NonNegative Matrix Factorization (NMF)
Find two nonnegative 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
Where:
A_Fro^2 = \sum_{i,j} A_{ij}^2 (Frobenius norm)
vec(A)_1 = \sum_{i,j} abs(A_{ij}) (Elementwise L1 norm)
For multiplicativeupdate (‘mu’) solver, the Frobenius norm (0.5 * X  WH_Fro^2) can be changed into another betadivergence loss, by changing the beta_loss parameter.
The objective function is minimized with an alternating minimization of W and H.
Read more in the User Guide.
Parameters: 

n_components : int or None 
Number of components, if n_components is not set all features are kept. 
init : ‘random’  ‘nndsvd’  ‘nndsvda’  ‘nndsvdar’  ‘custom’ 
Method used to initialize the procedure. Default: ‘nndsvd’ if n_components < n_features, otherwise random. Valid options: 
 ‘random’: nonnegative 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

solver : ‘cd’  ‘mu’ 
Numerical solver to use: ‘cd’ is a Coordinate Descent solver. ‘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’, ‘kullbackleibler’, ‘itakurasaito’}. Beta divergence to be minimized, measuring the distance between X and the dot product WH. Note that values different from ‘frobenius’ (or 2) and ‘kullbackleibler’ (or 1) lead to significantly slower fits. Note that for beta_loss <= 0 (or ‘itakurasaito’), the input matrix X cannot contain zeros. Used only in ‘mu’ solver. 
tol : float, default: 1e4 
Tolerance of the stopping condition. 
max_iter : integer, default: 200 
Maximum number of iterations before timing out. 
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 . 
alpha : double, default: 0. 
Constant that multiplies the regularization terms. Set it to zero to have no regularization. New in version 0.17: alpha used in the Coordinate Descent solver. 
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. New in version 0.17: Regularization parameter l1_ratio used in the Coordinate Descent solver. 
verbose : bool, default=False 
Whether to be verbose. 
shuffle : boolean, default: False 
If true, randomize the order of coordinates in the CD solver. New in version 0.17: shuffle parameter used in the Coordinate Descent solver. 
Attributes: 

components_ : array, [n_components, n_features] 
Factorization matrix, sometimes called ‘dictionary’. 
reconstruction_err_ : number 
Frobenius norm of the matrix difference, or betadivergence, between the training data X and the reconstructed data WH from the fitted model. 
n_iter_ : int 
Actual number of iterations. 
References
Cichocki, Andrzej, and P. H. A. N. AnhHuy. “Fast local algorithms for large scale nonnegative matrix and tensor factorizations.” IEICE transactions on fundamentals of electronics, communications and computer sciences 92.3: 708721, 2009.
Fevotte, C., & Idier, J. (2011). Algorithms for nonnegative matrix factorization with the betadivergence. Neural Computation, 23(9).
Examples
>>> 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 NMF
>>> model = NMF(n_components=2, init='random', random_state=0)
>>> W = model.fit_transform(X)
>>> H = model.components_
Methods
fit (X[, y])  Learn a NMF model for the data X. 
fit_transform (X[, y, W, H])  Learn a NMF model for the data X and returns the transformed data. 
get_params ([deep])  Get parameters for this estimator. 
inverse_transform (W)  Transform data back to its original space. 
set_params (**params)  Set the parameters of this estimator. 
transform (X)  Transform the data X according to the fitted NMF model 

__init__(n_components=None, init=None, solver=’cd’, beta_loss=’frobenius’, tol=0.0001, max_iter=200, random_state=None, alpha=0.0, l1_ratio=0.0, verbose=0, shuffle=False)
[source]

fit(X, y=None, **params)
[source]

Learn a NMF model for the data X.
Parameters: 

X : {arraylike, sparse matrix}, shape (n_samples, n_features) 
Data matrix to be decomposed 
y : Ignored 
Returns: 
 self


fit_transform(X, y=None, W=None, H=None)
[source]

Learn a NMF model for the data X and returns the transformed data.
This is more efficient than calling fit followed by transform.
Parameters: 

X : {arraylike, sparse matrix}, shape (n_samples, n_features) 
Data matrix to be decomposed 
y : Ignored 
W : arraylike, shape (n_samples, n_components) 
If init=’custom’, it is used as initial guess for the solution. 
H : arraylike, shape (n_components, n_features) 
If init=’custom’, it is used as initial guess for the solution. 
Returns: 

W : array, shape (n_samples, n_components) 
Transformed data. 

get_params(deep=True)
[source]

Get parameters for this estimator.
Parameters: 

deep : boolean, optional 
If True, will return the parameters for this estimator and contained subobjects that are estimators. 
Returns: 

params : mapping of string to any 
Parameter names mapped to their values. 

inverse_transform(W)
[source]

Transform data back to its original space.
Parameters: 

W : {arraylike, sparse matrix}, shape (n_samples, n_components) 
Transformed data matrix 
Returns: 

X : {arraylike, sparse matrix}, shape (n_samples, n_features) 
Data matrix of original shape  .. versionadded:: 0.18


set_params(**params)
[source]

Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter>
so that it’s possible to update each component of a nested object.

transform(X)
[source]

Transform the data X according to the fitted NMF model
Parameters: 

X : {arraylike, sparse matrix}, shape (n_samples, n_features) 
Data matrix to be transformed by the model 
Returns: 

W : array, shape (n_samples, n_components) 
Transformed data 