NMF: Non-negative Matrix Factorization Algorithms (NMF)

In nnTensor: Non-Negative Tensor Decomposition

Non-negative Matrix Factorization Algorithms (NMF)

Description

The input data is assumed to be non-negative matrix. NMF decompose the matrix to two low-dimensional factor matices. This function is also used as initialization step of tensor decomposition (see also NTF and NTD).

Usage

NMF(X, M=NULL, pseudocount=.Machine$double.eps, initU=NULL, initV=NULL,
  fixU=FALSE, fixV=FALSE,
  L1_U=1e-10, L1_V=1e-10, L2_U=1e-10, L2_V=1e-10, J = 3,
  rank.method=c("all", "ccc", "dispersion", "rss", "evar", "residuals",
    "sparseness.basis", "sparseness.coef", "sparseness2.basis",
    "sparseness2.coef", "norm.info.gain.basis", "norm.info.gain.coef",
    "singular",  "volume", "condition"), runtime=30,
  algorithm = c("Frobenius", "KL", "IS", "Pearson", "Hellinger", "Neyman",
    "Alpha", "Beta", "ALS", "PGD", "HALS", "GCD", "Projected", "NHR", "DTPP",
    "Orthogonal", "OrthReg", "t", "TV-GNMF"),
  Alpha = 1, Beta = 2, nu = 2,
  eta = 1e-04, thr1 = 1e-10, thr2 = 1e-10, tol = 1e-04,
  num.iter = 100,
  L_graph_U = NULL, L_graph_V = NULL,
  lambda_graph_U = 0, lambda_graph_V = 0,
  graph.method = c("Vicus", "LEM", "HLLE"),
  graph.K = 10, graph.alpha = 0.9,
  lambda_TV = 0, beta_TV = 0,
  viz = FALSE, figdir = NULL, verbose = FALSE)

Arguments

X	The input matrix which has N-rows and M-columns.
M	The mask matrix which has N-rows and M-columns. If the input matrix has missing values, specify the elements as 0 (otherwise 1).
pseudocount	The pseudo count to avoid zero division, when the element is zero (Default: Machine Epsilon).
initU	The initial values of factor matrix U, which has N-rows and J-columns (Default: NULL).
initV	The initial values of factor matrix V, which has M-rows and J-columns (Default: NULL).
fixU	Whether the factor matrix U is updated in each iteration step (Default: FALSE).
fixV	Whether the factor matrix V is updated in each iteration step (Default: FALSE).
L1_U	Paramter for L1 regularitation (Default: 1e-10). This also works as small positive constant to prevent division by zero, so should be set as 0.
L1_V	Paramter for L1 regularitation (Default: 1e-10). This also works as small positive constant to prevent division by zero, so should be set as 0.
L2_U	Paramter for L2 regularitation (Default: 1e-10).
L2_V	Paramter for L2 regularitation (Default: 1e-10).
J	The number of low-dimension (J < {N, M}). If a numerical vector is specified (e.g. 2:6), the appropriate rank is estimated.
rank.method	The rank estimation method (Default: "all"). Only if the J option is specified as a numerical vector longer than two, this option will be active.
runtime	The number of trials to estimate rank (Default: 10).
algorithm	NMF algorithms. "Frobenius", "KL", "IS", "Pearson", "Hellinger", "Neyman", "Alpha", "Beta", "ALS", "PGD", "HALS", "GCD", "Projected", "NHR", "DTPP", "Orthogonal", "OrthReg", "t", and "TV-GNMF" are available (Default: "Frobenius").
Alpha	The parameter of Alpha-divergence.
Beta	The parameter of Beta-divergence.
nu	The degrees of freedom parameter for t-NMF algorithm (Default: 2). Larger values approach Gaussian NMF. Only used when algorithm = "t".
eta	The stepsize for PGD algorithm (Default: 0.0001).
thr1	When error change rate is lower than thr1, the iteration is terminated (Default: 1E-10).
thr2	If the minus-value is generated, replaced as thr2 (Default: 1E-10). This value is used within the internal function .positive().
tol	The tolerance parameter used in GCD algorithm.
num.iter	The number of interation step (Default: 100).
L_graph_U	A pre-computed graph Laplacian matrix for rows (N x N). If NULL and lambda_graph_U > 0, it is computed automatically using Vicus::graphMatrix (Default: NULL).
L_graph_V	A pre-computed graph Laplacian matrix for columns (M x M). If NULL and lambda_graph_V > 0, it is computed automatically using Vicus::graphMatrix (Default: NULL).
lambda_graph_U	The weight for graph regularization on U (Default: 0). When set to a positive value, adds graph Laplacian regularization to the denominator of the multiplicative update rule for U.
lambda_graph_V	The weight for graph regularization on V (Default: 0). When set to a positive value, adds graph Laplacian regularization to the denominator of the multiplicative update rule for V.
graph.method	The method for computing the graph matrix via Vicus::graphMatrix. "Vicus", "LEM", and "HLLE" are available (Default: "Vicus").
graph.K	The number of nearest neighbors for graph construction (Default: 10).
graph.alpha	The alpha parameter for Vicus::graphMatrix (Default: 0.9).
lambda_TV	The weight for Total Variation regularization on V. Only used when algorithm = "TV-GNMF" (Default: 0).
beta_TV	The weight for graph regularization in TV-GNMF algorithm. Only used when algorithm = "TV-GNMF" (Default: 0).
viz	If viz == TRUE, internal reconstructed matrix can be visualized.
figdir	The directory for saving the figure, when viz == TRUE.
verbose	If verbose == TRUE, Error change rate is generated in console window.

