sklearn.decomposition.non_negative_factorization(X, W=None, H=None, n_components='warn', *, init=None, update_H=True, solver='cd', beta_loss='frobenius', tol=0.0001, max_iter=200, alpha_W=0.0, alpha_H='same', l1_ratio=0.0, 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:

\[ \begin{align}\begin{aligned}L(W, H) &= 0.5 * ||X - WH||_{loss}^2\\&+ alpha\_W * l1\_ratio * n\_features * ||vec(W)||_1\\&+ alpha\_H * l1\_ratio * n\_samples * ||vec(H)||_1\\&+ 0.5 * alpha\_W * (1 - l1\_ratio) * n\_features * ||W||_{Fro}^2\\&+ 0.5 * alpha\_H * (1 - l1\_ratio) * n\_samples * ||H||_{Fro}^2\end{aligned}\end{align} \]


\(||A||_{Fro}^2 = \sum_{i,j} A_{ij}^2\) (Frobenius norm)

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

The generic norm \(||X - WH||_{loss}^2\) may represent the Frobenius norm or another supported beta-divergence loss. The choice between options is controlled by the beta_loss parameter.

The regularization terms are scaled by n_features for W and by n_samples for H to keep their impact balanced with respect to one another and to the data fit term as independent as possible of the size n_samples of the training set.

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.

Note that the transformed data is named W and the components matrix is named H. In the NMF literature, the naming convention is usually the opposite since the data matrix X is transposed.

X{array-like, sparse matrix} of shape (n_samples, n_features)

Constant matrix.

Warray-like of shape (n_samples, n_components), default=None

If init='custom', it is used as initial guess for the solution. If update_H=False, it is initialised as an array of zeros, unless solver='mu', then it is filled with values calculated by np.sqrt(X.mean() / self._n_components). If None, uses the initialisation method specified in init.

Harray-like of shape (n_components, n_features), default=None

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. If None, uses the initialisation method specified in init.

n_componentsint or {‘auto’} or None, default=None

Number of components, if n_components is not set all features are kept. If n_components='auto', the number of components is automatically inferred from W or H shapes.

Changed in version 1.4: Added 'auto' value.

init{‘random’, ‘nndsvd’, ‘nndsvda’, ‘nndsvdar’, ‘custom’}, default=None

Method used to initialize the procedure.

Valid options:

  • None: ‘nndsvda’ 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’: If update_H=True, use custom matrices W and H which must both be provided. If update_H=False, then only custom matrix H is used.

Changed in version 0.23: The default value of init changed from ‘random’ to None in 0.23.

Changed in version 1.1: When init=None and n_components is less than n_samples and n_features defaults to nndsvda instead of nndsvd.

update_Hbool, 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’}, default=’cd’

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.

Added in version 0.17: Coordinate Descent solver.

Added in version 0.19: Multiplicative Update solver.

beta_lossfloat or {‘frobenius’, ‘kullback-leibler’, ‘itakura-saito’}, default=’frobenius’

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.

Added in version 0.19.

tolfloat, default=1e-4

Tolerance of the stopping condition.

max_iterint, default=200

Maximum number of iterations before timing out.

alpha_Wfloat, default=0.0

Constant that multiplies the regularization terms of W. Set it to zero (default) to have no regularization on W.

Added in version 1.0.

alpha_Hfloat or “same”, default=”same”

Constant that multiplies the regularization terms of H. Set it to zero to have no regularization on H. If “same” (default), it takes the same value as alpha_W.

Added in version 1.0.

l1_ratiofloat, default=0.0

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.

random_stateint, RandomState instance or None, default=None

Used for NMF initialisation (when init == ‘nndsvdar’ or ‘random’), and in Coordinate Descent. Pass an int for reproducible results across multiple function calls. See Glossary.

verboseint, default=0

The verbosity level.

shufflebool, default=False

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

Wndarray of shape (n_samples, n_components)

Solution to the non-negative least squares problem.

Hndarray of shape (n_components, n_features)

Solution to the non-negative least squares problem.


Actual number of iterations.



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


“Algorithms for nonnegative matrix factorization with the beta-divergence” Fevotte, C., & Idier, J. (2011). 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)