# `sklearn.decomposition`.NMF¶

class `sklearn.decomposition.``NMF`(n_components=None, init=None, solver='cd', tol=0.0001, max_iter=200, random_state=None, alpha=0.0, l1_ratio=0.0, verbose=0, shuffle=False, nls_max_iter=2000, sparseness=None, beta=1, eta=0.1)[source]

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
```

Where:

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

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: ‘nndsvdar’ if n_components < n_features, otherwise random. Valid options: ‘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 solver : ‘pg’ | ‘cd’ Numerical solver to use: ‘pg’ is a Projected Gradient solver (deprecated). ‘cd’ is a Coordinate Descent solver (recommended). New in version 0.17: Coordinate Descent solver. Changed in version 0.17: Deprecated Projected Gradient solver. tol : double, default: 1e-4 Tolerance value used in stopping conditions. max_iter : integer, default: 200 Number of iterations to compute. random_state : integer seed, RandomState instance, or None (default) Random number generator seed control. 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. 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. nls_max_iter : integer, default: 2000 Number of iterations in NLS subproblem. Used only in the deprecated ‘pg’ solver. Changed in version 0.17: Deprecated Projected Gradient solver. Use Coordinate Descent solver instead. sparseness : ‘data’ | ‘components’ | None, default: None Where to enforce sparsity in the model. Used only in the deprecated ‘pg’ solver. Changed in version 0.17: Deprecated Projected Gradient solver. Use Coordinate Descent solver instead. beta : double, default: 1 Degree of sparseness, if sparseness is not None. Larger values mean more sparseness. Used only in the deprecated ‘pg’ solver. Changed in version 0.17: Deprecated Projected Gradient solver. Use Coordinate Descent solver instead. eta : double, default: 0.1 Degree of correctness to maintain, if sparsity is not None. Smaller values mean larger error. Used only in the deprecated ‘pg’ solver. Changed in version 0.17: Deprecated Projected Gradient solver. Use Coordinate Descent solver instead. components_ : array, [n_components, n_features] Non-negative components of the data. reconstruction_err_ : number Frobenius norm of the matrix difference between the training data and the reconstructed data from the fit produced by the model. `|| X - WH ||_2` n_iter_ : int Actual number of iterations.

References

C.-J. Lin. Projected gradient methods for non-negative matrix factorization. Neural Computation, 19(2007), 2756-2779. http://www.csie.ntu.edu.tw/~cjlin/nmf/

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.

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)
>>> model.fit(X)
NMF(alpha=0.0, beta=1, eta=0.1, init='random', l1_ratio=0.0, max_iter=200,
n_components=2, nls_max_iter=2000, random_state=0, shuffle=False,
solver='cd', sparseness=None, tol=0.0001, verbose=0)
```
```>>> model.components_
array([[ 2.09783018,  0.30560234],
[ 2.13443044,  2.13171694]])
>>> model.reconstruction_err_
0.00115993...
```

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', tol=0.0001, max_iter=200, random_state=None, alpha=0.0, l1_ratio=0.0, verbose=0, shuffle=False, nls_max_iter=2000, sparseness=None, beta=1, eta=0.1)[source]
`fit`(X, y=None, **params)[source]

Learn a NMF model for the data X.

Parameters: X: {array-like, sparse matrix}, shape (n_samples, n_features) : Data matrix to be decomposed 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: {array-like, sparse matrix}, shape (n_samples, n_features) : Data matrix to be decomposed 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. 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. 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: {array-like, sparse matrix}, shape (n_samples, n_components) : Transformed data matrix X: {array-like, 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.

Returns: self :
`transform`(X)[source]

Transform the data X according to the fitted NMF model

Parameters: X: {array-like, sparse matrix}, shape (n_samples, n_features) : Data matrix to be transformed by the model W: array, shape (n_samples, n_components) : Transformed data