497 lines
14 KiB
Python
497 lines
14 KiB
Python
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"""SVD decomposition functions."""
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from __future__ import division, print_function, absolute_import
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import numpy
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from numpy import zeros, r_, diag, dot, arccos, arcsin, where, clip
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# Local imports.
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from .misc import LinAlgError, _datacopied
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from .lapack import get_lapack_funcs, _compute_lwork
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from .decomp import _asarray_validated
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from scipy._lib.six import string_types
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__all__ = ['svd', 'svdvals', 'diagsvd', 'orth', 'subspace_angles', 'null_space']
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def svd(a, full_matrices=True, compute_uv=True, overwrite_a=False,
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check_finite=True, lapack_driver='gesdd'):
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"""
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Singular Value Decomposition.
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Factorizes the matrix `a` into two unitary matrices ``U`` and ``Vh``, and
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a 1-D array ``s`` of singular values (real, non-negative) such that
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``a == U @ S @ Vh``, where ``S`` is a suitably shaped matrix of zeros with
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main diagonal ``s``.
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Parameters
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----------
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a : (M, N) array_like
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Matrix to decompose.
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full_matrices : bool, optional
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If True (default), `U` and `Vh` are of shape ``(M, M)``, ``(N, N)``.
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If False, the shapes are ``(M, K)`` and ``(K, N)``, where
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``K = min(M, N)``.
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compute_uv : bool, optional
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Whether to compute also ``U`` and ``Vh`` in addition to ``s``.
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Default is True.
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overwrite_a : bool, optional
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Whether to overwrite `a`; may improve performance.
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Default is False.
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check_finite : bool, optional
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Whether to check that the input matrix contains only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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lapack_driver : {'gesdd', 'gesvd'}, optional
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Whether to use the more efficient divide-and-conquer approach
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(``'gesdd'``) or general rectangular approach (``'gesvd'``)
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to compute the SVD. MATLAB and Octave use the ``'gesvd'`` approach.
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Default is ``'gesdd'``.
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.. versionadded:: 0.18
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Returns
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-------
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U : ndarray
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Unitary matrix having left singular vectors as columns.
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Of shape ``(M, M)`` or ``(M, K)``, depending on `full_matrices`.
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s : ndarray
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The singular values, sorted in non-increasing order.
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Of shape (K,), with ``K = min(M, N)``.
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Vh : ndarray
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Unitary matrix having right singular vectors as rows.
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Of shape ``(N, N)`` or ``(K, N)`` depending on `full_matrices`.
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For ``compute_uv=False``, only ``s`` is returned.
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Raises
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------
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LinAlgError
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If SVD computation does not converge.
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See also
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--------
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svdvals : Compute singular values of a matrix.
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diagsvd : Construct the Sigma matrix, given the vector s.
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Examples
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--------
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>>> from scipy import linalg
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>>> m, n = 9, 6
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>>> a = np.random.randn(m, n) + 1.j*np.random.randn(m, n)
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>>> U, s, Vh = linalg.svd(a)
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>>> U.shape, s.shape, Vh.shape
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((9, 9), (6,), (6, 6))
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Reconstruct the original matrix from the decomposition:
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>>> sigma = np.zeros((m, n))
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>>> for i in range(min(m, n)):
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... sigma[i, i] = s[i]
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>>> a1 = np.dot(U, np.dot(sigma, Vh))
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>>> np.allclose(a, a1)
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True
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Alternatively, use ``full_matrices=False`` (notice that the shape of
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``U`` is then ``(m, n)`` instead of ``(m, m)``):
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>>> U, s, Vh = linalg.svd(a, full_matrices=False)
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>>> U.shape, s.shape, Vh.shape
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((9, 6), (6,), (6, 6))
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>>> S = np.diag(s)
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>>> np.allclose(a, np.dot(U, np.dot(S, Vh)))
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True
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>>> s2 = linalg.svd(a, compute_uv=False)
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>>> np.allclose(s, s2)
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True
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"""
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a1 = _asarray_validated(a, check_finite=check_finite)
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if len(a1.shape) != 2:
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raise ValueError('expected matrix')
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m, n = a1.shape
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overwrite_a = overwrite_a or (_datacopied(a1, a))
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if not isinstance(lapack_driver, string_types):
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raise TypeError('lapack_driver must be a string')
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if lapack_driver not in ('gesdd', 'gesvd'):
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raise ValueError('lapack_driver must be "gesdd" or "gesvd", not "%s"'
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% (lapack_driver,))
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funcs = (lapack_driver, lapack_driver + '_lwork')
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gesXd, gesXd_lwork = get_lapack_funcs(funcs, (a1,))
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# compute optimal lwork
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lwork = _compute_lwork(gesXd_lwork, a1.shape[0], a1.shape[1],
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compute_uv=compute_uv, full_matrices=full_matrices)
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# perform decomposition
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u, s, v, info = gesXd(a1, compute_uv=compute_uv, lwork=lwork,
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full_matrices=full_matrices, overwrite_a=overwrite_a)
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if info > 0:
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raise LinAlgError("SVD did not converge")
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if info < 0:
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raise ValueError('illegal value in %d-th argument of internal gesdd'
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% -info)
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if compute_uv:
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return u, s, v
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else:
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return s
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def svdvals(a, overwrite_a=False, check_finite=True):
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"""
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Compute singular values of a matrix.
