sklearn.decomposition.non_negative_factorization(X, W=None, H=None, n_components=None, init=’warn’, update_H=True, solver=’cd’, beta_loss=’frobenius’, tol=0.0001, max_iter=200, alpha=0.0, l1_ratio=0.0, regularization=None, random_state=None, verbose=0, shuffle=False)[source]

Compute Non-negative Matrix Factorization (NMF)

Find two non-negative matrices (W, H) whose product approximates the non- negative matrix X. This factorization can be used for example for dimensionality reduction, source separation or topic extraction.

The objective function is:

0.5 * ||X - WH||_Fro^2
+ alpha * l1_ratio * ||vec(W)||_1
+ alpha * l1_ratio * ||vec(H)||_1
+ 0.5 * alpha * (1 - l1_ratio) * ||W||_Fro^2
+ 0.5 * alpha * (1 - l1_ratio) * ||H||_Fro^2


||A||_Fro^2 = \sum_{i,j} A_{ij}^2 (Frobenius norm)
||vec(A)||_1 = \sum_{i,j} abs(A_{ij}) (Elementwise L1 norm)

For multiplicative-update (‘mu’) solver, the Frobenius norm (0.5 * ||X - WH||_Fro^2) can be changed into another beta-divergence loss, by changing the beta_loss parameter.

The objective function is minimized with an alternating minimization of W and H. If H is given and update_H=False, it solves for W only.

X : array-like, shape (n_samples, n_features)

Constant matrix.

W : array-like, shape (n_samples, n_components)

If init=’custom’, it is used as initial guess for the solution.

H : array-like, shape (n_components, n_features)

If init=’custom’, it is used as initial guess for the solution. If update_H=False, it is used as a constant, to solve for W only.

n_components : integer

Number of components, if n_components is not set all features are kept.

init : None | ‘random’ | ‘nndsvd’ | ‘nndsvda’ | ‘nndsvdar’ | ‘custom’

Method used to initialize the procedure. Default: ‘random’.

The default value will change from ‘random’ to None in version 0.23 to make it consistent with decomposition.NMF.

Valid options:

  • None: ‘nndsvd’ if n_components < n_features, otherwise ‘random’.
  • ‘random’: non-negative random matrices, scaled with:
    sqrt(X.mean() / n_components)
  • ‘nndsvd’: Nonnegative Double Singular Value Decomposition (NNDSVD)
    initialization (better for sparseness)
  • ‘nndsvda’: NNDSVD with zeros filled with the average of X
    (better when sparsity is not desired)
  • ‘nndsvdar’: NNDSVD with zeros filled with small random values
    (generally faster, less accurate alternative to NNDSVDa for when sparsity is not desired)
  • ‘custom’: use custom matrices W and H
update_H : boolean, default: True

Set to True, both W and H will be estimated from initial guesses. Set to False, only W will be estimated.

solver : ‘cd’ | ‘mu’

Numerical solver to use: ‘cd’ is a Coordinate Descent solver that uses Fast Hierarchical

Alternating Least Squares (Fast HALS).

‘mu’ is a Multiplicative Update solver.

New in version 0.17: Coordinate Descent solver.

New in version 0.19: Multiplicative Update solver.

beta_loss : float or string, default ‘frobenius’

String must be in {‘frobenius’, ‘kullback-leibler’, ‘itakura-saito’}. Beta divergence to be minimized, measuring the distance between X and the dot product WH. Note that values different from ‘frobenius’ (or 2) and ‘kullback-leibler’ (or 1) lead to significantly slower fits. Note that for beta_loss <= 0 (or ‘itakura-saito’), the input matrix X cannot contain zeros. Used only in ‘mu’ solver.

New in version 0.19.

tol : float, default: 1e-4

Tolerance of the stopping condition.

max_iter : integer, default: 200

Maximum number of iterations before timing out.

alpha : double, default: 0.

Constant that multiplies the regularization terms.

l1_ratio : double, default: 0.

The regularization mixing parameter, with 0 <= l1_ratio <= 1. For l1_ratio = 0 the penalty is an elementwise L2 penalty (aka Frobenius Norm). For l1_ratio = 1 it is an elementwise L1 penalty. For 0 < l1_ratio < 1, the penalty is a combination of L1 and L2.

regularization : ‘both’ | ‘components’ | ‘transformation’ | None

Select whether the regularization affects the components (H), the transformation (W), both or none of them.

random_state : int, RandomState instance or None, optional, default: None

If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by np.random.

verbose : integer, default: 0

The verbosity level.

shuffle : boolean, default: False

If true, randomize the order of coordinates in the CD solver.

W : array-like, shape (n_samples, n_components)

Solution to the non-negative least squares problem.

H : array-like, shape (n_components, n_features)

Solution to the non-negative least squares problem.

n_iter : int

Actual number of iterations.


Cichocki, Andrzej, and P. H. A. N. Anh-Huy. “Fast local algorithms for large scale nonnegative matrix and tensor factorizations.” IEICE transactions on fundamentals of electronics, communications and computer sciences 92.3: 708-721, 2009.

Fevotte, C., & Idier, J. (2011). Algorithms for nonnegative matrix factorization with the beta-divergence. Neural Computation, 23(9).


>>> import numpy as np
>>> X = np.array([[1,1], [2, 1], [3, 1.2], [4, 1], [5, 0.8], [6, 1]])
>>> from sklearn.decomposition import non_negative_factorization
>>> W, H, n_iter = non_negative_factorization(X, n_components=2,
... init='random', random_state=0)