parafac(tensor, rank, n_iter_max=100, init='svd', svd='truncated_svd', normalize_factors=False, orthogonalise=False, tol=1e-08, random_state=None, verbose=0, return_errors=False, sparsity=None, l2_reg=0, mask=None, cvg_criterion='abs_rec_error', fixed_modes=None, svd_mask_repeats=5, linesearch=False, callback=None)[source]

CANDECOMP/PARAFAC decomposition via alternating least squares (ALS) Computes a rank-rank decomposition of tensor [1] such that:

tensor = [|weights; factors[0], ..., factors[-1] |].


Number of components.


Maximum number of iteration

init{‘svd’, ‘random’, CPTensor}, optional

Type of factor matrix initialization. If a CPTensor is passed, this is directly used for initalization. See initialize_factors.

svdstr, default is ‘truncated_svd’

function to use to compute the SVD, acceptable values in tensorly.SVD_FUNS

normalize_factorsif True, aggregate the weights of each factor in a 1D-tensor

of shape (rank, ), which will contain the norms of the factors

tolfloat, 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}
verboseint, optional

Level of verbosity

return_errorsbool, optional

Activate return of iteration errors


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.

sparsityfloat 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_modeslist, 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.

linesearchbool, default is False

Whether to perform line search as proposed by Bro [3].

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.


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



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.


R. Bro, “Multi-Way Analysis in the Food Industry: Models, Algorithms, and Applications”, PhD., University of Amsterdam, 1998