Source code for procrustes.generic

# -*- 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 <>
# --
"""Generic Procrustes Module."""

from typing import Optional

import numpy as np
from procrustes.utils import compute_error, ProcrustesResult, setup_input_arrays
from scipy.linalg import pinv, pinv2

[docs]def generic( 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, use_svd: bool = False, ) -> ProcrustesResult: r"""Perform generic one-sided Procrustes. Given matrix :math:`\mathbf{A}_{m \times n}` and a reference matrix :math:`\mathbf{B}_{m \times n}`, find the transformation matrix :math:`\mathbf{T}_{n \times n}` that makes :math:`\mathbf{AT}` as close as possible to :math:`\mathbf{B}`. In other words, .. math:: \underbrace{\text{min}}_{\mathbf{T}} \quad \|\mathbf{A} \mathbf{T} - \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}`. use_svd : bool, optional If True, the (Moore-Penrose) pseudo-inverse is computed by singular-value decomposition (SVD) including all 'large' singular values (using `scipy.linalg.pinv2`). If False, the the (Moore-Penrose) pseudo-inverse is computed by least-squares solver (using `scipy.linalg.pinv`). The least-squares implementation is less efficient, but more robust, than the SVD implementation. Returns ------- res : ProcrustesResult The Procrustes result represented as a class:`utils.ProcrustesResult` object. Notes ----- The optimal transformation matrix is obtained by solving the least-squares equations, .. math:: \mathbf{X}_\text{opt} = {(\mathbf{A}^{\top}\mathbf{A})}^{-1} \mathbf{A}^{\top} \mathbf{B} If :math:`m < n`, the transformation matrix :math:`\mathbf{T}_\text{opt}` is not unique, because the system of equations is underdetermined (i.e., there are fewer equations than unknowns). """ if not isinstance(use_svd, bool): raise TypeError(f"The use_svd parameter {type(use_svd)} should be type bool.") # check inputs new_a, new_b = setup_input_arrays( a, b, unpad_col, unpad_row, pad, translate, scale, check_finite, weight, ) # compute the generic solution if use_svd: # Use the singular value decomposition, much faster but less robust. a_inv = pinv2(, new_a)) else: # Uses the least-squared method. a_inv = pinv(, new_a)) array_x = np.linalg.multi_dot([a_inv, new_a.T, new_b]) # compute one-sided error e_opt = compute_error(new_a, new_b, array_x) return ProcrustesResult(error=e_opt, new_a=new_a, new_b=new_b, t=array_x, s=None)