File Content: _procrustes.py """ 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 Submit