Parafac2 tensor#
- class ivy.data_classes.factorized_tensor.parafac2_tensor.Parafac2Tensor(parafac2_tensor)[source]#
Bases:
FactorizedTensor
- _abc_impl = <_abc._abc_data object>#
- static apply_parafac2_projections(parafac2_tensor)[source]#
Apply the projection matrices to the evolving factor.
- Parameters:
parafac2_tensor (Parafac2Tensor) –
- Returns:
(weights, factors) – A tensor decomposition on the form A [B_i] C such that the \(X_{ijk}\) is given by \(sum_r A_{ir} [B_i]_{jr} C_{kr}\).
This is also equivalent to a coupled matrix factorisation, where each matrix, \(X_i = C diag([a_{i1}, ..., a_{ir}] B_i)\).
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.
- classmethod from_CPTensor(cp_tensor, parafac2_tensor_ok=False)[source]#
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 – Whether or not Parafac2Tensors can be used as input.
- Returns:
Parafac2Tensor with factor matrices and weights extracted from a CPTensor
- property n_param#
- static parafac2_normalise(parafac2_tensor)[source]#
Return 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
- static parafac2_to_slice(parafac2_tensor, slice_idx, validate=True)[source]#
Generate a single slice along the first mode from the PARAFAC2 tensor.
The decomposition is on the form \((A [B_i] C)\) such that the i-th frontal slice, \(X_i\), of \(X\) is given by
\[X_i = B_i diag(a_i) C^T,\]where \(diag(a_i)\) is the diagonal matrix whose nonzero entries are equal to the \(i\)-th row of the \(I times R\) factor matrix \(A\), \(B_i`is a :math:`J_i times R\) factor matrix such that the cross product matrix \(B_{i_1}^T B_{i_1}\) is constant for all \(i\), and \(C\) is a \(K times R\) factor matrix. To compute this decomposition, we reformulate the expression for \(B_i\) such that
\[B_i = P_i B,\]where \(P_i\) is a \(J_i times R\) orthogonal matrix and \(B\) is a \(R times R\) matrix.
An alternative formulation of the PARAFAC2 decomposition is that the tensor element \(X_{ijk}\) is given by
\[X_{ijk} = sum_{r=1}^R A_{ir} B_{ijr} C_{kr},\]with the same constraints hold for \(B_i\) as above.
- Parameters:
parafac2_tensor –
- weights
1D array of shape (rank, ) weights of the factors
- factors
List of factors of the PARAFAC2 decomposition Contains the matrices \(A\), \(B\) and \(C\) described above
- projection_matrices
List of projection matrices used to create evolving factors.
- Returns:
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.
- static parafac2_to_slices(parafac2_tensor, validate=True)[source]#
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.shapeThe decomposition is on the form \((A [B_i] C)\) such that the i-th frontal slice, \(X_i\), of \(X\) is given by
\[X_i = B_i diag(a_i) C^T,\]where \(diag(a_i)\) is the diagonal matrix whose nonzero entries are equal to the \(i\)-th row of the \(I times R\) factor matrix \(A\), \(B_i\) is a \(J_i times R\) factor matrix such that the cross product matrix \(B_{i_1}^T B_{i_1}\) is constant for all \(i\), and \(C\) is a \(K times R\) factor matrix.To compute this decomposition, we reformulate the expression for \(B_i\) such that
\[B_i = P_i B,\]where \(P_i\) is a \(J_i times R\) orthogonal matrix and \(B\) is a \(R times R\) matrix.
An alternative formulation of the PARAFAC2 decomposition is that the tensor element \(X_{ijk}\) is given by
\[X_{ijk} = sum_{r=1}^R A_{ir} B_{ijr} C_{kr},\]with the same constraints hold for \(B_i\) as above.
- Parameters:
parafac2_tensor (Parafac2Tensor - (weight, factors, projection_matrices)) –
- weights1D array of shape (rank, )
weights of the factors
- factorsList of factors of the PARAFAC2 decomposition
Contains the matrices \(A\), \(B\) and \(C\) described above
- projection_matricesList of projection matrices used to create evolving
factors.
- Returns:
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.
- static parafac2_to_tensor(parafac2_tensor)[source]#
Construct a full tensor from a PARAFAC2 decomposition.