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Parameters
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----------
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a : (M, N) array_like
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Matrix to decompose.
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overwrite_a : bool, optional
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Whether to overwrite `a`; may improve performance.
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Default is False.
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check_finite : bool, optional
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Whether to check that the input matrix contains only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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Returns
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-------
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s : (min(M, N),) ndarray
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The singular values, sorted in decreasing order.
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Raises
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------
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LinAlgError
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If SVD computation does not converge.
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Notes
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-----
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``svdvals(a)`` only differs from ``svd(a, compute_uv=False)`` by its
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handling of the edge case of empty ``a``, where it returns an
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empty sequence:
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>>> a = np.empty((0, 2))
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>>> from scipy.linalg import svdvals
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>>> svdvals(a)
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array([], dtype=float64)
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See Also
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--------
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svd : Compute the full singular value decomposition of a matrix.
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diagsvd : Construct the Sigma matrix, given the vector s.
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Examples
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--------
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>>> from scipy.linalg import svdvals
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>>> m = np.array([[1.0, 0.0],
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... [2.0, 3.0],
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... [1.0, 1.0],
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... [0.0, 2.0],
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... [1.0, 0.0]])
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>>> svdvals(m)
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array([ 4.28091555, 1.63516424])
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We can verify the maximum singular value of `m` by computing the maximum
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length of `m.dot(u)` over all the unit vectors `u` in the (x,y) plane.
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We approximate "all" the unit vectors with a large sample. Because
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of linearity, we only need the unit vectors with angles in [0, pi].
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>>> t = np.linspace(0, np.pi, 2000)
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>>> u = np.array([np.cos(t), np.sin(t)])
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>>> np.linalg.norm(m.dot(u), axis=0).max()
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4.2809152422538475
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`p` is a projection matrix with rank 1. With exact arithmetic,
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its singular values would be [1, 0, 0, 0].
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>>> v = np.array([0.1, 0.3, 0.9, 0.3])
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>>> p = np.outer(v, v)
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>>> svdvals(p)
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array([ 1.00000000e+00, 2.02021698e-17, 1.56692500e-17,
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8.15115104e-34])
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The singular values of an orthogonal matrix are all 1. Here we
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create a random orthogonal matrix by using the `rvs()` method of
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`scipy.stats.ortho_group`.
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>>> from scipy.stats import ortho_group
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>>> np.random.seed(123)
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>>> orth = ortho_group.rvs(4)
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>>> svdvals(orth)
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array([ 1., 1., 1., 1.])
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"""
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a = _asarray_validated(a, check_finite=check_finite)
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if a.size:
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return svd(a, compute_uv=0, overwrite_a=overwrite_a,
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check_finite=False)
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elif len(a.shape) != 2:
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raise ValueError('expected matrix')
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else:
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return numpy.empty(0)
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def diagsvd(s, M, N):
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"""
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Construct the sigma matrix in SVD from singular values and size M, N.
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Parameters
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----------
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s : (M,) or (N,) array_like
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Singular values
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M : int
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Size of the matrix whose singular values are `s`.
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N : int
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Size of the matrix whose singular values are `s`.
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Returns
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-------
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S : (M, N) ndarray
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The S-matrix in the singular value decomposition
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See Also
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--------
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svd : Singular value decomposition of a matrix
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svdvals : Compute singular values of a matrix.
