# NMF Algorithm/Updates for Kullback-Leibler Divergence

## Description

The built-in NMF algorithms described here minimise the Kullback-Leibler divergence (KL) between an NMF model and a target matrix. They use the updates for the basis and coefficient matrices (`W` and `H`) defined by Brunet et al. (2004), which are essentially those from Lee et al. (2001), with an stabilisation step that shift up all entries from zero every 10 iterations, to a very small positive value.

`nmf_update.brunet` implements in C++ an optimised version of the single update step.

Algorithms ‘brunet’ and ‘.R#brunet’ provide the complete NMF algorithm from Brunet et al. (2004), using the C++-optimised and pure R updates `nmf_update.brunet` and `nmf_update.brunet_R` respectively.

Algorithm ‘KL’ provides an NMF algorithm based on the C++-optimised version of the updates from Brunet et al. (2004), which uses the stationarity of the objective value as a stopping criterion `nmf.stop.stationary`, instead of the stationarity of the connectivity matrix `nmf.stop.connectivity` as used by ‘brunet’.

```file.show(system.file('matlab', 'brunet.m', package='NMF'))```

## Usage

```nmf_update.brunet_R(i, v, x, eps = .Machine\$double.eps, ...)

nmf_update.brunet(i, v, x, copy = FALSE, eps = .Machine\$double.eps, ...)

nmfAlgorithm.brunet_R(..., .stop = NULL, maxIter = 2000, eps = .Machine\$double.eps,
stopconv = 40, check.interval = 10)

nmfAlgorithm.brunet(..., .stop = NULL, maxIter = 2000, copy = FALSE, eps = .Machine\$double.eps,
stopconv = 40, check.interval = 10)

nmfAlgorithm.KL(..., .stop = NULL, maxIter = 2000, copy = FALSE, eps = .Machine\$double.eps,
stationary.th = .Machine\$double.eps, check.interval = 5 * check.niter, check.niter = 10L)

nmfAlgorithm.brunet_M(..., object, y, x)```

## Arguments

i
current iteration number.
v
target matrix.
x
current NMF model, as an `NMF-class` object.
eps
small numeric value used to ensure numeric stability, by shifting up entries from zero to this fixed value.
...
extra arguments. These are generally not used and present only to allow other arguments from the main call to be passed to the initialisation and stopping criterion functions (slots `onInit` and `Stop` respectively).
copy
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`.
.stop
specification of a stopping criterion, that is used instead of the one associated to the NMF algorithm. It may be specified as:
• the access key of a registered stopping criterion;
• a single integer that specifies the exact number of iterations to perform, which will be honoured unless a lower value is explicitly passed in argument `maxIter`.
• a single numeric value that specifies the stationnarity threshold for the objective function, used in with `nmf.stop.stationary`;
• a function with signature ```(object="NMFStrategy", i="integer", y="matrix", x="NMF", ...)```, where `object` is the `NMFStrategy` object that describes the algorithm being run, `i` is the current iteration, `y` is the target matrix and `x` is the current value of the NMF model.
maxIter
maximum number of iterations to perform.
object
an object computed using some algorithm, or that describes an algorithm itself.
y
data object, e.g. a target matrix
stopconv
number of iterations intervals over which the connectivity matrix must not change for stationarity to be achieved.
check.interval
interval (in number of iterations) on which the stopping criterion is computed.
stationary.th
maximum absolute value of the gradient, for the objective function to be considered stationary.
check.niter
number of successive iteration used to compute the stationnary criterion.

## Source

Original MATLAB files and references can be found at:

This software and its documentation are copyright 2004 by the Broad Institute/Massachusetts Institute of Technology. All rights are reserved. This software is supplied without any warranty or guaranteed support whatsoever. Neither the Broad Institute nor MIT can not be responsible for its use, misuse, or functionality.

## Details

`nmf_update.brunet_R` implements in pure R a single update step, i.e. it updates both matrices.

## References

Brunet J, Tamayo P, Golub TR and Mesirov JP (2004). "Metagenes and molecular pattern discovery using matrix factorization." _Proceedings of the National Academy of Sciences of the United States of America_, *101*(12), pp. 4164-9. ISSN 0027-8424, , .

Lee DD and Seung H (2001). "Algorithms for non-negative matrix factorization." _Advances in neural information processing systems_. .

## Author

Original implementation in MATLAB: Jean-Philippe Brunet brunet@broad.mit.edu Port to R and optimisation in C++: Renaud Gaujoux