# Source code for tensorly.decomposition._parafac2

from warnings import warn

import tensorly as tl
from ._base_decomposition import DecompositionMixin
from tensorly.random import random_parafac2
from tensorly import backend as T
from . import parafac, non_negative_parafac_hals
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)
A = T.ones((padded_tensor.shape[0], rank), **context)

unfolded_mode_2 = unfold(padded_tensor, 2)
if T.shape(unfolded_mode_2)[0] < rank:
raise ValueError("Cannot perform SVD init if rank ({}) is greater than the number of columns in each tensor slice ({})".format(
rank, T.shape(unfolded_mode_2)[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))

"""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)

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=2000, init='random', svd='numpy_svd', normalize_factors=False, tol=1e-8, absolute_tol=1e-13, nn_modes=None, 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, optional (Default: 2000) Maximum number of iteration .. versionchanged:: 0.6.1 Previously, the default maximum number of iterations was 100. 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 decrease tolerance. The algorithm is considered to have converged when :math:\left|\| X - \hat{X}_{n-1} \|^2 - \| X - \hat{X}_{n} \|^2\right| < \epsilon \| X - \hat{X}_{n-1} \|^2. That is, when the relative change in sum of squared error is less than the tolerance. .. versionchanged:: 0.6.1 Previously, the stopping condition was :math:\left|\| X - \hat{X}_{n-1} \| - \| X - \hat{X}_{n} \|\right| < \epsilon. absolute_tol : float, optional (Default: 1e-13) Absolute reconstruction error tolearnce. The algorithm is considered to have converged when :math:\left|\| X - \hat{X}_{n-1} \|^2 - \| X - \hat{X}_{n} \|^2\right| < \epsilon_\text{abs}. That is, when the relative sum of squared error is less than the specified tolerance. The absolute tolerance is necessary for stopping the algorithm when used on noise-free data that follows the PARAFAC2 constraint. If None, then the machine precision + 1000 will be used. nn_modes: None, 'all' or array of integers (Default: None) Used to specify which modes to impose non-negativity constraints on. We cannot impose non-negativity constraints on the the B-mode (mode 1) with the ALS algorithm, so if this mode is among the constrained modes, then a warning will be shown (see notes for more info). 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. Because of the reformulation above, :math:B_i = P_i B, the :math:B_i matrices cannot be constrained to be non-negative with ALS. If this mode is constrained to be non-negative, then :math:B will be non-negative, but not the orthogonal P_i matrices. Consequently, the B_i matrices are unlikely to be non-negative. """ 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) if absolute_tol is None: absolute_tol = tl.eps(factors[0].dtype) * 1000 # If nn_modes is set, we use HALS, otherwise, we use the standard parafac implementation. if nn_modes is None: def parafac_updates(X, w, f): return parafac(X, rank, n_iter_max=n_iter_parafac, init=(w, f), svd=svd, orthogonalise=False, verbose=verbose, return_errors=False, normalize_factors=False, mask=None, random_state=random_state, tol=1e-100)[1] else: if nn_modes == 'all' or 1 in nn_modes: warn("Mode 1 of PARAFAC2 fitted with ALS cannot be constrained to be truly non-negative. See the documentation for more info.") def parafac_updates(X, w, f): return non_negative_parafac_hals( X, rank, n_iter_max=n_iter_parafac, init=(w, f), svd=svd, nn_modes=nn_modes, verbose=verbose, return_errors=False, tol=1e-100)[1] 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_updates(projected_tensor, weights, factors) 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 abs(rec_errors[-2]**2 - rec_errors[-1]**2) < (tol * rec_errors[-2]**2) or rec_errors[-1]**2 < absolute_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): 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 ---------- rank : int Number of components. n_iter_max : int, optional (Default: 2000) Maximum number of iteration .. versionchanged:: 0.6.1 Previously, the default maximum number of iterations was 100. 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 decrease tolerance. The algorithm is considered to have converged when :math:\left|\| X - \hat{X}_{n-1} \|^2 - \| X - \hat{X}_{n} \|^2\right| < \epsilon \| X - \hat{X}_{n-1} \|^2. That is, when the relative change in sum of squared error is less than the tolerance. .. versionchanged:: 0.6.1 Previously, the stopping condition was :math:\left|\| X - \hat{X}_{n-1} \| - \| X - \hat{X}_{n} \|\right| < \epsilon. absolute_tol : float, optional (Default: 1e-13) Absolute reconstruction error tolearnce. The algorithm is considered to have converged when :math:\left|\| X - \hat{X}_{n-1} \|^2 - \| X - \hat{X}_{n} \|^2\right| < \epsilon_\text{abs}. That is, when the relative sum of squared error is less than the specified tolerance. The absolute tolerance is necessary for stopping the algorithm when used on noise-free data that follows the PARAFAC2 constraint. If None, then the machine precision + 1000 will be used. nn_modes: None, 'all' or array of integers (Default: None) Used to specify which modes to impose non-negativity constraints on. We cannot impose non-negativity constraints on the the B-mode (mode 1) with the ALS algorithm, so if this mode is among the constrained modes, then a warning will be shown (see notes for more info). 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. 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. """ def __init__(self, rank, n_iter_max=2000, init='random', svd='numpy_svd', normalize_factors=False, tol=1e-8, absolute_tol=1e-13, nn_modes=None, random_state=None, verbose=False, return_errors=False, n_iter_parafac=5,): 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.absolute_tol = absolute_tol self.nn_modes = nn_modes self.random_state = random_state self.verbose = verbose self.return_errors = return_errors self.n_iter_parafac = n_iter_parafac
[docs] def fit_transform(self, tensor): """Decompose an input tensor Parameters ---------- tensor : tensorly.tensor Returns ------- self """ 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, absolute_tol=self.absolute_tol, nn_modes=self.nn_modes, random_state=self.random_state, verbose=self.verbose, return_errors=self.return_errors, n_iter_parafac=self.n_iter_parafac,) return self.decomposition_