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Examples
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--------
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>>> from scipy.linalg import diagsvd
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>>> vals = np.array([1, 2, 3]) # The array representing the computed svd
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>>> diagsvd(vals, 3, 4)
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array([[1, 0, 0, 0],
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[0, 2, 0, 0],
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[0, 0, 3, 0]])
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>>> diagsvd(vals, 4, 3)
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array([[1, 0, 0],
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[0, 2, 0],
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[0, 0, 3],
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[0, 0, 0]])
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"""
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part = diag(s)
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typ = part.dtype.char
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MorN = len(s)
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if MorN == M:
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return r_['-1', part, zeros((M, N-M), typ)]
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elif MorN == N:
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return r_[part, zeros((M-N, N), typ)]
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else:
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raise ValueError("Length of s must be M or N.")
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# Orthonormal decomposition
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def orth(A, rcond=None):
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"""
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Construct an orthonormal basis for the range of A using SVD
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Parameters
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----------
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A : (M, N) array_like
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Input array
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rcond : float, optional
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Relative condition number. Singular values ``s`` smaller than
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``rcond * max(s)`` are considered zero.
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Default: floating point eps * max(M,N).
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Returns
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-------
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Q : (M, K) ndarray
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Orthonormal basis for the range of A.
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K = effective rank of A, as determined by rcond
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See also
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--------
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svd : Singular value decomposition of a matrix
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null_space : Matrix null space
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Examples
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--------
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>>> from scipy.linalg import orth
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>>> A = np.array([[2, 0, 0], [0, 5, 0]]) # rank 2 array
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>>> orth(A)
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array([[0., 1.],
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[1., 0.]])
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>>> orth(A.T)
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array([[0., 1.],
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[1., 0.],
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[0., 0.]])
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"""
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u, s, vh = svd(A, full_matrices=False)
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M, N = u.shape[0], vh.shape[1]
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if rcond is None:
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rcond = numpy.finfo(s.dtype).eps * max(M, N)
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tol = numpy.amax(s) * rcond
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num = numpy.sum(s > tol, dtype=int)
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Q = u[:, :num]
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return Q
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def null_space(A, rcond=None):
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"""
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Construct an orthonormal basis for the null space of A using SVD
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Parameters
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----------
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A : (M, N) array_like
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Input array
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rcond : float, optional
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Relative condition number. Singular values ``s`` smaller than
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``rcond * max(s)`` are considered zero.
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Default: floating point eps * max(M,N).
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Returns
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-------
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Z : (N, K) ndarray
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Orthonormal basis for the null space of A.
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K = dimension of effective null space, as determined by rcond
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See also
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--------
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svd : Singular value decomposition of a matrix
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orth : Matrix range
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Examples
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--------
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One-dimensional null space:
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>>> from scipy.linalg import null_space
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>>> A = np.array([[1, 1], [1, 1]])
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>>> ns = null_space(A)
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>>> ns * np.sign(ns[0,0]) # Remove the sign ambiguity of the vector
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array([[ 0.70710678],
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[-0.70710678]])
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Two-dimensional null space:
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>>> B = np.random.rand(3, 5)
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>>> Z = null_space(B)
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>>> Z.shape
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(5, 2)
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>>> np.allclose(B.dot(Z), 0)
|
||
|
|
True
|
||
|
|
|
||
|
|
The basis vectors are orthonormal (up to rounding error):
|
||
|
|
|
||
|
|
>>> Z.T.dot(Z)
|
||
|
|
array([[ 1.00000000e+00, 6.92087741e-17],
|
||
|
|
[ 6.92087741e-17, 1.00000000e+00]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
u, s, vh = svd(A, full_matrices=True)
|
||
|
|
M, N = u.shape[0], vh.shape[1]
|
||
|
|
if rcond is None:
|
||
|
|
rcond = numpy.finfo(s.dtype).eps * max(M, N)
|
||
|
|
tol = numpy.amax(s) * rcond
|
||
|
|
num = numpy.sum(s > tol, dtype=int)
|
||
|
|
Q = vh[num:,:].T.conj()
|
||
|
|
return Q
|
||
|
|
|
||
|
|
|
||
|
|
def subspace_angles(A, B):
|
||
|
|
r"""
|
||
|
|
Compute the subspace angles between two matrices.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
A : (M, N) array_like
|
||
|
|
The first input array.
