NAG CL Interface
f01sbc (real_nmf_rcomm)
1
Purpose
f01sbc computes a nonnegative matrix factorization for a real nonnegative $m$ by $n$ matrix $A$. It uses reverse communication for evaluating matrix products, so that the matrix $A$ is not accessed explicitly.
2
Specification
void 
f01sbc (Integer *irevcm,
Integer m,
Integer n,
Integer k,
double w[],
Integer pdw,
double h[],
Integer pdh,
double ht[],
Integer pdht,
Integer seed,
double errtol,
double comm[],
Integer icomm[],
NagError *fail) 

The function may be called by the names: f01sbc or nag_matop_real_nmf_rcomm.
3
Description
The matrix
$A$ is factorized into the product of an
$m$ by
$k$ matrix
$W$ and a
$k$ by
$n$ matrix
$H$, both with nonnegative elements. The factorization is approximate,
$A\approx WH$, with
$W$ and
$H$ chosen to minimize the functional
You are free to choose any value for $k$, provided $k<\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$. The product $WH$ will then be a lowrank approximation to $A$, with rank at most $k$.
f01sbc finds $W$ and $H$ using an iterative method known as the Hierarchical Alternating Least Squares algorithm. You may specify initial values for $W$ and $H$, or you may provide a seed value for f01sbc to generate the initial values using a random number generator.
f01sbc does not explicitly need to access the elements of $A$; it only requires the result of matrix multiplications of the form $AX$ or ${A}^{\mathrm{T}}Y$. A reverse communication interface is used, in which control is returned to the calling program whenever a matrix product is required.
4
References
Cichocki A and Phan A–H (2009) Fast local algorithms for large scale nonnegative matrix and tensor factorizations IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E92–A 708–721
Cichocki A, Zdunek R and Amari S–I (2007) Hierarchical ALS algorithms for nonnegative matrix and 3D tensor factorization Lecture Notes in Computer Science 4666 Springer 169–176
Ho N–D (2008) Nonnegative matrix factorization algorithms and applications PhD Thesis Univ. Catholique de Louvain
5
Arguments
Note: this function uses
reverse communication. Its use involves an initial entry, intermediate exits and reentries, and a final exit, as indicated by the argument
irevcm. Between intermediate exits and reentries,
all arguments other than w and
ht must remain unchanged.

1:
$\mathbf{irevcm}$ – Integer *
Input/Output

On initial entry: must be set to $0$.
On intermediate exit:
specifies what action you must take before reentering
f01sbc with
irevcm unchanged. The value of
irevcm should be interpreted as follows:
 ${\mathbf{irevcm}}=1$

Indicates the start of a new iteration. No action is required by you, but w and h are available for printing, and a limit on the number of iterations can be applied.
 ${\mathbf{irevcm}}=2$

Indicates that before reentry to f01sbc, the product ${A}^{\mathrm{T}}W$ must be computed and stored in ht.
 ${\mathbf{irevcm}}=3$

Indicates that before reentry to f01sbc, the product $A{H}^{\mathrm{T}}$ must be computed and stored in w.
Note: any values you return to f01sbc as part of the reverse communication procedure should not include floatingpoint NaN (Not a Number) or infinity values, since these are not handled by f01sbc. If your code inadvertently does return any NaNs or infinities, f01sbc is likely to produce unexpected results.

2:
$\mathbf{m}$ – Integer
Input

On entry: $m$, the number of rows of the matrix $A$. Also the number of rows of the matrix $W$.
Constraint:
${\mathbf{m}}\ge 2$.

3:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of columns of the matrix $A$. Also the number of columns of the matrix $H$.
Constraint:
${\mathbf{n}}\ge 2$.

4:
$\mathbf{k}$ – Integer
Input

On entry:
$k$, the number of columns of the matrix
$W$. Also, the number of rows of the matrix
$H$. See
Section 9.2 for further details.
Constraint:
$1\le {\mathbf{k}}<\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.

