# Source code for tensorly.parafac2_tensor

```
"""Core operations on PARAFAC2 tensors whose second mode evolve over their first.
"""
# Authors: Marie Roald
# Yngve Mardal Moe
from . import backend as T
from .base import unfold, tensor_to_vec
from ._factorized_tensor import FactorizedTensor
import warnings
class Parafac2Tensor(FactorizedTensor):
"""A wrapper class for the PARAFAC2 decomposition."""
def __init__(self, parafac2_tensor):
super().__init__()
shape, rank = _validate_parafac2_tensor(parafac2_tensor)
weights, factors, projections = parafac2_tensor
if weights is None:
weights = T.ones(rank, **T.context(factors[0]))
self.shape = shape
self.rank = rank
self.factors = factors
self.weights = weights
self.projections = projections
@classmethod
def from_CPTensor(self, cp_tensor, parafac2_tensor_ok=False):
"""Create a Parafac2Tensor from a CPTensor
Parameters:
-----------
cp_tensor: CPTensor or Parafac2Tensor
If it is a Parafac2Tensor, then the argument ``parafac2_tensor_ok`` must be True'
parafac2_tensor: bool (optional)
Whether or not Parafac2Tensors can be used as input.
Returns:
--------
Parafac2Tensor
Parafac2Tensor with factor matrices and weigths extracted from a CPTensor
"""
if parafac2_tensor_ok and len(cp_tensor) == 3:
return Parafac2Tensor(cp_tensor)
elif len(cp_tensor) == 3:
raise TypeError(
"Input is not a CPTensor. If it is a Parafac2Tensor, then the argument ``parafac2_tensor_ok`` must be True"
)
weights, (A, B, C) = cp_tensor
Q, R = T.qr(B)
projections = [Q for _ in range(T.shape(A)[0])]
B = R
return Parafac2Tensor((weights, (A, B, C), projections))
def __getitem__(self, index):
if index == 0:
return self.weights
elif index == 1:
return self.factors
elif index == 2:
return self.projections
else:
raise IndexError(
f"You tried to access index {index} of a PARAFAC2 tensor.\n"
"You can only access index 0, 1 and 2 of a PARAFAC2 tensor"
"(corresponding respectively to the weights, factors and projections)"
)
def __iter__(self):
yield self.weights
yield self.factors
yield self.projections
def __len__(self):
return 3
def __repr__(self):
message = f"(weights, factors, projections) : rank-{self.rank} Parafac2Tensor of shape {self.shape} "
return message
def to_tensor(self):
return parafac2_to_tensor(self)
def to_vec(self):
return parafac2_to_vec(self)
def to_unfolded(self, mode):
return parafac2_to_unfolded(self, mode)
def _validate_parafac2_tensor(parafac2_tensor):
"""Validates a parafac2_tensor in the form (weights, factors)
Returns the rank and shape of the validated tensor
Parameters
----------
parafac2_tensor : Parafac2Tensor or (weights, factors)
Returns
-------
(shape, rank) : (int tuple, int)
size of the full tensor and rank of the Kruskal tensor
"""
if isinstance(parafac2_tensor, Parafac2Tensor):
# it's already been validated at creation
return parafac2_tensor.shape, parafac2_tensor.rank
weights, factors, projections = parafac2_tensor
if len(factors) != 3:
raise ValueError(
"A PARAFAC2 tensor should be composed of exactly three factors."
f"However, {len(factors)} factors was given."
)
if len(projections) != factors[0].shape[0]:
raise ValueError(
"A PARAFAC2 tensor should have one projection matrix for each horisontal"
f" slice. However, {len(projections)} projection matrices was given and the first mode has"
f"length {factors[0].shape[0]}"
)
rank = int(T.shape(factors[0])[1])
shape = []
for i, projection in enumerate(projections):
current_mode_size, current_rank = T.shape(projection)
if current_rank != rank:
raise ValueError(
"All the projection matrices of a PARAFAC2 tensor should have the same number of "
f"columns as the rank. However, rank={rank} but projections[{i}].shape[1]={T.shape(projection)[1]}"
)
inner_product = T.dot(T.transpose(projection), projection)
if T.max(T.abs(inner_product - T.eye(rank, **T.context(inner_product)))) > 1e-5:
raise ValueError(
"All the projection matrices must be orthonormal, that is, P.T@P = I. "
f"However, T.norm(projection[{i}].T@projection[{i}] - T.eye(rank)) = "
f"{T.norm(inner_product - T.eye(rank, **T.context(inner_product)))}"
)
shape.append(
(current_mode_size, *[f.shape[0] for f in factors[2:]])
) # Tuple unpacking to possibly support higher order PARAFAC2 tensors in the future
