# -*- 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/>
#
# --
"""Symmetric Procrustes Module."""
from typing import Optional
import numpy as np
from procrustes.utils import _zero_padding, compute_error, ProcrustesResult, setup_input_arrays
import scipy
[docs]def symmetric(
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 symmetric Procrustes.
Given a matrix :math:`\mathbf{A}_{m \times n}` and a reference matrix :math:`\mathbf{B}_{m
\times n}` with :math:`m \geqslant n`, find the symmetrix transformation matrix
:math:`\mathbf{X}_{n \times n}` that makes :math:`\mathbf{AX}` as close as possible to
:math:`\mathbf{B}`. In other words,
.. math::
\underbrace{\text{min}}_{\left\{\mathbf{X} \left| \mathbf{X} = \mathbf{X}^\dagger
\right. \right\}} \|\mathbf{A} \mathbf{X} - \mathbf{B}\|_{F}^2
This Procrustes method requires the :math:`\mathbf{A}` and :math:`\mathbf{B}` matrices to
have the same shape with :math:`m \geqslant n`, which is guaranteed 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 symmetrix matrix is obtained by,
.. math::
\mathbf{X}_{\text{opt}} = \arg
\underbrace{\text{min}}_{\left\{\mathbf{X} \left| \mathbf{X} = \mathbf{X}^\dagger
\right. \right\}} \|\mathbf{A} \mathbf{X} - \mathbf{B}\|_{F}^2 =
\underbrace{\text{min}}_{\left\{\mathbf{X} \left| \mathbf{X} = \mathbf{X}^\dagger
\right. \right\}}
\text{Tr}\left[\left(\mathbf{A}\mathbf{X} - \mathbf{B} \right)^\dagger
\left(\mathbf{A}\mathbf{X} - \mathbf{B} \right)\right]
Considering the singular value decomposition of :math:`\mathbf{A}`,
.. math::
\mathbf{A}_{m \times n} = \mathbf{U}_{m \times m}
\mathbf{\Sigma}_{m \times n}
\mathbf{V}_{n \times n}^\dagger
where :math:`\mathbf{\Sigma}_{m \times n}` is a rectangular diagonal matrix with non-negative
singular values :math:`\sigma_i = [\mathbf{\Sigma}]_{ii}` listed in descending order, define
.. math::
\mathbf{C}_{m \times n} = \mathbf{U}_{m \times m}^\dagger
\mathbf{B}_{m \times n} \mathbf{V}_{n \times n}
with elements denoted by :math:`c_{ij}`.
Then we compute the symmetric matrix :math:`\mathbf{Y}_{n \times n}` with
.. math::
[\mathbf{Y}]_{ij} = \begin{cases}
0 && i \text{ and } j > \text{rank} \left(\mathbf{A}\right) \\
\frac{\sigma_i c_{ij} + \sigma_j c_{ji}}{\sigma_i^2 +
\sigma_j^2} && \text{otherwise} \end{cases}
It is worth noting that the first part of this definition only applies in the unusual case where
:math:`\mathbf{A}` has rank less than :math:`n`. The :math:`\mathbf{X}_\text{opt}` is given by
.. math::
\mathbf{X}_\text{opt} = \mathbf{V Y V}^{\dagger}
Examples
--------
>>> import numpy as np
>>> a = np.array([[5., 2., 8.],
... [2., 2., 3.],
... [1., 5., 6.],
... [7., 3., 2.]])
>>> b = np.array([[ 52284.5, 209138. , 470560.5],
... [ 22788.5, 91154. , 205096.5],
... [ 46139.5, 184558. , 415255.5],
... [ 22788.5, 91154. , 205096.5]])
>>> res = symmetric(a, b, pad=True, translate=True, scale=True)
>>> res.t # symmetric transformation array
array([[0.0166352 , 0.06654081, 0.14971682],
[0.06654081, 0.26616324, 0.59886729],
[0.14971682, 0.59886729, 1.34745141]])
>>> res.error # error
4.483083428047388e-31
"""
# check inputs
new_a, new_b = setup_input_arrays(
a,
b,
unpad_col,
unpad_row,
pad,
translate,
scale,
check_finite,
weight,
)
# if number of rows is less than column, the arrays are made square
if (new_a.shape[0] < new_a.shape[1]) or (new_b.shape[0] < new_b.shape[1]):
new_a, new_b = _zero_padding(new_a, new_b, "square")
# if new_a.shape[0] < new_a.shape[1]:
# raise ValueError(f"Shape of A {new_a.shape}=(m, n) needs to satisfy m >= n.")
#
# if new_b.shape[0] < new_b.shape[1]:
# raise ValueError(f"Shape of B {new_b.shape}=(m, n) needs to satisfy m >= n.")
# compute SVD of A & matrix C
u, s, vt = scipy.linalg.svd(new_a, lapack_driver=lapack_driver)
c = np.dot(np.dot(u.T, new_b), vt.T)
# compute intermediate matrix Y
n = new_a.shape[1]
y = np.zeros((n, n))
for i in range(n):
for j in range(n):
if s[i] ** 2 + s[j] ** 2 != 0:
y[i, j] = (s[i] * c[i, j] + s[j] * c[j, i]) / (s[i] ** 2 + s[j] ** 2)
# compute optimum symmetric transformation matrix X
x = np.dot(np.dot(vt.T, y), vt)
error = compute_error(new_a, new_b, x)
return ProcrustesResult(error=error, new_a=new_a, new_b=new_b, t=x, s=None)