f01sbc computes a non-negative matrix factorization for a real non-negative matrix . It uses reverse communication for evaluating matrix products, so that the matrix is not accessed explicitly.
The function may be called by the names: f01sbc or nag_matop_real_nmf_rcomm.
3Description
The matrix is factorized into the product of an matrix and a matrix , both with non-negative elements. The factorization is approximate, , with and chosen to minimize the functional
You are free to choose any value for , provided . The product will then be a low-rank approximation to , with rank at most .
f01sbc finds and using an iterative method known as the Hierarchical Alternating Least Squares algorithm. You may specify initial values for and , 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 ; it only requires the result of matrix multiplications of the form or . A reverse communication interface is used, in which control is returned to the calling program whenever a matrix product is required.
4References
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 SciencesE92–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 Science4666 Springer 169–176
Ho N–D (2008) Nonnegative matrix factorization algorithms and applications PhD Thesis Univ. Catholique de Louvain
5Arguments
Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument irevcm. Between intermediate exits and re-entries, all arguments other thanw and ht must remain unchanged.
1: – Integer *Input/Output
On initial entry: must be set to .
On intermediate exit:
specifies what action you must take before re-entering f01sbc with irevcm unchanged. The value of irevcm should be interpreted as follows:
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.
Indicates that before re-entry to f01sbc, the product must be computed and stored in ht.
Indicates that before re-entry to f01sbc, the product must be computed and stored in w.
On final exit: .
Note: any values you return to f01sbc as part of the reverse communication procedure should not include floating-point 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: – IntegerInput
On entry: , the number of rows of the matrix . Also the number of rows of the matrix .
Constraint:
.
3: – IntegerInput
On entry: , the number of columns of the matrix . Also the number of columns of the matrix .
Constraint:
.
4: – IntegerInput
On entry: , the number of columns of the matrix . Also, the number of rows of the matrix . See Section 9.2 for further details.
Constraint:
.
5: – doubleInput/Output
Note: the dimension, dim, of the array w
must be at least
.
The th element of the matrix is stored in .
On initial entry:
if , w should be set to an initial iterate for the non-negative matrix factor, .
If , w need not be set. f01sbc will generate a random initial iterate.
On intermediate exit:
if or , w contains the current iterate of the non-negative matrix .
On entry: the stride separating matrix row elements in the array ht.
Constraint:
.
11: – IntegerInput
On initial entry:
if , the supplied values of and are used for the initial iterate.
If , the value of seed is used to seed a random number generator for the initial iterates and . See Section 9.3 for further details.
12: – doubleInput
On entry: the convergence tolerance for when the Hierarchical Alternating Least Squares iteration has reached a stationary point. If , is used.
13: – doubleCommunication Array
Note: the dimension, dim, of the array comm
must be at least
.
14: – IntegerCommunication Array
15: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error 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 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, . Constraint: .
On entry, . Constraint: .
On initial entry, . Constraint: .
On intermediate re-entry, . Constraint: , or .
NE_INT_2
On entry, and . Constraint: .
On entry, and . Constraint: .
On entry, and . Constraint: .
NE_INT_3
On entry, , and . Constraint: .
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.
7Accuracy
The Hierarchical Alternating Least Squares algorithm used by f01sbc is locally convergent; it is guaranteed to converge to a stationary point of , but this may not be the global minimum. The iteration is deemed to have converged if the gradient of is less than errtol times the gradient at the initial values of and .
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 and each time, or by choosing a different random seed each time.
Note that even if f01sbc exits with NE_INT_2, the factorization given by and may still be a good enough approximation to be useful.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
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 implementation-specific information.
9Further Comments
f01sbc is designed to be used when 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 and will not, in general, be sparse even if is sparse.
If is small and dense, then f01sac can be used to compute and without the use of a reverse communication interface.
9.1Uniqueness
Note that non-negative matrix factorization is not unique. For a factorization given by the matrices and , an equally good solution is given by and , where is any real non-negative matrix whose inverse is also non-negative. In f01sbc, and are normalized so that the columns of have unit length.
9.2Choice of
The most appropriate choice of the factorization rank, , is often problem dependent. Details of your particular application may help in guiding your choice of , for example, it may be known a priori that the data in naturally falls into a certain number of categories.
Alternatively, trial and error can be used. Compute non-negative matrix factorizations for several different values of (typically with ) and select the one that performs the best.
Finally, it is also possible to use a singular value decomposition of to guide your choice of , by looking for an abrupt decay in the size of the singular values of . The singular value decomposition can be computed using f12fbc.
9.3Generating Random Initial Iterates
If on entry, then f01sbc uses the functions g05kfcandg05sac, 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.
9.4Use in Conjunction with NAG Library Functions
To compute the non-negative 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 is stored. If all the elements of are stored explicitly, then f16yac) can be used. If is triangular, then f16yfc should be used. If is symmetric, then f16ycc should be used. For sparse stored in coordinate storage format f11xacandf11xec can be used. Alternatively, if is stored in compressed column format f11mkc can be used.
10Example
This example finds a non-negative matrix factorization for the matrix