# Skip first factor matrix since the rank is extracted from it.
for i, factor in enumerate(factors[1:]):
current_mode_size, current_rank = T.shape(factor)
if current_rank != rank:
raise ValueError(
"All the factors of a PARAFAC2 tensor should have the same number of columns."
f"However, factors[0].shape[1]={rank} but factors[{i}].shape[1]={current_rank}."
)
if weights is not None and T.shape(weights)[0] != rank:
raise ValueError(
f"Given factors for a rank-{rank} PARAFAC2 tensor but len(weights)={T.shape(weights)[0]}."
)
return tuple(shape), rank
def parafac2_normalise(parafac2_tensor):
"""Returns parafac2_tensor with factors normalised to unit length
Turns ``factors = [|U_1, ... U_n|]`` into ``[weights; |V_1, ... V_n|]``,
where the columns of each `V_k` are normalized to unit Euclidean length
from the columns of `U_k` with the normalizing constants absorbed into
`weights`. In the special case of a symmetric tensor, `weights` holds the
eigenvalues of the tensor.
Parameters
----------
parafac2_tensor : Parafac2Tensor = (weight, factors, projections)
factors is list of matrices, all with the same number of columns
i.e.::
for u in U:
u[i].shape == (s_i, R)
where `R` is fixed while `s_i` can vary with `i`
Returns
-------
Parafac2Tensor = (normalisation_weights, normalised_factors, normalised_projections)
"""
# allocate variables for weights, and normalized factors
_, rank = _validate_parafac2_tensor(parafac2_tensor)
weights, factors, projections = parafac2_tensor
# if (not copy) and (weights is None):
# warnings.warn('Provided copy=False and weights=None: a new Parafac2Tensor'
# 'with new weights and factors normalised inplace will be returned.')
# weights = T.ones(rank, **T.context(factors[0]))
# The if test below was added to enable inplace edits
# however, TensorFlow does not support inplace edits
# so this is always set to True
if True:
factors = [T.copy(f) for f in factors]
projections = [T.copy(p) for p in projections]
if weights is not None:
factors[0] = factors[0] * weights
weights = T.ones(rank, **T.context(factors[0]))
for i, factor in enumerate(factors):
scales = T.norm(factor, axis=0)
weights = weights * scales
scales_non_zero = T.where(
scales == 0, T.ones(T.shape(scales), **T.context(factors[0])), scales
)
factors[i] = factor / scales_non_zero
return Parafac2Tensor((weights, factors, projections))
def apply_parafac2_projections(parafac2_tensor):
r"""Apply the projection matrices to the evolving factor.
Parameters
----------
parafac2_tensor : Parafac2Tensor
Returns
-------
(weights, factors) : ndarray, tuple
A tensor decomposition on the form A [B_i] C such that
the :math:`X_{ijk}` is given by :math:`\sum_r A_{ir} [B_i]_{jr} C_{kr}`.
This is also equivalent to a coupled matrix factorisation, where
each matrix, :math:`X_i = C diag([a_{i1}, ..., a_{ir}] B_i)`.
The first element of factors is the A matrix, the second element is
a list of B-matrices and the third element is the C matrix.
"""
_validate_parafac2_tensor(parafac2_tensor)
weights, factors, projections = parafac2_tensor
evolving_factor = [T.dot(projection, factors[1]) for projection in projections]
return weights, (factors[0], evolving_factor, factors[2])
[docs]def parafac2_to_slice(parafac2_tensor, slice_idx, validate=True):
r"""Generate a single slice along the first mode from the PARAFAC2 tensor.
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
----------
parafac2_tensor : Parafac2Tensor - (weight, factors, projection_matrices)
* weights : 1D array of shape (rank, )
weights of the factors
* factors : List of factors of the PARAFAC2 decomposition
Contains the matrices :math:`A`, :math:`B` and :math:`C` described above
* projection_matrices : List of projection matrices used to create evolving
factors.
