def kpsvd(A, bshape, cshape):
r"""Compute the Kronecker Product SVD.
Given matrix :math:`A`, find matrices :math:`B_i` and :math:`C_i`,
of the specified sizes, such that
.. math::
A = \sum_i \sigma_i B_i \otimes C_i
and :math:`\sum_i^n \sigma_i B_i \otimes C_i` is the best :math:`n`
term approximation to :math:`A` :cite:`loan-2000-ubiquitous`.
Parameters
----------
A : array_like
2D input array
bshape : tuple (Mb, Nb)
The desired shape of arrays :math:`B_i`
cshape : tuple (Mc, Nc)
The desired shape of arrays :math:`C_i`
Returns
-------
S : ndarray
1D array of :math:`\sigma_i` values
B : ndarray
3D array of :math:`B_i` matrices with index :math:`i` on the last
axis
C : ndarray
3D array of :math:`C_i` matrices with index :math:`i` on the last
axis
"""
ashape = A.shape
if ashape[0] != bshape[0] * cshape[0] or \
ashape[1] != bshape[1] * cshape[1]:
raise ValueError("Shape of A is not compatible with bshape and cshape")
atshape = (bshape[0] * bshape[1], cshape[0] * cshape[1])
Atilde = subsample_array(A, cshape).transpose().reshape(atshape)
U, S, Vt = np.linalg.svd(Atilde, full_matrices=False)
B = U.reshape(bshape + (U.shape[1], ), order='F')
C = Vt.T.reshape(cshape + (Vt.shape[0], ), order='F')
return S, B, C
def nkp(A, bshape, cshape):
r"""Solve the Nearest Kronecker Product problem.
Given matrix :math:`A`, find matrices :math:`B` and :math:`C`, of the
specified sizes, such that :math:`B` and :math:`C` solve the problem
:cite:`loan-2000-ubiquitous`
.. math::
\mathrm{argmin}_{B, C} \| A - B \otimes C \| \;.
Parameters
----------
A : array_like
2D input array
bshape : tuple (Mb, Nb)
The desired shape of returned array :math:`B`
cshape : tuple (Mc, Nc)
The desired shape of returned array :math:`C`
Returns
-------
B : ndarray
2D output array :math:`B`
C : ndarray
2D output array :math:`C`
"""
ashape = A.shape
if ashape[0] != bshape[0] * cshape[0] or \
ashape[1] != bshape[1] * cshape[1]:
raise ValueError("Shape of A is not compatible with bshape and cshape")
atshape = (bshape[0] * bshape[1], cshape[0] * cshape[1])
Atilde = subsample_array(A, cshape).transpose().reshape(atshape)
U, S, Vt = np.linalg.svd(Atilde, full_matrices=False)
B = np.sqrt(S[0]) * U[:, [0]].reshape(bshape, order='F')
C = np.sqrt(S[0]) * Vt[[0], :].reshape(cshape, order='F')
return B, C
def test_07(self):
x = np.arange(20).reshape((4, 5))
y = array.subsample_array(x, (2, 2), pad=True)
assert y.shape == (2, 2, 2, 3)
assert y[0, 0, -1, -1] == 14