##### https://github.com/tensorly/tensorly
Tip revision: 0fa8ab5
candecomp_parafac.py
``````import numpy as np
import warnings

import tensorly as tl
from ..random import check_random_state
from ..base import unfold
from ..kruskal_tensor import (kruskal_to_tensor, KruskalTensor,
unfolding_dot_khatri_rao, kruskal_norm)
from ..tenalg import khatri_rao

# Authors: Jean Kossaifi <jean.kossaifi+tensors@gmail.com>
#          Chris Swierczewski <csw@amazon.com>
#          Sam Schneider <samjohnschneider@gmail.com>
#          Aaron Meurer <asmeurer@gmail.com>

def initialize_factors(tensor, rank, init='svd', svd='numpy_svd', random_state=None,
non_negative=False, normalize_factors=False):
r"""Initialize factors used in `parafac`.

The type of initialization is set using `init`. If `init == 'random'` then
initialize factor matrices using `random_state`. If `init == 'svd'` then
initialize the `m`th factor matrix using the `rank` left singular vectors
of the `m`th unfolding of the input tensor.

Parameters
----------
tensor : ndarray
rank : int
init : {'svd', 'random'}, optional
svd : str, default is 'numpy_svd'
function to use to compute the SVD, acceptable values in tensorly.SVD_FUNS
non_negative : bool, default is False
if True, non-negative factors are returned

Returns
-------
factors : ndarray list
List of initialized factors of the CP decomposition where element `i`
is of shape (tensor.shape[i], rank)

"""
rng = check_random_state(random_state)

if init == 'random':
factors = [tl.tensor(rng.random_sample((tensor.shape[i], rank)), **tl.context(tensor)) for i in range(tl.ndim(tensor))]
if non_negative:
factors = [tl.abs(f) for f in factors]
if normalize_factors:
factors = [f/(tl.reshape(tl.norm(f, axis=0), (1, -1)) + 1e-12) for f in factors]
return factors

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)

factors = []
for mode in range(tl.ndim(tensor)):
U, _, _ = svd_fun(unfold(tensor, mode), n_eigenvecs=rank)

if tensor.shape[mode] < rank:
# TODO: this is a hack but it seems to do the job for now
# factor = tl.tensor(np.zeros((U.shape, rank)), **tl.context(tensor))
# factor[:, tensor.shape[mode]:] = tl.tensor(rng.random_sample((U.shape, rank - tl.shape(tensor)[mode])), **tl.context(tensor))
# factor[:, :tensor.shape[mode]] = U
random_part = tl.tensor(rng.random_sample((U.shape, rank - tl.shape(tensor)[mode])), **tl.context(tensor))
U = tl.concatenate([U, random_part], axis=1)

factor = U[:, :rank]
if non_negative:
factor = tl.abs(factor)
if normalize_factors:
factor = factor / (tl.reshape(tl.norm(factor, axis=0), (1, -1)) + 1e-12)
factors.append(factor)
return factors

raise ValueError('Initialization method "{}" not recognized'.format(init))

def parafac(tensor, rank, n_iter_max=100, init='svd', svd='numpy_svd', normalize_factors=False,
tol=1e-8, orthogonalise=False, random_state=None, verbose=0, return_errors=False,
"""CANDECOMP/PARAFAC decomposition via alternating least squares (ALS)
Computes a rank-`rank` decomposition of `tensor` _ such that,

``tensor = [|weights; factors, ..., factors[-1] |]``.

Parameters
----------
tensor : ndarray
rank  : int
Number of components.
n_iter_max : int
Maximum number of iteration
init : {'svd', 'random'}, optional
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 : if True, aggregate the weights of each factor in a 1D-tensor
of shape (rank, ), which will contain the norms of the factors
tol : float, optional
(Default: 1e-6) 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
non_negative : bool, optional
Perform non_negative PARAFAC. See :func:`non_negative_parafac`.
array of booleans with the same shape as ``tensor`` should be 0 where
the values are missing and 1 everywhere else. Note:  if tensor is
sparse, then mask should also be sparse with a fill value of 1 (or
True). Allows for missing values _

Returns
-------
KruskalTensor : (weight, factors)
* 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)

errors : list
A list of reconstruction errors at each iteration of the algorithms.

References
----------
..  T.G.Kolda and B.W.Bader, "Tensor Decompositions and Applications",
SIAM REVIEW, vol. 51, n. 3, pp. 455-500, 2009.

..  Tomasi, Giorgio, and Rasmus Bro. "PARAFAC and missing values."
Chemometrics and Intelligent Laboratory Systems 75.2 (2005): 163-180.

