import tensorly as tl
from ._base_decomposition import DecompositionMixin
from tensorly.random import random_parafac2
from tensorly import backend as T
from . import parafac
from ..parafac2_tensor import parafac2_to_slice, Parafac2Tensor, _validate_parafac2_tensor
from ..cp_tensor import CPTensor
from ..base import unfold
# Authors: Marie Roald
# Yngve Mardal Moe
def initialize_decomposition(tensor_slices, rank, init='random', svd='numpy_svd', random_state=None):
r"""Initiate a random PARAFAC2 decomposition given rank and tensor slices
Parameters
----------
tensor_slices : Iterable of ndarray
rank : int
init : {'random', 'svd', CPTensor, Parafac2Tensor}, optional
random_state : `np.random.RandomState`
Returns
-------
parafac2_tensor : Parafac2Tensor
List of initialized factors of the CP decomposition where element `i`
is of shape (tensor.shape[i], rank)
"""
context = tl.context(tensor_slices[0])
shapes = [m.shape for m in tensor_slices]
if init == 'random':
return random_parafac2(shapes, rank, full=False, random_state=random_state,
**context)
elif init == 'svd':
try:
svd_fun = tl.SVD_FUNS[svd]
except KeyError:
message = 'Got svd={}. However, for the current backend ({}), the possible choices are {}'.format(
svd, tl.get_backend(), tl.SVD_FUNS)
raise ValueError(message)
padded_tensor = _pad_by_zeros(tensor_slices)
A = svd_fun(unfold(padded_tensor, 0), n_eigenvecs=rank)[0]
C = svd_fun(unfold(padded_tensor, 2), n_eigenvecs=rank)[0]
B = T.eye(rank, **context)
projections = _compute_projections(tensor_slices, (A, B, C), svd_fun)
return Parafac2Tensor((None, (A, B, C), projections))
elif isinstance(init, (tuple, list, Parafac2Tensor, CPTensor)):
try:
decomposition = Parafac2Tensor.from_CPTensor(init, parafac2_tensor_ok=True)
except ValueError:
raise ValueError(
'If initialization method is a mapping, then it must '
'be possible to convert it to a Parafac2Tensor instance'
)
if decomposition.rank != rank:
raise ValueError('Cannot init with a decomposition of different rank')
return decomposition
raise ValueError('Initialization method "{}" not recognized'.format(init))
def _pad_by_zeros(tensor_slices):
"""Return zero-padded full tensor.
"""
I = len(tensor_slices)
J = max(tensor_slice.shape[0] for tensor_slice in tensor_slices)
K = tensor_slices[0].shape[1]
padded = T.zeros((I, J, K), **T.context(tensor_slices[0]))
for i, tensor_slice in enumerate(tensor_slices):
J_i = len(tensor_slice)
tl.index_update(padded, tl.index[i, :J_i], tensor_slice)
return padded
def _compute_projections(tensor_slices, factors, svd_fun, out=None):
A, B, C = factors
if out is None:
out = [T.zeros((tensor_slice.shape[0], C.shape[1]), **T.context(tensor_slice)) for tensor_slice in tensor_slices]
slice_idxes = range(T.shape(A)[0])
for projection, i, tensor_slice in zip(out, slice_idxes, tensor_slices):
a_i = A[i]
lhs = T.dot(B, T.transpose(a_i*C))
rhs = T.transpose(tensor_slice)
U, S, Vh = svd_fun(T.dot(lhs, rhs), n_eigenvecs=A.shape[1])
out[i] = tl.index_update(projection, tl.index[:], T.transpose(T.dot(U, Vh)))
return out
def _project_tensor_slices(tensor_slices, projections, out=None):
if out is None:
rank = projections[0].shape[1]
num_slices = len(tensor_slices)
num_cols = tensor_slices[0].shape[1]
out = T.zeros((num_slices, rank, num_cols), **T.context(tensor_slices[0]))
for i, (tensor_slice, projection) in enumerate(zip(tensor_slices, projections)):
slice_ = T.dot(T.transpose(projection), tensor_slice)
out = tl.index_update(out, tl.index[i, :], slice_)
return out
def _get_svd(svd):
if svd in tl.SVD_FUNS:
return tl.SVD_FUNS[svd]
else:
message = 'Got svd={}. However, for the current backend ({}), the possible choices are {}'.format(
svd, tl.get_backend(), tl.SVD_FUNS)
raise ValueError(message)
def _parafac2_reconstruction_error(tensor_slices, decomposition):
_validate_parafac2_tensor(decomposition)
squared_error = 0
for idx, tensor_slice in enumerate(tensor_slices):
reconstruction = parafac2_to_slice(decomposition, idx, validate=False)
squared_error += tl.sum((tensor_slice - reconstruction)**2)
return tl.sqrt(squared_error)
[docs]def parafac2(tensor_slices, rank, n_iter_max=100, init='random', svd='numpy_svd', normalize_factors=False,
tol=1e-8, random_state=None, verbose=False, return_errors=False, n_iter_parafac=5):
r"""PARAFAC2 decomposition [1]_ of a third order tensor via alternating least squares (ALS)
Computes a rank-`rank` PARAFAC2 decomposition of the third-order tensor defined by
`tensor_slices`. The decomposition is on the form :math:`(A [B_i] C)` such that the
i-th frontal slice, :math:`X_i`, of :math:`X` is given by
.. math::
X_i = B_i diag(a_i) C^T,
where :math:`diag(a_i)` is the diagonal matrix whose nonzero entries are equal to
the :math:`i`-th row of the :math:`I \times R` factor matrix :math:`A`, :math:`B_i`
is a :math:`J_i \times R` factor matrix such that the cross product matrix :math:`B_{i_1}^T B_{i_1}`
is constant for all :math:`i`, and :math:`C` is a :math:`K \times R` factor matrix.
