Source code for graspologic.align.orthogonal_procrustes

# Copyright (c) Microsoft Corporation and contributors.

import numpy as np
from scipy.linalg import orthogonal_procrustes

from .base import BaseAlign

[docs]class OrthogonalProcrustes(BaseAlign):
"""
Computes the matrix solution of the classical orthogonal Procrustes [1]_
problem, which is that given two matrices X and Y of equal shape
(n, d), find an orthogonal matrix that most closely maps X to
Y. Subsequently, uses that matrix to transform either the original X,
or a different dataset in the same space.

Note that when used to match two datasets, this method unlike
:class:~graspologic.align.SeedlessProcrustes, not only requires that the
datasets have the same number of entries, but also that there is some
correspondence between the entries. In graph embeddings, this usually
corresponds to the assumption that the vertex :math:i in graph X has the same
latent position as the vertex :math:i in graph Y.

Attributes
----------
Q_ : array, size (d, d)
Final orthogonal matrix, used to modify X.

score_ : float
Final value of the objective function: :math:|| X Q - Y ||_F
Lower means the datasets have been matched together better.

References
----------

.. [1] https://en.wikipedia.org/wiki/Orthogonal_Procrustes_problem

.. [2] Peter H. Schonemann, "A generalized solution of the orthogonal
Procrustes problem", Psychometrica -- Vol. 31, No. 1, March, 1996.

Notes
-----
Formally, minimizes :math:|| X Q - Y ||_F, which has a closed form
solution, whenever :math:Q is constrained to be an orthogonal matrix,
that is a matrix that satisfies :math:Q^T Q = Q Q^T = I. For the more
details, including the proof of the closed-form solution see [1]_.

Implementation-wise, this class is a wrapper of the
:func:scipy.linalg.orthogonal_procrustes, which itself uses an algorithm
described in find the optimal solution algorithm [2]_.

"""

def __init__(
self,
):

super().__init__()

[docs]    def fit(self, X, Y):
"""
Uses the two datasets to learn the matrix :attr:~graspologic.align.OrthogonalProcrustes.Q_ that aligns the
first dataset with the second.

Parameters
----------
X : np.ndarray, shape (n, d)
Dataset to be mapped to Y, must have the same shape as Y.

Y : np.ndarray, shape (m, d)
Target dataset, must have the same shape as X.

Returns
-------
self : returns an instance of self

"""
X, Y = self._check_datasets(X, Y)

_, d = X.shape
if X.shape[0] != Y.shape[0]:
msg = (
"Two datasets have different number of entries! "
"OrthogonalProcrustes assumes that entries of the two "
"datasets are matched. consider using SeedlessProcrustes "
)
raise ValueError(msg)

_, d = X.shape
self.Q_, _ = orthogonal_procrustes(X, Y)
self.score_ = np.linalg.norm(X @ self.Q_ - Y, ord="fro")
return self

[docs]    def fit_transform(self, X, Y):
"""
Uses the two datasets to learn the matrix :attr:~graspologic.align.OrthogonalProcrustes.Q_ that aligns the
first dataset with the second. Then, transforms the first dataset X
using the learned matrix :attr:~graspologic.align.OrthogonalProcrustes.Q_.

Parameters
----------
X : np.ndarray, shape (n, d)
Dataset to be mapped to Y, must have the same shape as Y.

Y : np.ndarray, shape (m, d)
Target dataset, must have the same shape as X.

Returns
-------
X_prime : np.ndarray, shape (n, d)
First dataset of vectors, aligned to second. Equal to
X @ :attr:~graspologic.align.BaseAlign.Q_.
"""
return super().fit_transform(X, Y)