"""
epsilon = 10e-12

if orthogonalise and not isinstance(orthogonalise, int):
orthogonalise = n_iter_max

factors = initialize_factors(tensor, rank, init=init, svd=svd,
random_state=random_state,
non_negative=non_negative,
normalize_factors=normalize_factors)
rec_errors = []
norm_tensor = tl.norm(tensor, 2)
weights = tl.ones(rank, **tl.context(tensor))

for iteration in range(n_iter_max):
if orthogonalise and iteration <= orthogonalise:
factors = [tl.qr(f) if min(tl.shape(f)) >= rank else f for i, f in enumerate(factors)]

if verbose > 1:
print("Starting iteration", iteration + 1)
for mode in range(tl.ndim(tensor)):
if verbose > 1:
print("Mode", mode, "of", tl.ndim(tensor))
if non_negative:
accum = 1
# khatri_rao(factors).tl.dot(khatri_rao(factors))
# simplifies to multiplications
sub_indices = [i for i in range(len(factors)) if i != mode]
for i, e in enumerate(sub_indices):
if i:
accum *= tl.dot(tl.transpose(factors[e]), factors[e])
else:
accum = tl.dot(tl.transpose(factors[e]), factors[e])

pseudo_inverse = tl.tensor(np.ones((rank, rank)), **tl.context(tensor))
for i, factor in enumerate(factors):
if i != mode:
pseudo_inverse = pseudo_inverse*tl.dot(tl.conj(tl.transpose(factor)), factor)

mttkrp = unfolding_dot_khatri_rao(tensor, (None, factors), mode)

if non_negative:
numerator = tl.clip(mttkrp, a_min=epsilon, a_max=None)
denominator = tl.dot(factors[mode], accum)
denominator = tl.clip(denominator, a_min=epsilon, a_max=None)
factor = factors[mode] * numerator / denominator
else:
factor = tl.transpose(tl.solve(tl.conj(tl.transpose(pseudo_inverse)),
tl.transpose(mttkrp)))

if normalize_factors:
weights = tl.norm(factor, order=2, axis=0)
weights = tl.where(tl.abs(weights) <= tl.eps(tensor.dtype),
tl.ones(tl.shape(weights), **tl.context(factors)),
weights)
factor = factor/(tl.reshape(weights, (1, -1)))

factors[mode] = factor

if tol:
# ||tensor - rec||^2 = ||tensor||^2 + ||rec||^2 - 2*<tensor, rec>
factors_norm = kruskal_norm((weights, factors))

# mttkrp and factor for the last mode. This is equivalent to the
# inner product <tensor, factorization>
iprod = tl.sum(tl.sum(mttkrp*factor, axis=0)*weights)
rec_error = tl.sqrt(tl.abs(norm_tensor**2 + factors_norm**2 - 2*iprod)) / norm_tensor
rec_errors.append(rec_error)

if iteration >= 1:
if verbose:
print('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('reconstruction error={}'.format(rec_errors[-1]))

kruskal_tensor = KruskalTensor((weights, factors))

if return_errors:
return kruskal_tensor, rec_errors
else:
return kruskal_tensor

def non_negative_parafac(tensor, rank, n_iter_max=100, init='svd', svd='numpy_svd',
tol=10e-7, random_state=None, verbose=0):
"""
Non-negative CP decomposition

This is the same as parafac(non_negative=True).

Parameters
----------
tensor : ndarray
rank   : int
number of components
n_iter_max : int
maximum number of iteration
init : {'svd', 'random'}, optional
svd : str, default is 'numpy_svd'
function to use to compute the SVD, acceptable values in tensorly.SVD_FUNS
tol : float, optional
tolerance: the algorithm stops when the variation in
the reconstruction error is less than the tolerance
random_state : {None, int, np.random.RandomState}
verbose : int, optional
level of verbosity

Returns
-------
factors : ndarray list
list of positive factors of the CP decomposition
element `i` is of shape ``(tensor.shape[i], rank)``

References
----------
..  Amnon Shashua and Tamir Hazan,
"Non-negative tensor factorization with applications to statistics and computer vision",
In Proceedings of the International Conference on Machine Learning (ICML),
pp 792-799, ICML, 2005
"""
return parafac(tensor, rank, n_iter_max=n_iter_max, init=init, svd=svd,
tol=tol, random_state=random_state, verbose=verbose, non_negative=True)

def sample_khatri_rao(matrices, n_samples, skip_matrix=None,
return_sampled_rows=False, random_state=None):
"""Random subsample of the Khatri-Rao product of the given list of matrices

If one matrix only is given, that matrix is directly returned.