5:
$\mathbf{w}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, of the array
w
must be at least
${\mathbf{pdw}}\times {\mathbf{k}}$.
The $\left(i,j\right)$th element of the matrix $W$ is stored in ${\mathbf{w}}\left[\left(j1\right)\times {\mathbf{pdw}}+i1\right]$.
On initial entry:
 if ${\mathbf{seed}}\le 0$, w should be set to an initial iterate for the nonnegative matrix factor, $W$.
 If ${\mathbf{seed}}\ge 1$, w need not be set. f01sbc will generate a random initial iterate.
On intermediate exit:
if
${\mathbf{irevcm}}=1$ or
$2$,
w contains the current iterate of the
$m\times k$ nonnegative matrix
$W$.
On intermediate reentry:
 if ${\mathbf{irevcm}}=3$, w must contain $A{H}^{\mathrm{T}}$, where ${H}^{\mathrm{T}}$ is stored in $\mathit{ht}$.
 If ${\mathbf{irevcm}}=0$, $1$ or $2$, w must not be changed.
On final exit:
w contains the
$m\times k$ nonnegative matrix
$W$.

6:
$\mathbf{pdw}$ – Integer
Input

On entry: the stride separating matrix row elements in the array
w.
Constraint:
${\mathbf{pdw}}\ge {\mathbf{m}}$.

7:
$\mathbf{h}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, of the array
h
must be at least
${\mathbf{pdh}}\times {\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $H$ is stored in ${\mathbf{h}}\left[\left(j1\right)\times {\mathbf{pdh}}+i1\right]$.
On initial entry:
 if ${\mathbf{seed}}\le 0$, h should be set to an initial iterate for the nonnegative matrix factor, $H$.
 If ${\mathbf{seed}}\ge 1$, h need not be set. f01sbc will generate a random initial iterate.
On intermediate exit:
if
${\mathbf{irevcm}}=1$,
h contains the current iterate of the
$k\times n$ nonnegative matrix
$H$.
On intermediate reentry:
h must not be changed.
On final exit:
h contains the
$k\times n$ nonnegative matrix
$H$.

8:
$\mathbf{pdh}$ – Integer
Input

On entry: the stride separating matrix row elements in the array
h.
Constraint:
${\mathbf{pdh}}\ge {\mathbf{k}}$.

9:
$\mathbf{ht}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, of the array
ht
must be at least
${\mathbf{pdht}}\times {\mathbf{k}}$.
The $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{ht}}\left[\left(j1\right)\times {\mathbf{pdht}}+i1\right]$.
On initial entry:
ht need not be set.
On intermediate exit:
if
${\mathbf{irevcm}}=3$,
ht contains the
$n\times k$ nonnegative matrix
${H}^{\mathrm{T}}$, which is required in order to from
$A{H}^{\mathrm{T}}$.
On intermediate reentry: if
${\mathbf{irevcm}}=2$,
ht must contain
${A}^{\mathrm{T}}W$.
If
${\mathbf{irevcm}}=0$,
$1$ or
$3$,
ht must not be changed.
On final exit:
ht is undefined.

10:
$\mathbf{pdht}$ – Integer
Input

On entry: the stride separating matrix row elements in the array
ht.
Constraint:
${\mathbf{pdht}}\ge {\mathbf{n}}$.

11:
$\mathbf{seed}$ – Integer
Input

On initial entry:
 if ${\mathbf{seed}}\le 0$, the supplied values of $W$ and $H$ are used for the initial iterate.
 If ${\mathbf{seed}}\ge 1$, the value of seed is used to seed a random number generator for the initial iterates $W$ and $H$. See Section 9.3 for further details.

12:
$\mathbf{errtol}$ – double
Input

On entry: the convergence tolerance for when the Hierarchical Alternating Least Squares iteration has reached a stationary point. If ${\mathbf{errtol}}\le 0.0$, $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\times \sqrt{\mathit{machineprecision}}$ is used.

13:
$\mathbf{comm}\left[\left(2\times {\mathbf{m}}+{\mathbf{n}}\right)\times {\mathbf{k}}+3\right]$ – double
Communication Array

Note: the dimension,
dim, of the array
comm
must be at least
$\left(2\times {\mathbf{m}}+{\mathbf{n}}\right)\times {\mathbf{k}}+3$.

