single: bool = True,
pad: bool = True,
translate: bool = False,
scale: bool = False,
unpad_col: bool = False,
unpad_row: bool = False,
check_finite: bool = True,
weight: Optional[np.ndarray] = None,
lapack_driver: str = "gesvd",
) -> ProcrustesResult:
r"""Perform two-sided orthogonal Procrustes with one- or two-transformations.
**Two Transformations:** Given a matrix :math:`\mathbf{A}_{m \times n}` and a reference matrix
:math:`\mathbf{B}_{m \times n}`, find two :math:`n \times n` orthogonal
transformation matrices :math:`\mathbf{Q}_1^\dagger` and :math:`\mathbf{Q}_2` that makes
:math:`\mathbf{Q}_1^\dagger\mathbf{A}\mathbf{Q}_2` as close as possible to :math:`\mathbf{B}`.
In other words,
.. math::
\underbrace{\text{min}}_{\left\{ {\mathbf{Q}_1 \atop \mathbf{Q}_2} \left|
{\mathbf{Q}_1^{-1} = \mathbf{Q}_1^\dagger \atop \mathbf{Q}_2^{-1} =
\mathbf{Q}_2^\dagger} \right. \right\}}
\|\mathbf{Q}_1^\dagger \mathbf{A} \mathbf{Q}_2 - \mathbf{B}\|_{F}^2
**Single Transformations:** Given a **symmetric** matrix :math:`\mathbf{A}_{n \times n}` and
a reference :math:`\mathbf{B}_{n \times n}`, find one orthogonal transformation
matrix :math:`\mathbf{Q}_{n \times n}` that makes :math:`\mathbf{A}` as close as possible to
:math:`\mathbf{B}`. In other words,
.. math::
\underbrace{\min}_{\left\{\mathbf{Q} | \mathbf{Q}^{-1} = {\mathbf{Q}}^\dagger \right\}}
\|\mathbf{Q}^\dagger\mathbf{A}\mathbf{Q} - \mathbf{B}\|_{F}^2
This Procrustes method requires the :math:`\mathbf{A}` and :math:`\mathbf{B}` matrices to
have the same shape, which is gauranteed with the default ``pad`` argument for any given
:math:`\mathbf{A}` and :math:`\mathbf{B}` matrices. In preparing the :math:`\mathbf{A}` and
:math:`\mathbf{B}` matrices, the (optional) order of operations is: **1)** unpad zero
rows/columns, **2)** translate the matrices to the origin, **3)** weight entries of
:math:`\mathbf{A}`, **4)** scale the matrices to have unit norm, **5)** pad matrices with zero
rows/columns so they have the same shape.
Parameters
----------
a : ndarray
The 2D-array :math:`\mathbf{A}` which is going to be transformed.
b : ndarray
The 2D-array :math:`\mathbf{B}` representing the reference matrix.
single : bool, optional
If True, single transformation is used (i.e., :math:`\mathbf{Q}_1=\mathbf{Q}_2=\mathbf{Q}`),
otherwise, two transformations are used.
pad : bool, optional
Add zero rows (at the bottom) and/or columns (to the right-hand side) of matrices
:math:`\mathbf{A}` and :math:`\mathbf{B}` so that they have the same shape.
translate : bool, optional
If True, both arrays are centered at origin (columns of the arrays will have mean zero).
scale : bool, optional
If True, both arrays are normalized with respect to the Frobenius norm, i.e.,
:math:`\text{Tr}\left[\mathbf{A}^\dagger\mathbf{A}\right] = 1` and
:math:`\text{Tr}\left[\mathbf{B}^\dagger\mathbf{B}\right] = 1`.
unpad_col : bool, optional
If True, zero columns (with values less than 1.0e-8) on the right-hand side of the intial
:math:`\mathbf{A}` and :math:`\mathbf{B}` matrices are removed.
unpad_row : bool, optional
If True, zero rows (with values less than 1.0e-8) at the bottom of the intial
:math:`\mathbf{A}` and :math:`\mathbf{B}` matrices are removed.
check_finite : bool, optional
If True, convert the input to an array, checking for NaNs or Infs.
weight : ndarray, optional
The 1D-array representing the weights of each row of :math:`\mathbf{A}`. This defines the
elements of the diagonal matrix :math:`\mathbf{W}` that is multiplied by :math:`\mathbf{A}`
matrix, i.e., :math:`\mathbf{A} \rightarrow \mathbf{WA}`.
lapack_driver : {"gesvd", "gesdd"}, optional
Used in the singular value decomposition function from SciPy. Only allowed two options,
with "gesvd" being less-efficient than "gesdd" but is more robust. Default is "gesvd".
Returns
-------
res : ProcrustesResult
The Procrustes result represented as a class:`utils.ProcrustesResult` object.
