# procrustes.orthogonal

Orthogonal Procrustes Module.

procrustes.orthogonal.orthogonal(a: numpy.ndarray, b: numpy.ndarray, pad: bool = True, translate: bool = False, scale: bool = False, unpad_col: bool = False, unpad_row: bool = False, check_finite: bool = True, weight: Optional[numpy.ndarray] = None, lapack_driver: str = 'gesvd') [source]

Perform orthogonal Procrustes.

Given a matrix $$\mathbf{A}_{m \times n}$$ and a reference matrix $$\mathbf{B}_{m \times n}$$, find the orthogonal transformation matrix $$\mathbf{Q}_{n \times n}$$ that makes $$\mathbf{AQ}$$ as close as possible to $$\mathbf{B}$$. In other words,

$\underbrace{\min}_{\left\{\mathbf{Q} | \mathbf{Q}^{-1} = {\mathbf{Q}}^\dagger \right\}} \|\mathbf{A}\mathbf{Q} - \mathbf{B}\|_{F}^2$

This Procrustes method requires the $$\mathbf{A}$$ and $$\mathbf{B}$$ matrices to have the same shape, which is gauranteed with the default pad argument for any given $$\mathbf{A}$$ and $$\mathbf{B}$$ matrices. In preparing the $$\mathbf{A}$$ and $$\mathbf{B}$$ matrices, the (optional) order of operations is: 1) unpad zero rows/columns, 2) translate the matrices to the origin, 3) weight entries of $$\mathbf{A}$$, 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 $$\mathbf{A}$$ which is going to be transformed.

• b (ndarray) -- The 2D-array $$\mathbf{B}$$ representing the reference matrix.

• pad (bool, optional) -- Add zero rows (at the bottom) and/or columns (to the right-hand side) of matrices $$\mathbf{A}$$ and $$\mathbf{B}$$ 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., $$\text{Tr}\left[\mathbf{A}^\dagger\mathbf{A}\right] = 1$$ and $$\text{Tr}\left[\mathbf{B}^\dagger\mathbf{B}\right] = 1$$.

• unpad_col (bool, optional) -- If True, zero columns (with values less than 1.0e-8) on the right-hand side of the intial $$\mathbf{A}$$ and $$\mathbf{B}$$ matrices are removed.

• unpad_row (bool, optional) -- If True, zero rows (with values less than 1.0e-8) at the bottom of the intial $$\mathbf{A}$$ and $$\mathbf{B}$$ 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 $$\mathbf{A}$$. This defines the elements of the diagonal matrix $$\mathbf{W}$$ that is multiplied by $$\mathbf{A}$$ matrix, i.e., $$\mathbf{A} \rightarrow \mathbf{WA}$$.

• 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 -- The Procrustes result represented as a class:utils.ProcrustesResult object.

Return type

ProcrustesResult

Notes

The optimal orthogonal matrix is obtained by,

$\mathbf{Q}^{\text{opt}} = \arg \underbrace{\min}_{\left\{\mathbf{Q} \left| {\mathbf{Q}^{-1} = {\mathbf{Q}}^\dagger} \right. \right\}} \|\mathbf{A}\mathbf{Q} - \mathbf{B}\|_{F}^2 = \arg \underbrace{\max}_{\left\{\mathbf{Q} \left| {\mathbf{Q}^{-1} = {\mathbf{Q}}^\dagger} \right. \right\}} \text{Tr}\left[\mathbf{Q^\dagger}\mathbf{A^\dagger}\mathbf{B}\right]$

The solution is obtained using the singular value decomposition (SVD) of the $$\mathbf{A}^\dagger \mathbf{B}$$ matrix,

$\begin{split}\mathbf{A}^\dagger \mathbf{B} &= \tilde{\mathbf{U}} \tilde{\mathbf{\Sigma}} \tilde{\mathbf{V}}^{\dagger} \\ \mathbf{Q}^{\text{opt}} &= \tilde{\mathbf{U}} \tilde{\mathbf{V}}^{\dagger}\end{split}$

The singular values are always listed in decreasing order, with the smallest singular value in the bottom-right-hand corner of $$\tilde{\mathbf{\Sigma}}$$.

