Multiplicative updates from Lee et al. (2001) for standard Nonnegative Matrix Factorization models \(V \approx W H\), where the distance between the target matrix and its NMF estimate is measured by the -- euclidean -- Frobenius norm.
nmf_update.euclidean.w
and
nmf_update.euclidean.h
compute the updated basis
and coefficient matrices respectively. They use a
C++ implementation which is optimised for speed
and memory usage.
nmf_update.euclidean.w_R
and
nmf_update.euclidean.h_R
implement the same
updates in plain R.
nmf_update.euclidean.h(v, w, h, eps = 10^-9,
nbterms = 0L, ncterms = 0L, copy = TRUE) nmf_update.euclidean.h_R(v, w, h, wh = NULL, eps = 10^-9)
nmf_update.euclidean.w(v, w, h, eps = 10^-9,
nbterms = 0L, ncterms = 0L, weight = NULL, copy = TRUE)
nmf_update.euclidean.w_R(v, w, h, wh = NULL, eps = 10^-9)
a matrix of the same dimension as the input matrix to
update (i.e. w
or h
). If copy=FALSE
,
the returned matrix uses the same memory as the input
object.
small numeric value used to ensure numeric stability, by shifting up entries from zero to this fixed value.
already computed NMF estimate used to compute the denominator term.
numeric vector of sample weights, e.g.,
used to normalise samples coming from multiple datasets.
It must be of the same length as the number of
samples/columns in v
-- and h
.
target matrix
current basis matrix
current coefficient matrix
number of fixed basis terms
number of fixed coefficient terms
logical that indicates if the update should
be made on the original matrix directly (FALSE
) or
on a copy (TRUE
- default). With copy=FALSE
the memory footprint is very small, and some speed-up may
be achieved in the case of big matrices. However, greater
care should be taken due the side effect. We recommend
that only experienced users use copy=TRUE
.
Update definitions by Lee2001.
C++ optimised implementation by Renaud Gaujoux.
The coefficient matrix (H
) is updated as follows:
$$ H_{kj} \leftarrow \frac{\max(H_{kj} W^T V)_{kj},
\varepsilon) }{(W^T W H)_{kj} + \varepsilon} $$
These updates are used by the built-in NMF algorithms
Frobenius
and
lee
.
The basis matrix (W
) is updated as follows: $$
W_ik \leftarrow \frac{\max(W_ik (V H^T)_ik, \varepsilon)
}{ (W H H^T)_ik + \varepsilon} $$
Lee DD and Seung H (2001). "Algorithms for non-negative matrix factorization." _Advances in neural information processing systems_. <URL: http://scholar.google.com/scholar?q=intitle:Algorithms+for+non-negative+matrix+factorization>.