Revision c6d4fdb44a004f804c489c8cb1739ba17d0b0939 authored by TUNA Caglayan on 20 October 2021, 10:09:49 UTC, committed by TUNA Caglayan on 20 October 2021, 10:09:49 UTC
1 parent 2b16764
_nn_cp.py
import numpy as np
import warnings
import tensorly as tl
from ._base_decomposition import DecompositionMixin
from ..random import random_cp
from ..base import unfold
from ..tenalg.proximal import soft_thresholding,hals_nnls
from ..cp_tensor import (cp_to_tensor, CPTensor,
unfolding_dot_khatri_rao, cp_norm,
cp_normalize, validate_cp_rank)
# Authors: Jean Kossaifi <jean.kossaifi+tensors@gmail.com>
# Chris Swierczewski <csw@amazon.com>
# Sam Schneider <samjohnschneider@gmail.com>
# Aaron Meurer <asmeurer@gmail.com>
# Aaron Meyer <tensorly@ameyer.me>
# Jeremy Cohen <jeremy.cohen@irisa.fr>
# Axel Marmoret <axel.marmoret@inria.fr>
# Caglayan TUna <caglayantun@gmail.com>
# License: BSD 3 clause
def make_svd_non_negative(tensor, U, S, V, nntype):
""" Use NNDSVD method to transform SVD results into a non-negative form. This
method leads to more efficient solving with NNMF [1].
Parameters
----------
tensor : tensor being decomposed
U, S, V: SVD factorization results
nntype : {'nndsvd', 'nndsvda'}
Whether to fill small values with 0.0 (nndsvd), or the tensor mean (nndsvda, default).
[1]: Boutsidis & Gallopoulos. Pattern Recognition, 41(4): 1350-1362, 2008.
"""
# NNDSVD initialization
W = tl.zeros_like(U)
H = tl.zeros_like(V)
# The leading singular triplet is non-negative
# so it can be used as is for initialization.
W = tl.index_update(W, tl.index[:, 0], tl.sqrt(S[0]) * tl.abs(U[:, 0]))
H = tl.index_update(H, tl.index[0, :], tl.sqrt(S[0]) * tl.abs(V[0, :]))
for j in range(1, tl.shape(U)[1]):
x, y = U[:, j], V[j, :]
# extract positive and negative parts of column vectors
x_p, y_p = tl.clip(x, a_min=0.0), tl.clip(y, a_min=0.0)
x_n, y_n = tl.abs(tl.clip(x, a_max=0.0)), tl.abs(tl.clip(y, a_max=0.0))
# and their norms
x_p_nrm, y_p_nrm = tl.norm(x_p), tl.norm(y_p)
x_n_nrm, y_n_nrm = tl.norm(x_n), tl.norm(y_n)
m_p, m_n = x_p_nrm * y_p_nrm, x_n_nrm * y_n_nrm
# choose update
if m_p > m_n:
u = x_p / x_p_nrm
v = y_p / y_p_nrm
sigma = m_p
else:
u = x_n / x_n_nrm
v = y_n / y_n_nrm
sigma = m_n
lbd = tl.sqrt(S[j] * sigma)
W = tl.index_update(W, tl.index[:, j], lbd * u)
H = tl.index_update(H, tl.index[j, :], lbd * v)
# After this point we no longer need H
eps = tl.eps(tensor.dtype)
if nntype == "nndsvd":
W = soft_thresholding(W, eps)
elif nntype == "nndsvda":
avg = tl.mean(tensor)
W = tl.where(W < eps, tl.ones(tl.shape(W), **tl.context(W)) * avg, W)
else:
raise ValueError(
'Invalid nntype parameter: got %r instead of one of %r' %
(nntype, ('nndsvd', 'nndsvda')))
return W
def initialize_nn_cp(tensor, rank, init='svd', svd='numpy_svd', random_state=None,
normalize_factors=False, nntype='nndsvda'):
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
nntype : {'nndsvd', 'nndsvda'}
Whether to fill small values with 0.0 (nndsvd), or the tensor mean (nndsvda, default).
