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"""
Solve the orthogonal Procrustes problem.
"""
from __future__ import division, print_function, absolute_import
import numpy as np
from .decomp_svd import svd
__all__ = ['orthogonal_procrustes']
def orthogonal_procrustes(A, B, check_finite=True):
"""
Compute the matrix solution of the orthogonal Procrustes problem.
Given matrices A and B of equal shape, find an orthogonal matrix R
that most closely maps A to B [1]_.
Note that unlike higher level Procrustes analyses of spatial data,
this function only uses orthogonal transformations like rotations
and reflections, and it does not use scaling or translation.
Parameters
----------
A : (M, N) array_like
Matrix to be mapped.
B : (M, N) array_like
Target matrix.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
R : (N, N) ndarray
The matrix solution of the orthogonal Procrustes problem.
Minimizes the Frobenius norm of dot(A, R) - B, subject to
dot(R.T, R) == I.
scale : float
Sum of the singular values of ``dot(A.T, B)``.
Raises
------
ValueError
If the input arrays are incompatibly shaped.
This may also be raised if matrix A or B contains an inf or nan
and check_finite is True, or if the matrix product AB contains
an inf or nan.
Notes
-----
.. versionadded:: 0.15.0
References
----------
.. [1] Peter H. Schonemann, "A generalized solution of the orthogonal
Procrustes problem", Psychometrica -- Vol. 31, No. 1, March, 1996.
"""
if check_finite:
A = np.asarray_chkfinite(A)
B = np.asarray_chkfinite(B)
else:
A = np.asanyarray(A)
B = np.asanyarray(B)
if A.ndim != 2:
raise ValueError('expected ndim to be 2, but observed %s' % A.ndim)
if A.shape != B.shape:
raise ValueError('the shapes of A and B differ (%s vs %s)' % (
A.shape, B.shape))
# Be clever with transposes, with the intention to save memory.
u, w, vt = svd(B.T.dot(A).T)
R = u.dot(vt)
scale = w.sum()
return R, scale
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