Parameters
----------
matrices : ndarray list
list of matrices with the same number of columns, i.e.::

for i in len(matrices):
matrices[i].shape = (n_i, m)

n_samples : int
number of samples to be taken from the Khatri-Rao product

skip_matrix : None or int, optional, default is None
if not None, index of a matrix to skip

random_state : None, int or numpy.random.RandomState
if int, used to set the seed of the random number generator
if numpy.random.RandomState, used to generate random_samples

returned_sampled_rows : bool, default is False
if True, also returns a list of the rows sampled from the full
khatri-rao product

Returns
-------
sampled_Khatri_Rao : ndarray
The sampled matricised tensor Khatri-Rao with `n_samples` rows

indices : tuple list
a list of indices sampled for each mode

indices_kr : int list
list of length `n_samples` containing the sampled row indices
"""
if random_state is None or not isinstance(random_state, np.random.RandomState):
rng = check_random_state(random_state)
warnings.warn('You are creating a new random number generator at each call.\n'
'If you are calling sample_khatri_rao inside a loop this will be slow:'
' best to create a rng outside and pass it as argument (random_state=rng).')
else:
rng = random_state

if skip_matrix is not None:
matrices = [matrices[i] for i in range(len(matrices)) if i != skip_matrix]

rank = tl.shape(matrices)
sizes = [tl.shape(m) for m in matrices]

# For each matrix, randomly choose n_samples indices for which to compute the khatri-rao product
indices_list = [rng.randint(0, tl.shape(m), size=n_samples, dtype=int) for m in matrices]
if return_sampled_rows:
# Compute corresponding rows of the full khatri-rao product
indices_kr = np.zeros((n_samples), dtype=int)
for size, indices in zip(sizes, indices_list):
indices_kr = indices_kr*size + indices

# Compute the Khatri-Rao product for the chosen indices
sampled_kr = tl.ones((n_samples, rank), **tl.context(matrices))
for indices, matrix in zip(indices_list, matrices):
sampled_kr = sampled_kr*matrix[indices, :]

if return_sampled_rows:
return sampled_kr, indices_list, indices_kr
else:
return sampled_kr, indices_list

def randomised_parafac(tensor, rank, n_samples, n_iter_max=100, init='random', svd='numpy_svd',
tol=10e-9, max_stagnation=20, random_state=None, verbose=1):
"""Randomised CP decomposition via sampled ALS

Parameters
----------
tensor : ndarray
rank   : int
number of components
n_samples : int
number of samples per ALS step
n_iter_max : int
maximum number of iteration
init : {'svd', 'random'}, optional
svd : str, default is 'numpy_svd'
function to use to compute the SVD, acceptable values in tensorly.SVD_FUNS
tol : float, optional
tolerance: the algorithm stops when the variation in
the reconstruction error is less than the tolerance
max_stagnation: int, optional, default is 0
if not zero, the maximum allowed number
of iterations with no decrease in fit
random_state : {None, int, np.random.RandomState}, default is None
verbose : int, optional
level of verbosity

Returns
-------
factors : ndarray list
list of positive factors of the CP decomposition
element `i` is of shape ``(tensor.shape[i], rank)``

References
----------
..  Casey Battaglino, Grey Ballard and Tamara G. Kolda,
"A Practical Randomized CP Tensor Decomposition",
"""
rng = check_random_state(random_state)
factors = initialize_factors(tensor, rank, init=init, svd=svd, random_state=random_state)
rec_errors = []
n_dims = tl.ndim(tensor)
norm_tensor = tl.norm(tensor, 2)
min_error = 0

weights = tl.ones(rank, **tl.context(tensor))
for iteration in range(n_iter_max):
for mode in range(n_dims):
kr_prod, indices_list = sample_khatri_rao(factors, n_samples, skip_matrix=mode, random_state=rng)
indices_list = [i.tolist() for i in indices_list]
# Keep all the elements of the currently considered mode
indices_list.insert(mode, slice(None, None, None))
# MXNet will not be happy if this is a list insteaf of a tuple
indices_list = tuple(indices_list)
if mode:
sampled_unfolding = tensor[indices_list]
else:
sampled_unfolding = tl.transpose(tensor[indices_list])

pseudo_inverse = tl.dot(tl.transpose(kr_prod), kr_prod)
factor = tl.dot(tl.transpose(kr_prod), sampled_unfolding)
factor = tl.transpose(tl.solve(pseudo_inverse, factor))
factors[mode] = factor

if max_stagnation or tol:
rec_error = tl.norm(tensor - kruskal_to_tensor((weights, factors)), 2) / norm_tensor
if not min_error or rec_error < min_error:
min_error = rec_error
stagnation = -1
stagnation += 1

rec_errors.append(rec_error)

if iteration > 1:
if verbose:
print('reconstruction error={}, variation={}.'.format(
rec_errors[-1], rec_errors[-2] - rec_errors[-1]))

if (tol and abs(rec_errors[-2] - rec_errors[-1]) < tol) or \
(stagnation and (stagnation > max_stagnation)):
if verbose:
print('converged in {} iterations.'.format(iteration))
break

return KruskalTensor((weights, factors))
``````