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

procrustes.orthogonal.orthogonal_2sided(a: numpy.ndarray, b: numpy.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[numpy.ndarray] = None, lapack_driver: str = 'gesvd') [source]

Perform two-sided orthogonal Procrustes with one- or two-transformations.

Two Transformations: Given a matrix $$\mathbf{A}_{m \times n}$$ and a reference matrix $$\mathbf{B}_{m \times n}$$, find two $$n \times n$$ orthogonal transformation matrices $$\mathbf{Q}_1^\dagger$$ and $$\mathbf{Q}_2$$ that makes $$\mathbf{Q}_1^\dagger\mathbf{A}\mathbf{Q}_2$$ as close as possible to $$\mathbf{B}$$. In other words,

$\underbrace{\text{min}}_{\left\{ {\mathbf{Q}_1 \atop \mathbf{Q}_2} \left| {\mathbf{Q}_1^{-1} = \mathbf{Q}_1^\dagger \atop \mathbf{Q}_2^{-1} = \mathbf{Q}_2^\dagger} \right. \right\}} \|\mathbf{Q}_1^\dagger \mathbf{A} \mathbf{Q}_2 - \mathbf{B}\|_{F}^2$

Single Transformations: Given a symmetric matrix $$\mathbf{A}_{n \times n}$$ and a reference $$\mathbf{B}_{n \times n}$$, find one orthogonal transformation matrix $$\mathbf{Q}_{n \times n}$$ that makes $$\mathbf{A}$$ as close as possible to $$\mathbf{B}$$. In other words,

$\underbrace{\min}_{\left\{\mathbf{Q} | \mathbf{Q}^{-1} = {\mathbf{Q}}^\dagger \right\}} \|\mathbf{Q}^\dagger\mathbf{A}\mathbf{Q} - \mathbf{B}\|_{F}^2$

This Procrustes method requires the $$\mathbf{A}$$ and $$\mathbf{B}$$ matrices to have the same shape, which is gauranteed with the default pad argument for any given $$\mathbf{A}$$ and $$\mathbf{B}$$ matrices. In preparing the $$\mathbf{A}$$ and $$\mathbf{B}$$ matrices, the (optional) order of operations is: 1) unpad zero rows/columns, 2) translate the matrices to the origin, 3) weight entries of $$\mathbf{A}$$, 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 $$\mathbf{A}$$ which is going to be transformed.

• b (ndarray) -- The 2D-array $$\mathbf{B}$$ representing the reference matrix.

• single (bool, optional) -- If True, single transformation is used (i.e., $$\mathbf{Q}_1=\mathbf{Q}_2=\mathbf{Q}$$), otherwise, two transformations are used.

• pad (bool, optional) -- Add zero rows (at the bottom) and/or columns (to the right-hand side) of matrices $$\mathbf{A}$$ and $$\mathbf{B}$$ 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., $$\text{Tr}\left[\mathbf{A}^\dagger\mathbf{A}\right] = 1$$ and $$\text{Tr}\left[\mathbf{B}^\dagger\mathbf{B}\right] = 1$$.

• unpad_col (bool, optional) -- If True, zero columns (with values less than 1.0e-8) on the right-hand side of the intial $$\mathbf{A}$$ and $$\mathbf{B}$$ matrices are removed.

• unpad_row (bool, optional) -- If True, zero rows (with values less than 1.0e-8) at the bottom of the intial $$\mathbf{A}$$ and $$\mathbf{B}$$ 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 $$\mathbf{A}$$. This defines the elements of the diagonal matrix $$\mathbf{W}$$ that is multiplied by $$\mathbf{A}$$ matrix, i.e., $$\mathbf{A} \rightarrow \mathbf{WA}$$.