|
||
|
|
B : (M, K) array_like
|
||
|
|
The second input array.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
angles : ndarray, shape (min(N, K),)
|
||
|
|
The subspace angles between the column spaces of `A` and `B` in
|
||
|
|
descending order.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
orth
|
||
|
|
svd
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This computes the subspace angles according to the formula
|
||
|
|
provided in [1]_. For equivalence with MATLAB and Octave behavior,
|
||
|
|
use ``angles[0]``.
|
||
|
|
|
||
|
|
.. versionadded:: 1.0
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Knyazev A, Argentati M (2002) Principal Angles between Subspaces
|
||
|
|
in an A-Based Scalar Product: Algorithms and Perturbation
|
||
|
|
Estimates. SIAM J. Sci. Comput. 23:2008-2040.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
A Hadamard matrix, which has orthogonal columns, so we expect that
|
||
|
|
the suspace angle to be :math:`\frac{\pi}{2}`:
|
||
|
|
|
||
|
|
>>> from scipy.linalg import hadamard, subspace_angles
|
||
|
|
>>> H = hadamard(4)
|
||
|
|
>>> print(H)
|
||
|
|
[[ 1 1 1 1]
|
||
|
|
[ 1 -1 1 -1]
|
||
|
|
[ 1 1 -1 -1]
|
||
|
|
[ 1 -1 -1 1]]
|
||
|
|
>>> np.rad2deg(subspace_angles(H[:, :2], H[:, 2:]))
|
||
|
|
array([ 90., 90.])
|
||
|
|
|
||
|
|
And the subspace angle of a matrix to itself should be zero:
|
||
|
|
|
||
|
|
>>> subspace_angles(H[:, :2], H[:, :2]) <= 2 * np.finfo(float).eps
|
||
|
|
array([ True, True], dtype=bool)
|
||
|
|
|
||
|
|
The angles between non-orthogonal subspaces are in between these extremes:
|
||
|
|
|
||
|
|
>>> x = np.random.RandomState(0).randn(4, 3)
|
||
|
|
>>> np.rad2deg(subspace_angles(x[:, :2], x[:, [2]]))
|
||
|
|
array([ 55.832])
|
||
|
|
"""
|
||
|
|
# Steps here omit the U and V calculation steps from the paper
|
||
|
|
|
||
|
|
# 1. Compute orthonormal bases of column-spaces
|
||
|
|
A = _asarray_validated(A, check_finite=True)
|
||
|
|
if len(A.shape) != 2:
|
||
|
|
raise ValueError('expected 2D array, got shape %s' % (A.shape,))
|
||
|
|
QA = orth(A)
|
||
|
|
del A
|
||
|
|
|
||
|
|
B = _asarray_validated(B, check_finite=True)
|
||
|
|
if len(B.shape) != 2:
|
||
|
|
raise ValueError('expected 2D array, got shape %s' % (B.shape,))
|
||
|
|
if len(B) != len(QA):
|
||
|
|
raise ValueError('A and B must have the same number of rows, got '
|
||
|
|
'%s and %s' % (QA.shape[0], B.shape[0]))
|
||
|
|
QB = orth(B)
|
||
|
|
del B
|
||
|
|
|
||
|
|
# 2. Compute SVD for cosine
|
||
|
|
QA_H_QB = dot(QA.T.conj(), QB)
|
||
|
|
sigma = svdvals(QA_H_QB)
|
||
|
|
|
||
|
|
# 3. Compute matrix B
|
||
|
|
if QA.shape[1] >= QB.shape[1]:
|
||
|
|
B = QB - dot(QA, QA_H_QB)
|
||
|
|
else:
|
||
|
|
B = QA - dot(QB, QA_H_QB.T.conj())
|
||
|
|
del QA, QB, QA_H_QB
|
||
|
|
|
||
|
|
# 4. Compute SVD for sine
|
||
|
|
mask = sigma ** 2 >= 0.5
|
||
|
|
if mask.any():
|
||
|
|
mu_arcsin = arcsin(clip(svdvals(B, overwrite_a=True), -1., 1.))
|
||
|
|
else:
|
||
|
|
mu_arcsin = 0.
|
||
|
|
|
||
|
|
# 5. Compute the principal angles
|
||
|
|
# with reverse ordering of sigma because smallest sigma belongs to largest
|
||
|
|
# angle theta
|
||
|
|
theta = where(mask, mu_arcsin, arccos(clip(sigma[::-1], -1., 1.)))
|
||
|
|
return theta
|