Returns
-------
factors : CPTensor
An initial cp tensor.
"""
rng = tl.check_random_state(random_state)
if init == 'random':
kt = random_cp(tl.shape(tensor), rank, normalise_factors=False, random_state=rng, **tl.context(tensor))
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, S, V = svd_fun(unfold(tensor, mode), n_eigenvecs=rank)
# Apply nnsvd to make non-negative
U = make_svd_non_negative(tensor, U, S, V, nntype)
if tensor.shape[mode] < rank:
# TODO: this is a hack but it seems to do the job for now
random_part = tl.tensor(rng.random_sample((U.shape[0], rank - tl.shape(tensor)[mode])), **tl.context(tensor))
U = tl.concatenate([U, random_part], axis=1)
factors.append(U[:, :rank])
kt = CPTensor((None, factors))
# If the initialisation is a precomputed decomposition, we double check its validity and return it
elif isinstance(init, (tuple, list, CPTensor)):
# TODO: Test this
try:
kt = CPTensor(init)
except ValueError:
raise ValueError(
'If initialization method is a mapping, then it must '
'be possible to convert it to a CPTensor instance'
)
return kt
else:
raise ValueError('Initialization method "{}" not recognized'.format(init))
# Make decomposition feasible by taking the absolute value of all factor matrices
kt.factors = [tl.abs(f) for f in kt[1]]
if normalize_factors:
kt = cp_normalize(kt)
return kt
def non_negative_parafac(tensor, rank, n_iter_max=100, init='svd', svd='numpy_svd',
tol=10e-7, random_state=None, verbose=0, normalize_factors=False,
return_errors=False, mask=None, cvg_criterion='abs_rec_error',
fixed_modes=None):
"""
Non-negative CP decomposition
Uses multiplicative updates, see [2]_
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
fixed_modes : list, default is None
A list of modes for which the initial value is not modified.
The last mode cannot be fixed due to error computation.
Returns
-------
factors : ndarray list
list of positive factors of the CP decomposition
element `i` is of shape ``(tensor.shape[i], rank)``
References
----------
.. [2] 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
"""
epsilon = tl.eps(tensor.dtype)
rank = validate_cp_rank(tl.shape(tensor), rank=rank)
if mask is not None and init == "svd":
message = "Masking occurs after initialization. Therefore, random initialization is recommended."
warnings.warn(message, Warning)
weights, factors = initialize_nn_cp(tensor, rank, init=init, svd=svd,
random_state=random_state,
normalize_factors=normalize_factors)
rec_errors = []
norm_tensor = tl.norm(tensor, 2)
if fixed_modes is None:
fixed_modes = []
if tl.ndim(tensor) - 1 in fixed_modes:
warnings.warn('You asked for fixing the last mode, which is not supported while tol is fixed.\n The last mode will not be fixed. Consider using tl.moveaxis()')
fixed_modes.remove(tl.ndim(tensor) - 1)
modes_list = [mode for mode in range(tl.ndim(tensor)) if mode not in fixed_modes]
for iteration in range(n_iter_max):
if verbose > 1:
print("Starting iteration", iteration + 1)
for mode in modes_list:
if verbose > 1:
print("Mode", mode, "of", tl.ndim(tensor))
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])
if mask is not None:
tensor = tensor * mask + tl.cp_to_tensor((None, factors), mask=1 - mask)
mttkrp = unfolding_dot_khatri_rao(tensor, (None, factors), mode)
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
factors[mode] = factor
if normalize_factors:
weights, factors = cp_normalize((weights, factors))
if tol:
# ||tensor - rec||^2 = ||tensor||^2 + ||rec||^2 - 2*<tensor, rec>
factors_norm = cp_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:
rec_error_decrease = rec_errors[-2] - rec_errors[-1]
if verbose:
print("iteration {}, reconstraction error: {}, decrease = {}".format(iteration, rec_error, rec_error_decrease))
if cvg_criterion == 'abs_rec_error':
stop_flag = abs(rec_error_decrease) < tol
elif cvg_criterion == 'rec_error':
stop_flag = rec_error_decrease < tol
else:
raise TypeError("Unknown convergence criterion")
if stop_flag:
if verbose:
print("PARAFAC converged after {} iterations".format(iteration))
break
else:
if verbose:
print('reconstruction error={}'.format(rec_errors[-1]))
cp_tensor = CPTensor((weights, factors))
if return_errors:
return cp_tensor, rec_errors
else:
return cp_tensor
def non_negative_parafac_hals(tensor, rank, n_iter_max=100, init="svd", svd='numpy_svd', tol=10e-8,
sparsity_coefficients=None, fixed_modes=None, nn_modes='all', exact=False,
verbose=False, return_errors=False, cvg_criterion='abs_rec_error'):
"""
Non-negative CP decomposition via HALS
Uses Hierarchical ALS (Alternating Least Squares) which updates each factor column-wise (one column at a time while keeping all other columns fixed), see [1]_
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
Default: 1e-8
sparsity_coefficients: array of float (of length the number of modes)
The sparsity coefficients on each factor.
