def test_zero_padding_rows_columns():
r"""Test _zero_padding with random array padding rows and columns."""
array1 = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12]])
array2 = np.array([[1, 2.5], [9, 5], [4, 8.5]])
padded2, padded1 = _zero_padding(array2, array1, pad_mode='row-col')
array2_test = np.array([[1, 2.5, 0], [9, 5, 0], [4, 8.5, 0], [0, 0, 0]])
assert padded1.shape == (4, 3)
assert padded2.shape == (4, 3)
assert (abs(padded1 - array1) < 1.e-10).all()
assert (abs(padded2 - array2_test) < 1.e-10).all()
# Test in the scenario they have the same shape but fat rectangular.
array1 = np.array([[60, 85, 86, 1.], [85, 151, 153, 2.],
[86, 153, 158, 10.]])
padded2, padded1 = _zero_padding(array1, array1, pad_mode='row-col')
assert np.all(np.abs(array1 - padded2)) < 1e-5
assert np.all(np.abs(array1 - padded1)) < 1e-5
assert padded1.shape == padded2.shape
assert padded1.shape == (3, 4)
# Test in the scenario they have the same shape but tall rectangular.
array1 = np.random.random((2, 10))
array2 = np.random.random((2, 10))
padded2, padded1 = _zero_padding(array1, array2, pad_mode='row-col')
assert np.all(np.abs(array1 - padded2)) < 1e-5
assert np.all(np.abs(array2 - padded1)) < 1e-5
assert padded1.shape == padded2.shape
assert padded1.shape == (2, 10)
def test_zero_padding_columns():
r"""Test _zero_padding with random array padding columns."""
array1 = np.array([[4, 7, 2], [1, 3, 5]])
array2 = np.array([[5], [2]])
# match the number of columns of the 1st array
padded2, padded1 = _zero_padding(array2, array1, pad_mode='col')
assert padded1.shape == (2, 3)
assert padded2.shape == (2, 3)
assert (abs(padded1 - array1) < 1.e-10).all()
assert (abs(padded2 - np.array([[5, 0, 0], [2, 0, 0]])) < 1.e-10).all()
# match the number of columns of the 1st array
array3 = np.arange(8).reshape(8, 1)
array4 = np.arange(8).reshape(2, 4)
padded3, padded4 = _zero_padding(array3, array4, pad_mode='col')
assert padded3.shape == (8, 4)
assert padded4.shape == (2, 4)
assert (abs(array4 - padded4) < 1.e-10).all()
expected = list(range(8))
expected.extend([0] * 24)
expected = np.array(expected).reshape(4, 8).T
assert (abs(expected - padded3) < 1.e-10).all()
# padding the padded_arrays should not change anything
padded5, padded6 = _zero_padding(padded3, padded4, pad_mode='col')
assert padded3.shape == (8, 4)
assert padded4.shape == (2, 4)
assert padded5.shape == (8, 4)
assert padded6.shape == (2, 4)
assert (abs(padded5 - padded3) < 1.e-10).all()
assert (abs(padded6 - padded4) < 1.e-10).all()
def test_zero_padding_rows():
r"""Test _zero_padding with random array padding rows."""
array1 = np.array([[1, 2], [3, 4]])
array2 = np.array([[5, 6]])
# match the number of rows of the 1st array
padded2, padded1 = _zero_padding(array2, array1, pad_mode='row')
assert padded1.shape == (2, 2)
assert padded2.shape == (2, 2)
assert (abs(padded1 - array1) < 1.e-10).all()
assert (abs(padded2 - np.array([[5, 6], [0, 0]])) < 1.e-10).all()
# match the number of rows of the 1st array
array3 = np.arange(8).reshape(2, 4)
array4 = np.arange(8).reshape(4, 2)
padded3, padded4 = _zero_padding(array3, array4, pad_mode='row')
assert padded3.shape == (4, 4)
assert padded4.shape == (4, 2)
assert (abs(array4 - padded4) < 1.e-10).all()
expected = list(range(8))
expected.extend([0] * 8)
expected = np.array(expected).reshape(4, 4)
assert (abs(expected - padded3) < 1.e-10).all()
# padding the padded_arrays should not change anything
padded5, padded6 = _zero_padding(padded3, padded4, pad_mode='row')
assert padded3.shape == (4, 4)
assert padded4.shape == (4, 2)
assert padded5.shape == (4, 4)
assert padded6.shape == (4, 2)
assert (abs(padded5 - padded3) < 1.e-10).all()
assert (abs(padded6 - padded4) < 1.e-10).all()
def test_zero_padding_square():
r"""Test _zero_padding with squared array."""