Value

U : A matrix which has N-rows and J-columns (J < {N, M}). V : A matrix which has M-rows and J-columns (J < {N, M}). J : The number of dimension (J < {N, M}). RecError : The reconstruction error between data tensor and reconstructed tensor from U and V. TrainRecError : The reconstruction error calculated by training set (observed values specified by M). TestRecError : The reconstruction error calculated by test set (missing values specified by M). RelChange : The relative change of the error. Trial : All the results of the trials to estimate the rank. Runtime : The number of the trials to estimate the rank. RankMethod : The rank estimation method.

Author(s)

Koki Tsuyuzaki

References

Andrzej CICHOCK, et. al., (2009). Nonnegative Matrix and Tensor Factorizations. John Wiley & Sons, Ltd

Keigo Kimura, (2017). A Study on Efficient Algorithms for Nonnegative Matrix/ Tensor Factorization. Hokkaido University Collection of Scholarly and Academic Papers

K. Yoshii, K. Itoyama, and M. Goto (2016). Student's t Nonnegative Matrix Factorization and Positive Semidefinite Tensor Factorization for Single-Channel Audio Source Separation. ICASSP, pp.51-55.

C. Leng, Z. Chen, G. Cai, I. Cheng, Z. Xiong, J. Tian, and A. Basu (2016). Total Variation Constrained Graph Regularized NMF for Medical Image Registration. IEEE ICIP.

D. Cai, X. He, J. Han, and T. S. Huang (2011). Graph Regularized Nonnegative Matrix Factorization for Data Representation. IEEE Transactions on PAMI, 33(8), pp.1548-1560.

Examples

  if(interactive()){
    # Test data
    matdata <- toyModel(model = "NMF")

    # Simple usage
    out <- NMF(matdata, J=5)

    # Rank estimation mode (single method)
    out2 <- NMF(matdata, J=2:10, rank.method="ccc", runtime=3)
    plot(out2)

    # Rank estimation mode (all method)
    out3 <- NMF(matdata, J=2:10, rank.method="all", runtime=10)
    plot(out3)

    # t-NMF (robust to outliers)
    out4 <- NMF(matdata, J=5, algorithm="t", nu=2)

    # Graph-regularized NMF
    out5 <- NMF(matdata, J=5, algorithm="Frobenius",
      lambda_graph_V=0.1, graph.K=5)

    # TV-GNMF (Total Variation + Graph regularization)
    out6 <- NMF(matdata, J=5, algorithm="TV-GNMF",
      lambda_TV=0.1, beta_TV=0.1,
      lambda_graph_V=0.1, graph.K=5)
  }

nnTensor documentation built on May 8, 2026, 5:07 p.m.