To compute this decomposition, we reformulate the expression for :math:`B_i` such that
.. math::
B_i = P_i B,
where :math:`P_i` is a :math:`J_i \times R` orthogonal matrix and :math:`B` is a
:math:`R \times R` matrix.
An alternative formulation of the PARAFAC2 decomposition is that the tensor element
:math:`X_{ijk}` is given by
.. math::
X_{ijk} = \sum_{r=1}^R A_{ir} B_{ijr} C_{kr},
with the same constraints hold for :math:`B_i` as above.
Parameters
----------
tensor_slices : ndarray or list of ndarrays
Either a third order tensor or a list of second order tensors that may have different number of rows.
Note that the second mode factor matrices are allowed to change over the first mode, not the
third mode as some other implementations use (see note below).
rank : int
Number of components.
n_iter_max : int
Maximum number of iteration
init : {'svd', 'random', CPTensor, Parafac2Tensor}
Type of factor matrix initialization. See `initialize_factors`.
svd : str, default is 'numpy_svd'
function to use to compute the SVD, acceptable values in tensorly.SVD_FUNS
normalize_factors : bool (optional)
If True, aggregate the weights of each factor in a 1D-tensor
of shape (rank, ), which will contain the norms of the factors. Note that
there may be some inaccuracies in the component weights.
tol : float, optional
(Default: 1e-8) Relative reconstruction error tolerance. The
algorithm is considered to have found the global minimum when the
reconstruction error is less than `tol`.
random_state : {None, int, np.random.RandomState}
verbose : int, optional
Level of verbosity
return_errors : bool, optional
Activate return of iteration errors
n_iter_parafac: int, optional
Number of PARAFAC iterations to perform for each PARAFAC2 iteration
Returns
-------
Parafac2Tensor : (weight, factors, projection_matrices)
* weights : 1D array of shape (rank, )
all ones if normalize_factors is False (default),
weights of the (normalized) factors otherwise
* factors : List of factors of the CP decomposition element `i` is of shape
(tensor.shape[i], rank)
* projection_matrices : List of projection matrices used to create evolving
factors.
errors : list
A list of reconstruction errors at each iteration of the algorithms.
References
----------
.. [1] Kiers, H.A.L., ten Berge, J.M.F. and Bro, R. (1999),
PARAFAC2—Part I. A direct fitting algorithm for the PARAFAC2 model.
J. Chemometrics, 13: 275-294.
Notes
-----
This formulation of the PARAFAC2 decomposition is slightly different from the one in [1]_.
The difference lies in that here, the second mode changes over the first mode, whereas in
[1]_, the second mode changes over the third mode. We made this change since that means
that the function accept both lists of matrices and a single nd-array as input without
any reordering of the modes.
"""
weights, factors, projections = initialize_decomposition(tensor_slices, rank, init=init, svd=svd, random_state=random_state)
rec_errors = []
norm_tensor = tl.sqrt(sum(tl.norm(tensor_slice, 2)**2 for tensor_slice in tensor_slices))
svd_fun = _get_svd(svd)
projected_tensor = tl.zeros([factor.shape[0] for factor in factors], **T.context(factors[0]))
for iteration in range(n_iter_max):
if verbose:
print("Starting iteration", iteration)
factors[1] = factors[1]*T.reshape(weights, (1, -1))
weights = T.ones(weights.shape, **tl.context(tensor_slices[0]))
projections = _compute_projections(tensor_slices, factors, svd_fun, out=projections)
projected_tensor = _project_tensor_slices(tensor_slices, projections, out=projected_tensor)
_, factors = parafac(projected_tensor, rank, n_iter_max=n_iter_parafac, init=(weights, factors),
svd=svd, orthogonalise=False, verbose=verbose, return_errors=False,
normalize_factors=False, mask=None, random_state=random_state, tol=1e-100)
if normalize_factors:
new_factors = []
for factor in factors:
norms = T.norm(factor, axis=0)
norms = tl.where(tl.abs(norms) <= tl.eps(factor.dtype),
tl.ones(tl.shape(norms), **tl.context(factors[0])),
norms)
weights = weights*norms
new_factors.append(factor/(tl.reshape(norms, (1, -1))))
factors = new_factors
if tol:
rec_error = _parafac2_reconstruction_error(tensor_slices, (weights, factors, projections))
rec_error /= norm_tensor
rec_errors.append(rec_error)
if iteration >= 1:
if verbose:
print('PARAFAC2 reconstruction error={}, variation={}.'.format(
rec_errors[-1], rec_errors[-2] - rec_errors[-1]))
if tol and abs(rec_errors[-2] - rec_errors[-1]) < tol:
