Source code for procrustes.rotational

# -*- 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
# 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 <>
# --
"""Rotational-Orthogonal Procrustes Module."""

from typing import Optional

import numpy as np
from procrustes.utils import compute_error, ProcrustesResult, setup_input_arrays
import scipy

[docs]def rotational( 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 rotational Procrustes. Given a matrix :math:`\mathbf{A}_{m \times n}` and a reference matrix :math:`\mathbf{B}_{m \times n}`, find the rotational transformation matrix :math:`\mathbf{R}_{n \times n}` that makes :math:`\mathbf{A}` as close as possible to :math:`\mathbf{B}`. In other words, .. math:: \underbrace{\min}_{\left\{\mathbf{R} \left| {\mathbf{R}^{-1} = {\mathbf{R}}^\dagger \atop \left| \mathbf{R} \right| = 1} \right. \right\}} \|\mathbf{A}\mathbf{R} - \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 rotational matrix is obtained by, .. math:: \mathbf{R}_{\text{opt}} = \arg \underbrace{\min}_{\left\{\mathbf{R} \left| {\mathbf{R}^{-1} = {\mathbf{R}}^\dagger \atop \left| \mathbf{R} \right| = 1} \right. \right\}} \|\mathbf{A}\mathbf{R} - \mathbf{B}\|_{F}^2 = \arg \underbrace{\max}_{\left\{\mathbf{R} \left| {\mathbf{R}^{-1} = {\mathbf{R}}^\dagger \atop \left| \mathbf{R} \right| = 1} \right. \right\}} \text{Tr}\left[\mathbf{R}^\dagger {\mathbf{A}}^\dagger \mathbf{B} \right] The solution is obtained by taking 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{R}_{\text{opt}} &= \tilde{\mathbf{U}} \tilde{\mathbf{S}} \tilde{\mathbf{V}}^{\dagger} where :math:`\tilde{\mathbf{S}}_{n \times m}` is almost an identity matrix, .. math:: \tilde{\mathbf{S}}_{m \times n} \equiv \begin{bmatrix} 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \ddots & \vdots &0 \\ 0 & \ddots &\ddots & 0 &\vdots \\ \vdots&0 & 0 & 1 &0 \\ 0 & 0 & 0 \cdots &0 &\operatorname{sgn} \left(\left|\mathbf{U}\mathbf{V}^\dagger\right|\right) \end{bmatrix} in which the smallest singular value is replaced by .. math:: \operatorname{sgn} \left(\left|\tilde{\mathbf{U}} \tilde{\mathbf{V}}^\dagger\right|\right) = \begin{cases} +1 \qquad \left|\tilde{\mathbf{U}} \tilde{\mathbf{V}}^\dagger\right| \geq 0 \\ -1 \qquad \left|\tilde{\mathbf{U}} \tilde{\mathbf{V}}^\dagger\right| < 0 \end{cases} Examples -------- >>> import numpy as np >>> array_a = np.array([[1.5, 7.4], [8.5, 4.5]]) >>> array_b = np.array([[6.29325035, 4.17193001, 0., 0,], ... [9.19238816, -2.82842712, 0., 0.], ... [0., 0., 0., 0.]]) >>> res = rotational(array_a,array_b,translate=False,scale=False) >>> res.t # rotational transformation array([[ 0.70710678, -0.70710678], [ 0.70710678, 0.70710678]]) >>> res.error # one-sided Procrustes error 1.483808210011695e-17 """ # 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." ) # compute SVD of A.T * B u, _, vt = scipy.linalg.svd(, new_b), lapack_driver=lapack_driver) # construct S: an identity matrix with the smallest singular value replaced by sgn(|U*V^t|) s = np.eye(new_a.shape[1]) s[-1, -1] = np.sign(np.linalg.det(, vt))) # compute optimal rotational transformation r_opt =, s), vt) # compute one-sided error error = compute_error(new_a, new_b, r_opt) return ProcrustesResult(error=error, new_a=new_a, new_b=new_b, t=r_opt, s=None)