• 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 -- The Procrustes result represented as a class:utils.ProcrustesResult object.

Return type

ProcrustesResult

Notes

Two-Sided Orthogonal Procrustes with Two Transformations: The optimal orthogonal transformations are obtained by:

$\mathbf{Q}_{1}^{\text{opt}}, \mathbf{Q}_{2}^{\text{opt}} = \arg \underbrace{\text{min}}_{\left\{ {\mathbf{Q}_1 \atop \mathbf{Q}_2} \left| {\mathbf{Q}_1^{-1} = \mathbf{Q}_1^\dagger \atop \mathbf{Q}_2^{-1} = \mathbf{Q}_2^\dagger} \right. \right\}} \|\mathbf{Q}_1^\dagger \mathbf{A} \mathbf{Q}_2 - \mathbf{B}\|_{F}^2 = \arg \underbrace{\text{max}}_{\left\{ {\mathbf{Q}_1 \atop \mathbf{Q}_2} \left| {\mathbf{Q}_1^{-1} = \mathbf{Q}_1^\dagger \atop \mathbf{Q}_2^{-1} = \mathbf{Q}_2^\dagger} \right. \right\}} \text{Tr}\left[\mathbf{Q}_2^\dagger\mathbf{A}^\dagger\mathbf{Q}_1\mathbf{B} \right]$

This is solved by taking the singular value decomposition (SVD) of $$\mathbf{A}$$ and $$\mathbf{B}$$,

$\begin{split}\mathbf{A} = \mathbf{U}_A \mathbf{\Sigma}_A \mathbf{V}_A^\dagger \\ \mathbf{B} = \mathbf{U}_B \mathbf{\Sigma}_B \mathbf{V}_B^\dagger\end{split}$

Then the two optimal orthogonal matrices are given by,

$\begin{split}\mathbf{Q}_1^{\text{opt}} = \mathbf{U}_A \mathbf{U}_B^\dagger \\ \mathbf{Q}_2^{\text{opt}} = \mathbf{V}_A \mathbf{V}_B^\dagger\end{split}$

Two-Sided Orthogonal Procrustes with Single-Transformation: The optimal orthogonal transformation is obtained by:

$\mathbf{Q}^{\text{opt}} = \arg \underbrace{\min}_{\left\{\mathbf{Q} | \mathbf{Q}^{-1} = {\mathbf{Q}}^\dagger \right\}} \|\mathbf{Q}^\dagger\mathbf{A}\mathbf{Q} - \mathbf{B}\|_{F}^2 = \arg \underbrace{\text{max}}_{\left\{\mathbf{Q} | \mathbf{Q}^{-1} = {\mathbf{Q}}^\dagger\right\}} \text{Tr}\left[\mathbf{Q}^\dagger\mathbf{A}^\dagger\mathbf{Q}\mathbf{B} \right]$

Using the singular value decomposition (SVD) of $$\mathbf{A}$$ and $$\mathbf{B}$$,

$\begin{split}\mathbf{A} = \mathbf{U}_A \mathbf{\Lambda}_A \mathbf{U}_A^\dagger \\ \mathbf{B} = \mathbf{U}_B \mathbf{\Lambda}_B \mathbf{U}_B^\dagger\end{split}$

The optimal orthogonal matrix $$\mathbf{Q}^\text{opt}$$ is obtained through,

$\mathbf{Q}^\text{opt} = \mathbf{U}_A \mathbf{S} \mathbf{U}_B^\dagger$

where $$\mathbf{S}$$ is a diagonal matrix with $$\pm{1}$$ elements,

$\begin{split}\mathbf{S} = \begin{bmatrix} { \pm 1} & 0 &\cdots &0 \\ 0 &{ \pm 1} &\ddots &\vdots \\ \vdots &\ddots &\ddots &0\\ 0 &\cdots &0 &{ \pm 1} \end{bmatrix}\end{split}$

The matrix $$\mathbf{S}$$ 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.]])