Returns
-------
ndarray
Full tensor of shape [P[slice_idx].shape[1], C.shape[1]], where
P is the projection matrices and C is the last factor matrix of
the Parafac2Tensor.
"""
if validate:
_validate_parafac2_tensor(parafac2_tensor)
weights, (A, B, C), projections = parafac2_tensor
a = A[slice_idx]
if weights is not None:
a = a * weights
Ct = T.transpose(C)
B_i = T.dot(projections[slice_idx], B)
return T.dot(B_i * a, Ct)
[docs]def parafac2_to_slices(parafac2_tensor, validate=True):
r"""Generate all slices along the first mode from a PARAFAC2 tensor.
Generates a list of all slices from a PARAFAC2 tensor. A list is returned
since the tensor might have varying size along the second mode. To return
a tensor, see the ``parafac2_to_tensor`` function instead.shape
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
----------
parafac2_tensor : Parafac2Tensor - (weight, factors, projection_matrices)
* weights : 1D array of shape (rank, )
weights of the factors
* factors : List of factors of the PARAFAC2 decomposition
Contains the matrices :math:`A`, :math:`B` and :math:`C` described above
* projection_matrices : List of projection matrices used to create evolving
factors.
Returns
-------
List[ndarray]
A list of full tensors of shapes [P[i].shape[1], C.shape[1]], where
P is the projection matrices and C is the last factor matrix of the
Parafac2Tensor.
"""
if validate:
_validate_parafac2_tensor(parafac2_tensor)
weights, (A, B, C), projections = parafac2_tensor
if weights is not None:
A = A * weights
weights = None
decomposition = weights, (A, B, C), projections
I, _ = A.shape
return [parafac2_to_slice(decomposition, i, validate=False) for i in range(I)]
[docs]def parafac2_to_tensor(parafac2_tensor):
r"""Construct a full tensor from a PARAFAC2 decomposition.
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
----------
parafac2_tensor : Parafac2Tensor - (weight, factors, projection_matrices)
* weights : 1D array of shape (rank, )
weights of the factors
* factors : List of factors of the PARAFAC2 decomposition
Contains the matrices :math:`A`, :math:`B` and :math:`C` described above
* projection_matrices : List of projection matrices used to create evolving
factors.
Returns
-------
ndarray
Full constructed tensor. Uneven slices are padded with zeros.
"""
_, (A, _, C), projections = parafac2_tensor
slices = parafac2_to_slices(parafac2_tensor)
lengths = [projection.shape[0] for projection in projections]
tensor = T.zeros((A.shape[0], max(lengths), C.shape[0]), **T.context(slices[0]))
for i, (slice_, length) in enumerate(zip(slices, lengths)):
tensor = T.index_update(tensor, T.index[i, :length], slice_)
return tensor
[docs]def parafac2_to_unfolded(parafac2_tensor, mode):
r"""Construct an unfolded tensor from a PARAFAC2 decomposition. Uneven slices are padded by zeros.
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
----------
parafac2_tensor : Parafac2Tensor - (weight, factors, projection_matrices)
* weights : 1D array of shape (rank, )
weights of the factors
* factors : List of factors of the PARAFAC2 decomposition
Contains the matrices :math:`A`, :math:`B` and :math:`C` described above
* projection_matrices : List of projection matrices used to create evolving
factors.
Returns
-------
ndarray
Full constructed tensor. Uneven slices are padded with zeros.
"""
return unfold(parafac2_to_tensor(parafac2_tensor), mode)
[docs]def parafac2_to_vec(parafac2_tensor):
r"""Construct a vectorized tensor from a PARAFAC2 decomposition. Uneven slices are padded by zeros.
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
----------
parafac2_tensor : Parafac2Tensor - (weight, factors, projection_matrices)
* weights : 1D array of shape (rank, )
weights of the factors
* factors : List of factors of the PARAFAC2 decomposition
Contains the matrices :math:`A`, :math:`B` and :math:`C` described above
* projection_matrices : List of projection matrices used to create evolving
factors.
Returns
-------
ndarray
Full constructed tensor. Uneven slices are padded with zeros.
"""
return tensor_to_vec(parafac2_to_tensor(parafac2_tensor))
```