if verbose:
print('converged in {} iterations.'.format(iteration))
break
else:
if verbose:
print('PARAFAC2 reconstruction error={}'.format(rec_errors[-1]))
parafac2_tensor = Parafac2Tensor((weights, factors, projections))
if return_errors:
return parafac2_tensor, rec_errors
else:
return parafac2_tensor
[docs]class Parafac2(DecompositionMixin):
def __init__(self, rank, n_iter_max=100, init='random', svd='numpy_svd', normalize_factors=False,
tol=1e-8, random_state=None, verbose=False, n_iter_parafac=5):
r"""PARAFAC2 decomposition [1]_ of a third order tensor via alternating least squares (ALS)
Computes a rank-`rank` PARAFAC2 decomposition of the third-order tensor defined by
`tensor_slices`. The decomposition is on the form :math:`(A [B_i] C)` such that the
i-th frontal slice, :math:`X_i`, of :math:`X` is given by
.. math::
X_i = B_i diag(a_i) C^T,
where :math:`diag(a_i)` is the diagonal matrix whose nonzero entries are equal to
the :math:`i`-th row of the :math:`I \times R` factor matrix :math:`A`, :math:`B_i`
is a :math:`J_i \times R` factor matrix such that the cross product matrix :math:`B_{i_1}^T B_{i_1}`
is constant for all :math:`i`, and :math:`C` is a :math:`K \times R` factor matrix.
To compute this decomposition, we reformulate the expression for :math:`B_i` such that
.. math::
B_i = P_i B,
where :math:`P_i` is a :math:`J_i \times R` orthogonal matrix and :math:`B` is a
:math:`R \times R` matrix.
An alternative formulation of the PARAFAC2 decomposition is that the tensor element
:math:`X_{ijk}` is given by
.. math::
X_{ijk} = \sum_{r=1}^R A_{ir} B_{ijr} C_{kr},
with the same constraints hold for :math:`B_i` as above.
Parameters
----------
tensor_slices : ndarray or list of ndarrays
Either a third order tensor or a list of second order tensors that may have different number of rows.
Note that the second mode factor matrices are allowed to change over the first mode, not the
third mode as some other implementations use (see note below).
rank : int
Number of components.
n_iter_max : int
Maximum number of iteration
init : {'svd', 'random', CPTensor, Parafac2Tensor}
Type of factor matrix initialization. See `initialize_factors`.
svd : str, default is 'numpy_svd'
function to use to compute the SVD, acceptable values in tensorly.SVD_FUNS
normalize_factors : bool (optional)
If True, aggregate the weights of each factor in a 1D-tensor
of shape (rank, ), which will contain the norms of the factors. Note that
there may be some inaccuracies in the component weights.
tol : float, optional
(Default: 1e-8) Relative reconstruction error tolerance. The
algorithm is considered to have found the global minimum when the
reconstruction error is less than `tol`.
random_state : {None, int, np.random.RandomState}
verbose : int, optional
Level of verbosity
n_iter_parafac: int, optional
Number of PARAFAC iterations to perform for each PARAFAC2 iteration
Returns
-------
Parafac2Tensor : (weight, factors, projection_matrices)
* weights : 1D array of shape (rank, )
all ones if normalize_factors is False (default),
weights of the (normalized) factors otherwise
* factors : List of factors of the CP decomposition element `i` is of shape
(tensor.shape[i], rank)
* projection_matrices : List of projection matrices used to create evolving
factors.
References
----------
.. [1] Kiers, H.A.L., ten Berge, J.M.F. and Bro, R. (1999),
PARAFAC2—Part I. A direct fitting algorithm for the PARAFAC2 model.
J. Chemometrics, 13: 275-294.
Notes
-----
This formulation of the PARAFAC2 decomposition is slightly different from the one in [1]_.
The difference lies in that here, the second mode changes over the first mode, whereas in
[1]_, the second mode changes over the third mode. We made this change since that means
that the function accept both lists of matrices and a single nd-array as input without
any reordering of the modes.
"""
self.rank = rank
self.n_iter_max=n_iter_max
self.init=init
self.svd=svd
self.normalize_factors=normalize_factors
self.tol=tol
self.random_state=random_state
self.verbose=verbose
self.n_iter_parafac = n_iter_parafac
def fit_transform(self, tensor):
self.decomposition_, self.errors_ = parafac2(tensor, rank = self.rank,
n_iter_max=self.n_iter_max,
init=self.init,
svd=self.svd,
normalize_factors=self.normalize_factors,
tol=self.tol,
random_state=self.random_state,
verbose=self.verbose,
return_errors=True,
n_iter_parafac = self.n_iter_parafac)
return self.decomposition_