If set to None, the algorithm is computed without sparsity
Default: None,
fixed_modes: array of integers (between 0 and the number of modes)
Has to be set not to update a factor, 0 and 1 for U and V respectively
Default: None
nn_modes: None, 'all' or array of integers (between 0 and the number of modes)
Used to specify which modes to impose non-negativity constraints on.
If 'all', then non-negativity is imposed on all modes.
Default: 'all'
exact: If it is True, the algorithm gives a results with high precision but it needs high computational cost.
If it is False, the algorithm gives an approximate solution
Default: False
verbose: boolean
Indicates whether the algorithm prints the successive
reconstruction errors or not
Default: False
return_errors: boolean
Indicates whether the algorithm should return all reconstruction errors
and computation time of each iteration or not
Default: False
cvg_criterion : {'abs_rec_error', 'rec_error'}, optional
Stopping criterion for ALS, works if `tol` is not None.
If 'rec_error', ALS stops at current iteration if ``(previous rec_error - current rec_error) < tol``.
If 'abs_rec_error', ALS terminates when `|previous rec_error - current rec_error| < tol`.
sparsity : float or int
Returns
-------
factors : ndarray list
list of positive 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 algorithm.
References
----------
.. [1]: N. Gillis and F. Glineur, Accelerated Multiplicative Updates and
Hierarchical ALS Algorithms for Nonnegative Matrix Factorization,
Neural Computation 24 (4): 1085-1105, 2012.
"""
weights, factors = initialize_nn_cp(tensor, rank, init=init, svd=svd,
random_state=None,
normalize_factors=False)
norm_tensor = tl.norm(tensor, 2)
n_modes = tl.ndim(tensor)
if sparsity_coefficients is None or isinstance(sparsity_coefficients, float):
sparsity_coefficients = [sparsity_coefficients] * n_modes
if fixed_modes is None:
fixed_modes = []
if nn_modes == 'all':
nn_modes = set(range(n_modes))
elif nn_modes is None:
nn_modes = set()
# Avoiding errors
for fixed_value in fixed_modes:
sparsity_coefficients[fixed_value] = None
for mode in range(n_modes):
if sparsity_coefficients[mode] is not None:
warnings.warn("Sparsity coefficient is ignored in unconstrained modes.")
