# NMF Algorithm/Updates for Frobenius Norm

## Description

The built-in NMF algorithms described here minimise the Frobenius norm (Euclidean distance) between an NMF model and a target matrix. They use the updates for the basis and coefficient matrices (`W` and `H`) defined by Lee et al. (2001).

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

Algorithms ‘lee’ and ‘.R#lee’ provide the complete NMF algorithm from Lee et al. (2001), using the C++-optimised and pure R updates `nmf_update.lee` and `nmf_update.lee_R` respectively.

Algorithm ‘Frobenius’ provides an NMF algorithm based on the C++-optimised version of the updates from Lee et al. (2001), 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 ‘lee’.

## Usage

```nmf_update.lee_R(i, v, x, rescale = TRUE, eps = 10^-9, ...)

nmf_update.lee(i, v, x, rescale = TRUE, copy = FALSE, eps = 10^-9, weight = NULL,
...)

nmfAlgorithm.lee_R(..., .stop = NULL, maxIter = 2000, rescale = TRUE, eps = 10^-9,
stopconv = 40, check.interval = 10)

nmfAlgorithm.lee(..., .stop = NULL, maxIter = 2000, rescale = TRUE, copy = FALSE,
eps = 10^-9, weight = NULL, stopconv = 40, check.interval = 10)

nmfAlgorithm.Frobenius(..., .stop = NULL, maxIter = 2000, rescale = TRUE, copy = FALSE,
eps = 10^-9, weight = NULL, stationary.th = .Machine\$double.eps, check.interval = 5 *
check.niter, check.niter = 10L)```

## Arguments

rescale
logical that indicates if the basis matrix `W` should be rescaled so that its columns sum up to one.
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.
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.
weight
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`.

## Details

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

## References

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

## Author

Original update definition: D D Lee and HS Seung Port to R and optimisation in C++: Renaud Gaujoux