This function solves the following nonnegative least square linear problem using normal equations and the fast combinatorial strategy from Van Benthem et al. (2004):
$$ \begin{array}{l} \min \|Y - X K\|_F\\ \mbox{s.t. } K>=0 \end{array} $$
where \(Y\) and \(X\) are two real matrices of dimension \(n \times p\) and \(n \times r\) respectively, and \(\|.\|_F\) is the Frobenius norm.
The algorithm is very fast compared to other approaches, as it is optimised for handling multiple right-hand sides.
fcnnls(x, y, ...) # S4 method for matrix,matrix
fcnnls(x, y, verbose = FALSE,
pseudo = TRUE, ...)
A list containing the following components:
the estimated optimal matrix \(K\).
the fitted matrix \(X K\).
the residual matrix \(Y - X K\).
the residual sum of squares between the fitted matrix \(X K\) and the target matrix \(Y\). That is the sum of the square residuals.
a \(r x p\) logical matrix containing the passive set, that is the set of entries in \(K\) that are not null (i.e. strictly positive).
a logical that
is TRUE if the computation was performed using the
pseudoinverse. See argument pseudo.
extra arguments passed to the internal
function .fcnnls. Currently not used.
toggle verbosity (default is
FALSE).
the coefficient matrix
the target matrix to be approximated by \(X K\).
By default (pseudo=FALSE) the
algorithm uses Gaussian elimination to solve the
successive internal linear problems, using the
solve function. If pseudo=TRUE the
algorithm uses Moore-Penrose generalized
pseudoinverse from the
corpcor package instead of solve.
signature(x = "matrix", y =
"matrix"): This method wraps a call to the internal
function .fcnnls, and formats the results in a
similar way as other lest-squares methods such as
lm.
signature(x = "numeric", y =
"matrix"): Shortcut for fcnnls(as.matrix(x), y,
...).
signature(x = "ANY", y = "numeric"):
Shortcut for fcnnls(x, as.matrix(y), ...).
Original MATLAB code : Van Benthem and Keenan
Adaption of MATLAB code for SNMF/R(L): H. Kim
Adaptation to the NMF package framework: Renaud Gaujoux
Within the NMF package, this algorithm is used
internally by the SNMF/R(L) algorithm from Kim et
al. (2007) to solve general Nonnegative Matrix
Factorization (NMF) problems, using alternating
nonnegative constrained least-squares. That is by
iteratively and alternatively estimate each matrix
factor.
The algorithm is an active/passive set method, which rearrange the right-hand side to reduce the number of pseudo-inverse calculations. It uses the unconstrained solution \(K_u\) obtained from the unconstrained least squares problem, i.e. \(\min \|Y - X K\|_F^2\) , so as to determine the initial passive sets.
The function fcnnls is provided separately so that
it can be used to solve other types of nonnegative least
squares problem. For faster computation, when multiple
nonnegative least square fits are needed, it is
recommended to directly use the function
.fcnnls.
The code of this function is a port from the original MATLAB code provided by Kim et al. (2007).
Original MATLAB code from Van Benthem and Keenan, slightly modified by H. Kim:(http://www.cc.gatech.edu/~hpark/software/fcnnls.m)
Van Benthem M and Keenan MR (2004). "Fast algorithm for the solution of large-scale non-negativity-constrained least squares problems." _Journal of Chemometrics_, *18*(10), pp. 441-450. ISSN 0886-9383, <URL: http://dx.doi.org/10.1002/cem.889>, <URL: http://doi.wiley.com/10.1002/cem.889>.
Kim H and Park H (2007). "Sparse non-negative matrix factorizations via alternating non-negativity-constrained least squares for microarray data analysis." _Bioinformatics (Oxford, England)_, *23*(12), pp. 1495-502. ISSN 1460-2059, <URL: http://dx.doi.org/10.1093/bioinformatics/btm134>, <URL: http://www.ncbi.nlm.nih.gov/pubmed/17483501>.
nmf
# roxygen generated flag
options(R_CHECK_RUNNING_EXAMPLES_=TRUE)
## Define a random nonnegative matrix matrix
n <- 200; p <- 20; r <- 3
V <- rmatrix(n, p)
## Compute the optimal matrix K for a given X matrix
X <- rmatrix(n, r)
res <- fcnnls(X, V)
## Compute the same thing using the Moore-Penrose generalized pseudoinverse
res <- fcnnls(X, V, pseudo=TRUE)
## It also works in the case of single vectors
y <- runif(n)
res <- fcnnls(X, y)
# or
res <- fcnnls(X[,1], y)
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