# Generating the mode update sequence
modes = [mode for mode in range(n_modes) if mode not in fixed_modes]
# initialisation - declare local varaibles
rec_errors = []
# Iteratation
for iteration in range(n_iter_max):
# One pass of least squares on each updated mode
for mode in modes:
# Computing Hadamard of cross-products
pseudo_inverse = tl.tensor(tl.ones((rank, rank)), **tl.context(tensor))
for i, factor in enumerate(factors):
if i != mode:
pseudo_inverse = pseudo_inverse * tl.dot(tl.transpose(factor), factor)
if not iteration and weights is not None:
# Take into account init weights
mttkrp = unfolding_dot_khatri_rao(tensor, (weights, factors), mode)
else:
mttkrp = unfolding_dot_khatri_rao(tensor, (None, factors), mode)
if mode in nn_modes:
# Call the hals resolution with nnls, optimizing the current mode
nn_factor, _, _, _ = hals_nnls(tl.transpose(mttkrp), pseudo_inverse, tl.transpose(factors[mode]),
n_iter_max=100, sparsity_coefficient=sparsity_coefficients[mode],
exact=exact)
factors[mode] = tl.transpose(nn_factor)
else:
factor = tl.solve(tl.transpose(pseudo_inverse), tl.transpose(mttkrp))
factors[mode] = tl.transpose(factor)
if tol:
factors_norm = cp_norm((weights, factors))
iprod = tl.sum(tl.sum(mttkrp * factors[-1], 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:
rec_error_decrease = rec_errors[-2] - rec_errors[-1]
if verbose:
print("iteration {}, reconstruction error: {}, decrease = {}".format(iteration, rec_error, rec_error_decrease))
if cvg_criterion == 'abs_rec_error':
stop_flag = abs(rec_error_decrease) < tol
elif cvg_criterion == 'rec_error':
stop_flag = rec_error_decrease < tol
else:
raise TypeError("Unknown convergence criterion")
if stop_flag:
if verbose:
print("PARAFAC converged after {} iterations".format(iteration))
break
else:
if verbose:
print('reconstruction error={}'.format(rec_errors[-1]))
cp_tensor = CPTensor((weights, factors))
if return_errors:
return cp_tensor, rec_errors
else:
return cp_tensor
class CP_NN(DecompositionMixin):
"""
Non-Negative Candecomp-Parafac decomposition via Alternating-Least Square
Computes a rank-`rank` decomposition of `tensor` [1]_ such that,
``tensor = [|weights; factors[0], ..., 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
mask : ndarray
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 [2]_
cvg_criterion : {'abs_rec_error', 'rec_error'}, optional
Stopping criterion for ALS, works if `tol` is not None.
If 'rec_error', ALS stops at current iteration if (previous rec_error - current rec_error) < tol.
If 'abs_rec_error', ALS terminates when |previous rec_error - current rec_error| < tol.
sparsity : float or int
If `sparsity` is not None, we approximate tensor as a sum of low_rank_component and sparse_component, where low_rank_component = cp_to_tensor((weights, factors)). `sparsity` denotes desired fraction or number of non-zero elements in the sparse_component of the `tensor`.
fixed_modes : list, default is None
A list of modes for which the initial value is not modified.
The last mode cannot be fixed due to error computation.
svd_mask_repeats: int
If using a tensor with masked values, this initializes using SVD multiple times to
remove the effect of these missing values on the initialization.
Returns
-------
CPTensor : (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)
* sparse_component : nD array of shape tensor.shape. Returns only if `sparsity` is not None.
errors : list
A list of reconstruction errors at each iteration of the algorithms.
References
----------
.. [1] T.G.Kolda and B.W.Bader, "Tensor Decompositions and Applications",
SIAM REVIEW, vol. 51, n. 3, pp. 455-500, 2009.
.. [2] Tomasi, Giorgio, and Rasmus Bro. "PARAFAC and missing values."
Chemometrics and Intelligent Laboratory Systems 75.2 (2005): 163-180.
.. [3] R. Bro, "Multi-Way Analysis in the Food Industry: Models, Algorithms, and
Applications", PhD., University of Amsterdam, 1998
"""
def __init__(self, rank, n_iter_max=100, init='svd', svd='numpy_svd',
tol=10e-7, random_state=None, verbose=0, normalize_factors=False,
mask=None, cvg_criterion='abs_rec_error',
fixed_modes=None):
self.n_iter_max = n_iter_max
self.init = init
self.svd = svd
self.tol = tol
self.random_state = random_state
self.verbose = verbose
self.normalize_factors = normalize_factors
self.mask = mask
self.cvg_criterion = cvg_criterion
self.fixed_modes = fixed_modes
def fit_transform(self, tensor):
"""Decompose an input tensor
Parameters
----------
tensor : tensorly tensor
input tensor to decompose
Returns
-------
CPTensor
decomposed tensor
"""
cp_tensor, errors = non_negative_parafac(
tensor,
n_iter_max=self.n_iter_max,
init=self.init,
svd=self.svd,
tol=self.tol,
random_state=self.random_state,
verbose=self.verbose,
normalize_factors=self.normalize_factors,
mask=self.mask,
cvg_criterion=self.cvg_criterion,
fixed_modes=self.fixed_modes,
return_errors=True,
)
self.decomposition_ = cp_tensor
self.errors_ = errors
return self.decomposition_
def __repr__(self):
return f'Rank-{self.rank} Non-Negative CP decomposition.'