14:
$\mathbf{icomm}\left[9\right]$ – Integer
Communication Array


15:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INIT_ESTIMATE

An internal error occurred when generating initial values for
w and
h. Please contact
NAG.
 NE_INT

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}\ge 2$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 2$.
On initial entry, ${\mathbf{irevcm}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{irevcm}}=0$.
On intermediate reentry, ${\mathbf{irevcm}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{irevcm}}=1$, $2$ or $3$.
 NE_INT_2

On entry, ${\mathbf{pdh}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{k}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pdh}}\ge {\mathbf{k}}$.
On entry, ${\mathbf{pdht}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pdht}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdw}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pdw}}\ge {\mathbf{m}}$.
 NE_INT_3

On entry, ${\mathbf{k}}=\u2329\mathit{\text{value}}\u232a$, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $1\le {\mathbf{k}}<\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_INVALID_ARRAY

On entry, one of more of the elements of
w or
h were negative.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
The Hierarchical Alternating Least Squares algorithm used by
f01sbc is locally convergent; it is guaranteed to converge to a stationary point of
$f\left(W,H\right)$, but this may not be the global minimum. The iteration is deemed to have converged if the gradient of
$f\left(W,H\right)$ is less than
errtol times the gradient at the initial values of
$W$ and
$H$.
Due to the local convergence property, you may wish to run f01sbc multiple times with different starting iterates. This can be done by explicitly providing the starting values of $W$ and $H$ each time, or by choosing a different random seed each time.
Note that even if
f01sbc exits with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_2, the factorization given by
$W$ and
$H$ may still be a good enough approximation to be useful.
8
Parallelism and Performance
f01sbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01sbc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
f01sbc is designed to be used when $A$ is large and sparse. Whenever a matrix multiplication is required, the function will return control to the calling program so that the multiplication can be done in the most efficient way possible. Note that $W$ and $H$ will not, in general, be sparse even if $A$ is sparse.
If
$A$ is small and dense, then
f01sac can be used to compute
$W$ and
$H$ without the use of a reverse communication interface.
9.1
Uniqueness
Note that nonnegative matrix factorization is not unique. For a factorization given by the matrices $W$ and $H$, an equally good solution is given by $WD$ and ${D}^{1}H$, where $D$ is any real nonnegative $k\times k$ matrix whose inverse is also nonnegative. In f01sbc, $W$ and $H$ are normalized so that the columns of $W$ have unit length.
9.2
Choice of $k$
The most appropriate choice of the factorization rank, $k$, is often problem dependent. Details of your particular application may help in guiding your choice of $k$, for example, it may be known a priori that the data in $A$ naturally falls into a certain number of categories.
Alternatively, trial and error can be used. Compute nonnegative matrix factorizations for several different values of $k$ (typically with $k\ll \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$) and select the one that performs the best.
Finally, it is also possible to use a singular value decomposition of
$A$ to guide your choice of
$k$, by looking for an abrupt decay in the size of the singular values of
$A$. The singular value decomposition can be computed using
f12fbc.
9.3
Generating Random Initial Iterates
If
${\mathbf{seed}}\ge 1$ on entry, then
f01sbc uses the functions
g05kfc and
g05sac, with the NAG basic generator, to populate
w and
h. For further information on this random number generator see
Section 2.1.1 in the
G05 Chapter Introduction.
Note that this generator gives a repeatable sequence of random numbers, so if the value of
seed is not changed between function calls, then the same initial iterates will be generated.
9.4
Use in Conjunction with NAG Library Functions
To compute the nonnegative matrix factorization, the following skeleton code can normally be used:
do {
f01sbc(&irevcm,m,n,k,w,ldw,h,ldh,ht,ldht,seed,errtol,comm,icomm,&ifail)
if (irevcm == 1) {
.. Print W and H if required and check number of iterations ..
}
else if (irevcm == 2) {
.. Compute A^TW and store in ht..
}
else if (irevcm == 3) {
.. Compute AH^T and store in w ..
}
} (while irevcm !=0)
The code used to compute the matrix products will vary depending on the way
$A$ is stored. If all the elements of
$A$ are stored explicitly, then
f16yac) can be used. If
$A$ is triangular, then
f16yfc should be used. If
$A$ is symmetric, then
f16ycc should be used. For sparse
$A$ stored in coordinate storage format
f11xac and
f11xec can be used. Alternatively, if
$A$ is stored in compressed column format
f11mkc can be used.
10
Example
This example finds a nonnegative matrix factorization for the matrix
10.1
Program Text
10.2
Program Data
10.3
Program Results