Notes
-----
**Two-Sided Orthogonal Procrustes with Two Transformations:**
The optimal orthogonal transformations are obtained by:
.. math::
\mathbf{Q}_{1}^{\text{opt}}, \mathbf{Q}_{2}^{\text{opt}} = \arg
\underbrace{\text{min}}_{\left\{ {\mathbf{Q}_1 \atop \mathbf{Q}_2} \left|
{\mathbf{Q}_1^{-1} = \mathbf{Q}_1^\dagger \atop \mathbf{Q}_2^{-1} =
\mathbf{Q}_2^\dagger} \right. \right\}}
\|\mathbf{Q}_1^\dagger \mathbf{A} \mathbf{Q}_2 - \mathbf{B}\|_{F}^2 = \arg
\underbrace{\text{max}}_{\left\{ {\mathbf{Q}_1 \atop \mathbf{Q}_2} \left|
{\mathbf{Q}_1^{-1} = \mathbf{Q}_1^\dagger \atop \mathbf{Q}_2^{-1} =
\mathbf{Q}_2^\dagger} \right. \right\}}
\text{Tr}\left[\mathbf{Q}_2^\dagger\mathbf{A}^\dagger\mathbf{Q}_1\mathbf{B} \right]
This is solved by taking the singular value decomposition (SVD) of :math:`\mathbf{A}` and
:math:`\mathbf{B}`,
.. math::
\mathbf{A} = \mathbf{U}_A \mathbf{\Sigma}_A \mathbf{V}_A^\dagger \\
\mathbf{B} = \mathbf{U}_B \mathbf{\Sigma}_B \mathbf{V}_B^\dagger
Then the two optimal orthogonal matrices are given by,
.. math::
\mathbf{Q}_1^{\text{opt}} = \mathbf{U}_A \mathbf{U}_B^\dagger \\
\mathbf{Q}_2^{\text{opt}} = \mathbf{V}_A \mathbf{V}_B^\dagger
**Two-Sided Orthogonal Procrustes with Single-Transformation:**
The optimal orthogonal transformation is obtained by:
.. math::
\mathbf{Q}^{\text{opt}} = \arg
\underbrace{\min}_{\left\{\mathbf{Q} | \mathbf{Q}^{-1} = {\mathbf{Q}}^\dagger \right\}}
\|\mathbf{Q}^\dagger\mathbf{A}\mathbf{Q} - \mathbf{B}\|_{F}^2 = \arg
\underbrace{\text{max}}_{\left\{\mathbf{Q} | \mathbf{Q}^{-1} = {\mathbf{Q}}^\dagger\right\}}
\text{Tr}\left[\mathbf{Q}^\dagger\mathbf{A}^\dagger\mathbf{Q}\mathbf{B} \right]
Using the singular value decomposition (SVD) of :math:`\mathbf{A}` and :math:`\mathbf{B}`,
.. math::
\mathbf{A} = \mathbf{U}_A \mathbf{\Lambda}_A \mathbf{U}_A^\dagger \\
\mathbf{B} = \mathbf{U}_B \mathbf{\Lambda}_B \mathbf{U}_B^\dagger
The optimal orthogonal matrix :math:`\mathbf{Q}^\text{opt}` is obtained through,
.. math::
\mathbf{Q}^\text{opt} = \mathbf{U}_A \mathbf{S} \mathbf{U}_B^\dagger
where :math:`\mathbf{S}` is a diagonal matrix with :math:`\pm{1}` elements,
.. math::
\mathbf{S} =
\begin{bmatrix}
{ \pm 1} & 0 &\cdots &0 \\
0 &{ \pm 1} &\ddots &\vdots \\
\vdots &\ddots &\ddots &0\\
0 &\cdots &0 &{ \pm 1}
\end{bmatrix}
The matrix :math:`\mathbf{S}` is chosen to be the identity matrix.
Examples
--------
>>> import numpy as np
>>> a = np.array([[30, 33, 20], [33, 53, 43], [20, 43, 46]])
>>> b = np.array([[ 22.78131838, -0.58896768,-43.00635291, 0., 0.],
... [ -0.58896768, 16.77132475, 0.24289990, 0., 0.],
... [-43.00635291, 0.2428999 , 89.44735687, 0., 0.],
... [ 0. , 0. , 0. , 0., 0.]])
>>> res = orthogonal_2sided(a, b, single=True, pad=True, unpad_col=True)
>>> res.t
array([[ 0.25116633, 0.76371527, 0.59468855],
[-0.95144277, 0.08183302, 0.29674906],
[ 0.17796663, -0.64034549, 0.74718507]])
>>> res.error
1.9646186414076689e-26
"""
# if translate:
# warnings.warn(
# "The translation matrix was not well defined. \
# Two sided rotation and translation don't commute.",
# stacklevel=2,
# )
# Check inputs
new_a, new_b = setup_input_arrays(
a,
b,
unpad_col,
unpad_row,
pad,
translate,
scale,
check_finite,
weight,
)
# check symmetry if single_transform=True
if single:
if not np.allclose(new_a.T, new_a):
raise ValueError(
f"Array A with {new_a.shape} shape is not symmetric. "
"Check pad, unpad_col, and unpad_row arguments."
)
if not np.allclose(new_b.T, new_b):
raise ValueError(
f"Array B with {new_b.shape} shape is not symmetric. "
"Check pad, unpad_col, and unpad_row arguments."
)
# two-sided orthogonal Procrustes with one-transformations
if single:
_, ua = np.linalg.eigh(new_a)
_, ub = np.linalg.eigh(new_b)
u_opt = np.dot(ua, ub.T)
# compute one-sided error
error = compute_error(new_a, new_b, u_opt, u_opt.T)
return ProcrustesResult(error=error, new_a=new_a, new_b=new_b, t=u_opt, s=u_opt.T)
# two-sided orthogonal Procrustes with two-transformations
ua, _, vta = scipy.linalg.svd(new_a, lapack_driver=lapack_driver)
ub, _, vtb = scipy.linalg.svd(new_b, lapack_driver=lapack_driver)
u_opt1 = np.dot(ua, ub.T)
u_opt2 = np.dot(vta.T, vtb)
error = compute_error(new_a, new_b, u_opt2, u_opt1.T)
return ProcrustesResult(error=error, new_a=new_a, new_b=new_b, t=u_opt2, s=u_opt1.T)