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# `sklearn.decomposition`.non_negative_factorization

sklearn.decomposition.non_negative_factorization(X, W=None, H=None, n_components=None, *, 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)

  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:

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

  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 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.

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

      Constant matrix.

    W array-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`.

    H array-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_components int, default=None

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

    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_H bool, 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.

      New in version 0.17: Coordinate Descent solver.

      New in version 0.19: Multiplicative Update solver.

    beta_loss float 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.

      New in version 0.19.

    tol float, default=1e-4

      Tolerance of the stopping condition.

    max_iter int, default=200

      Maximum number of iterations before timing out.

    alpha_W float, default=0.0

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

      New in version 1.0.

    alpha_H float 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`.

      New in version 1.0.

    l1_ratio float, 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_state int, 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.

    verbose int, default=0

      The verbosity level.

    shuffle bool, default=False

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

  Returns:
    W ndarray of shape (n_samples, n_components)

      Solution to the non-negative least squares problem.

    H ndarray of shape (n_components, n_features)

      Solution to the non-negative least squares problem.

    n_iter int

      Actual number of iterations.

  References

  [1]

  “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.

  [2]

  “Algorithms for nonnegative matrix factorization with the beta-divergence” Fevotte, C., & Idier, J. (2011). 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 non_negative_factorization
      >>> W, H, n_iter = non_negative_factorization(
      ...     X, n_components=2, init='random', random_state=0)