The decomposition is on the form \((A [B_i] C)\) such that the i-th frontal slice, \(X_i\), of \(X\) is given by
\[X_i = B_i diag(a_i) C^T,\]where \(diag(a_i)\) is the diagonal matrix whose nonzero entries are equal to the \(i\)-th row of the \(I times R\) factor matrix \(A\), \(B_i\) is a \(J_i times R\) factor matrix such that the cross product matrix \(B_{i_1}^T B_{i_1}\) is constant for all \(i\), and \(C\) is a \(K times R\) factor matrix. To compute this decomposition, we reformulate the expression for \(B_i\) such that
\[B_i = P_i B,\]where \(P_i\) is a \(J_i times R\) orthogonal matrix and \(B\) is a \(R times R\) matrix.
An alternative formulation of the PARAFAC2 decomposition is that the tensor element \(X_{ijk}\) is given by
\[X_{ijk} = sum_{r=1}^R A_{ir} B_{ijr} C_{kr},\]with the same constraints hold for \(B_i\) as above.
- Parameters:
parafac2_tensor (Parafac2Tensor - (weight, factors, projection_matrices)) –
- weights1D array of shape (rank, )
weights of the factors
- factorsList of factors of the PARAFAC2 decomposition
Contains the matrices \(A\), \(B\) and \(C\) described above
- projection_matricesList of projection matrices used to create evolving
factors.
- Returns:
ndarray – Full constructed tensor. Uneven slices are padded with zeros.
- static parafac2_to_unfolded(parafac2_tensor, mode)[source]#
Construct an unfolded tensor from a PARAFAC2 decomposition. Uneven slices are padded by zeros.
The decomposition is on the form \((A [B_i] C)\) such that the i-th frontal slice, \(X_i\), of \(X\) is given by
\[X_i = B_i diag(a_i) C^T,\]where \(diag(a_i)\) is the diagonal matrix whose nonzero entries are equal to the \(i\)-th row of the \(I times R\) factor matrix \(A\), \(B_i\) is a \(J_i times R\) factor matrix such that the cross product matrix \(B_{i_1}^T B_{i_1}\) is constant for all \(i\), and \(C\) is a \(K times R\) factor matrix. To compute this decomposition, we reformulate the expression for \(B_i\) such that
\[B_i = P_i B,\]where \(P_i\) is a \(J_i times R\) orthogonal matrix and \(B\) is a \(R times R\) matrix.
An alternative formulation of the PARAFAC2 decomposition is that the tensor element \(X_{ijk}\) is given by
\[X_{ijk} = sum_{r=1}^R A_{ir} B_{ijr} C_{kr},\]with the same constraints hold for \(B_i\) as above.
- Parameters:
parafac2_tensor (Parafac2Tensor - (weight, factors, projection_matrices)) –
- weights
weights of the factors
- factors
Contains the matrices \(A\), \(B\) and \(C\) described above
- projection_matrices
factors
- Returns:
Full constructed tensor. Uneven slices are padded with zeros.
- static parafac2_to_vec(parafac2_tensor)[source]#
Construct a vectorized tensor from a PARAFAC2 decomposition. Uneven slices are padded by zeros.
The decomposition is on the form \((A [B_i] C)\) such that the i-th frontal slice, \(X_i\), of \(X\) is given by
\[X_i = B_i diag(a_i) C^T,\]where \(diag(a_i)\) is the diagonal matrix whose nonzero entries are equal to the \(i\)-th row of the \(I times R\) factor matrix \(A\), \(B_i\) is a \(J_i times R\) factor matrix such that the cross product matrix :math: B_{i_1}^T B_{i_1}`is constant for all :math:`i, and \(C\) is a \(K times R\) factor matrix. To compute this decomposition, we reformulate the expression for \(B_i\) such that
\[B_i = P_i B,\]where \(P_i\) is a \(J_i times R\) orthogonal matrix and \(B\) is a \(R times R\) matrix.
An alternative formulation of the PARAFAC2 decomposition is that the tensor element \(X_{ijk}\) is given by
\[X_{ijk} = sum_{r=1}^R A_{ir} B_{ijr} C_{kr},\]with the same constraints hold for \(B_i\) 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 \(A, :math:`B\) and \(C\) described above * projection_matrices
List of projection matrices used to create evolving factors.
- Returns:
Full constructed tensor. Uneven slices are padded with zeros.6
- static validate_parafac2_tensor(parafac2_tensor)[source]#
Validate a parafac2_tensor in the form (weights, factors) Return the rank and shape of the validated tensor.
- Parameters:
parafac2_tensor – Parafac2Tensor or (weights, factors)
- Returns:
(shape, rank) – size of the full tensor and rank of the CP tensor
This should have hopefully given you an overview of the parafac2_tensor submodule, if you have any questions, please feel free to reach out on our discord!