class CP_NN_HALS(DecompositionMixin):
"""
Non-Negative Candecomp-Parafac decomposition via Alternating-Least Square
Computes a rank-`rank` decomposition of `tensor` [1]_ such that::
``tensor = [|weights; factors[0], ..., 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
mask : ndarray
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 [2]_
cvg_criterion : {'abs_rec_error', 'rec_error'}, optional
Stopping criterion for ALS, works if `tol` is not None.
If 'rec_error', ALS stops at current iteration if (previous rec_error - current rec_error) < tol.
If 'abs_rec_error', ALS terminates when |previous rec_error - current rec_error| < tol.
sparsity : float or int
If `sparsity` is not None, we approximate tensor as a sum of low_rank_component and sparse_component, where low_rank_component = cp_to_tensor((weights, factors)). `sparsity` denotes desired fraction or number of non-zero elements in the sparse_component of the `tensor`.
fixed_modes : list, default is None
A list of modes for which the initial value is not modified.
The last mode cannot be fixed due to error computation.
svd_mask_repeats: int
If using a tensor with masked values, this initializes using SVD multiple times to
remove the effect of these missing values on the initialization.
Returns
-------
CPTensor : (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)
* sparse_component : nD array of shape tensor.shape. Returns only if `sparsity` is not None.
errors : list
A list of reconstruction errors at each iteration of the algorithms.
References
----------
.. [1] T.G.Kolda and B.W.Bader, "Tensor Decompositions and Applications",
SIAM REVIEW, vol. 51, n. 3, pp. 455-500, 2009.
.. [2] Tomasi, Giorgio, and Rasmus Bro. "PARAFAC and missing values."
Chemometrics and Intelligent Laboratory Systems 75.2 (2005): 163-180.
.. [3] R. Bro, "Multi-Way Analysis in the Food Industry: Models, Algorithms, and
Applications", PhD., University of Amsterdam, 1998
"""
def __init__(self, rank, n_iter_max=100, init="svd", svd='numpy_svd', tol=10e-8,
sparsity_coefficients=None, fixed_modes=None, nn_modes='all', exact=False,
verbose=False, cvg_criterion='abs_rec_error'):
self.rank = rank
self.n_iter_max = n_iter_max
self.init = init
self.svd = svd
self.tol = tol
self.sparsity_coefficients = sparsity_coefficients
self.fixed_modes = fixed_modes
self.nn_modes = nn_modes
self.exact = exact
self.verbose = verbose
self.cvg_criterion = cvg_criterion
def fit_transform(self, tensor):
"""Decompose an input tensor
Parameters
----------
tensor : tensorly tensor
input tensor to decompose
Returns
-------
CPTensor
decomposed tensor
"""
cp_tensor, errors = non_negative_parafac_hals(
tensor,
rank=self.rank,
n_iter_max=self.n_iter_max,
init=self.init,
svd=self.svd,
tol=self.tol,
sparsity_coefficients=self.sparsity_coefficients,
fixed_modes=self.fixed_modes,
nn_modes=self.nn_modes,
exact=self.exact,
verbose=self.verbose,
return_errors=True,
cvg_criterion=self.cvg_criterion,
)
self.decomposition_ = cp_tensor
self.errors_ = errors
return self.decomposition_
def __repr__(self):
return f'Rank-{self.rank} Non-Negative CP decomposition.'
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