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scipy.linalg.

# orthogonal_procrustes

scipy.linalg.orthogonal_procrustes(A, B, check_finite=True)

  Compute the matrix solution of the orthogonal (or unitary) Procrustes problem.

  Given matrices A and B of the same shape, find an orthogonal (or unitary in the case of complex input) matrix R that most closely maps A to B using the algorithm given in [1].

  The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details. Note that calls with zero-size batches are unsupported and will raise a `ValueError`.

  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 `(A @ R) - B`, subject to `R.conj().T @ R = I`.

    scale float

      Sum of the singular values of `A.conj().T @ B`.

  Raises:
    ValueError

      If the input array shapes don’t match or if check_finite is True and the arrays contain Inf or NaN.

  Notes

  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.

  Array API Standard Support

  `orthogonal_procrustes` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ✅
  JAX | ⚠️ no JIT | ⚠️ no JIT
  Dask | ⛔ | n/a

  > See Support for the array API standard for more information.

  References

  [1]

  Peter H. Schonemann, “A generalized solution of the orthogonal Procrustes problem”, Psychometrica – Vol. 31, No. 1, March, 1966. DOI:10.1007/BF02289451

  Examples

      >>> import numpy as np
      >>> from scipy.linalg import orthogonal_procrustes
      >>> A = np.array([[ 2,  0,  1], [-2,  0,  0]])

  Flip the order of columns and check for the anti-diagonal mapping

      >>> R, sca = orthogonal_procrustes(A, np.fliplr(A))
      >>> R
      array([[-5.34384992e-17,  0.00000000e+00,  1.00000000e+00],
             [ 0.00000000e+00,  1.00000000e+00,  0.00000000e+00],
             [ 1.00000000e+00,  0.00000000e+00, -7.85941422e-17]])
      >>> sca
      9.0

  As an example of the unitary Procrustes problem, generate a random complex matrix `A`, a random unitary matrix `Q`, and their product `B`.

      >>> shape = (4, 4)
      >>> rng = np.random.default_rng()
      >>> A = rng.random(shape) + rng.random(shape)*1j
      >>> Q = rng.random(shape) + rng.random(shape)*1j
      >>> Q, _ = np.linalg.qr(Q)
      >>> B = A @ Q

  `orthogonal_procrustes` recovers the unitary matrix `Q` from `A` and `B`.

      >>> R, _ = orthogonal_procrustes(A, B)
      >>> np.allclose(R, Q)
      True