Source code for procrustes.symmetric

# -*- 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 <>
# --
"""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 =, 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 =, 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)