# Try two equivalent (but different sized) symmetric arrays
array1 = np.array([[60, 85, 86], [85, 151, 153], [86, 153, 158]])
array2 = np.array([[60, 85, 86, 0, 0], [85, 151, 153, 0, 0],
[86, 153, 158, 0, 0], [0, 0, 0, 0, 0]])
square1, square2 = _zero_padding(array1, array2, pad_mode='square')
assert square1.shape == square2.shape
assert square1.shape[0] == square1.shape[1]
# Test in the scenario they have the same shape but rectangular.
array1 = np.array([[60, 85, 86, 1.], [85, 151, 153, 2.],
[86, 153, 158, 10.]])
array2 = np.array([[60, 85, 86, 1.], [85, 151, 153, 2.],
[86, 153, 158, 10.]])
square1, square2 = _zero_padding(array1, array2, pad_mode='square')
assert square1.shape == square2.shape
assert square1.shape[0] == square1.shape[1]
assert square1.shape[0] == 4
# Performing the analysis on equally sized square arrays should return the same input arrays
sym_part = np.array([[1, 7, 8, 4], [6, 4, 8, 1]])
array1 = np.dot(sym_part, sym_part.T)
array2 = array1
assert array1.shape == array2.shape
square1, square2 = _zero_padding(array1, array2, pad_mode='square')
assert square1.shape == square2.shape
assert square1.shape[0] == square1.shape[1]
assert (abs(array2 - array1) < 1.e-10).all()
def permutation(
a,
b,
pad=True,
translate=False,
scale=False,
unpad_col=False,
unpad_row=False,
check_finite=True,
weight=None,
):
r"""Perform one-sided permutation Procrustes.
Given matrix :math:`\mathbf{A}_{m \times n}` and a reference matrix :math:`\mathbf{B}_{m \times
n}`, find the permutation transformation matrix :math:`\mathbf{P}_{n \times n}`
that makes :math:`\mathbf{AP}` as close as possible to :math:`\mathbf{B}`. In other words,
.. math::
\underbrace{\text{min}}_{\left\{\mathbf{P} \left| {[\mathbf{P}]_{ij} \in \{0, 1\} \atop
\sum_{i=1}^n [\mathbf{P}]_{ij} = \sum_{j=1}^n [\mathbf{P}]_{ij} = 1} \right. \right\}}
\|\mathbf{A} \mathbf{P} - \mathbf{B}\|_{F}^2
This Procrustes method requires the :math:`\mathbf{A}` and :math:`\mathbf{B}` matrices to
have the same shape, which is guaranteed with the default ``pad=True`` 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 are removed.
unpad_row : bool, optional
If True, zero rows (with values less than 1.0e-8) at the bottom 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}`.
Returns
-------
res : ProcrustesResult
The Procrustes result represented as a class:`utils.ProcrustesResult` object.
Notes
-----
The optimal :math:`n \times n` permutation matrix is obtained by,
.. math::
\mathbf{P}^{\text{opt}} =
\arg \underbrace{\text{min}}_{\left\{\mathbf{P} \left| {[\mathbf{P}]_{ij} \in \{0, 1\}
\atop \sum_{i=1}^n [\mathbf{P}]_{ij} = \sum_{j=1}^n [\mathbf{P}]_{ij} = 1} \right. \right\}}
\|\mathbf{A} \mathbf{P} - \mathbf{B}\|_{F}^2
= \underbrace{\text{max}}_{\left\{\mathbf{P} \left| {[\mathbf{P}]_{ij} \in \{0, 1\}
\atop \sum_{i=1}^n [\mathbf{P}]_{ij} = \sum_{j=1}^n [\mathbf{P}]_{ij} = 1} \right. \right\}}
\text{Tr}\left[\mathbf{P}^\dagger\mathbf{A}^\dagger\mathbf{B} \right]
The solution is found by relaxing the problem into a linear programming problem. The solution
to a linear programming problem is always at the boundary of the allowed region. So,
.. math::
\underbrace{\text{max}}_{\left\{\mathbf{P} \left| {[\mathbf{P}]_{ij} \in \{0, 1\}
\atop \sum_{i=1}^n [\mathbf{P}]_{ij} = \sum_{j=1}^n [\mathbf{P}]_{ij} = 1} \right. \right\}}
\text{Tr}\left[\mathbf{P}^\dagger\mathbf{A}^\dagger\mathbf{B} \right] =
\underbrace{\text{max}}_{\left\{\mathbf{P} \left| {[\mathbf{P}]_{ij} \geq 0
\atop \sum_{i=1}^n [\mathbf{P}]_{ij} = \sum_{j=1}^n [\mathbf{P}]_{ij} = 1} \right. \right\}}
\text{Tr}\left[\mathbf{P}^\dagger\left(\mathbf{A}^\dagger\mathbf{B}\right) \right]
This is a matching problem and can be solved by the Hungarian algorithm. The cost matrix is
defined as :math:`\mathbf{A}^\dagger\mathbf{B}` and the `scipy.optimize.linear_sum_assignment`
is used to solve for the permutation that maximizes the linear sum assignment problem.
"""
# 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")
# compute cost matrix C = A.T B
c = np.dot(new_a.T, new_b)
# compute permutation matrix using Hungarian algorithm
p = _compute_permutation_hungarian(c)
# compute one-sided permutation error
error = compute_error(new_a, new_b, p)
return ProcrustesResult(new_a=new_a, new_b=new_b, t=p, error=error)
def symmetric(a,
b,
pad=True,
translate=False,
scale=False,
unpad_col=False,
unpad_row=False,
check_finite=True,
weight=None,
lapack_driver="gesvd"):
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)