void	f01sac (Integer m, Integer n, Integer k, const double a[], Integer pda, double w[], Integer pdw, double h[], Integer pdh, Integer seed, double errtol, Integer maxit, NagError *fail)

The function may be called by the names: f01sac or nag_matop_real_nmf.

3 Description

The matrix

A

is factorized into the product of an

m \times k

matrix

W

and a

k \times n

matrix

H

, both with non-negative elements. The factorization is approximate,

A \approx W H

, with

W

and

H

chosen to minimize the functional

f (W, H) = {‖ A - W H ‖}_{F}^{2} .

You are free to choose any value for

k

, provided

k < \min (m, n)

. The product

W H

will then be a low-rank approximation to

A

, with rank at most

k

f01sac 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 f01sac to generate the initial values using a random number generator.

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

1: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

. Also the number of rows of the matrix

W

Constraint:

m \geq 2

2: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

. Also the number of columns of the matrix

H

Constraint:

n \geq 2

3: $k$ – Integer Input

On entry:

k

, the number of columns of the matrix

W

; the number of rows of the matrix

H

. See Section 9.2 for further details.

Constraint:

1 \leq k < \min (m, n)

4: $a [\dim]$ – const double Input

Note: the dimension, dim, of the array a must be at least

pda \times n

The

(i, j)

th element of the matrix

A

is stored in

a [(j - 1) \times pda + i - 1]

On entry: the

m \times n

non-negative matrix

A

5: $pda$ – Integer Input

On entry: the stride separating matrix row elements in the array a.

Constraint:

pda \geq m

6: $w [\dim]$ – double Input/Output

Note: the dimension, dim, of the array w must be at least

pdw \times k

The

(i, j)

th element of the matrix

W

is stored in

w [(j - 1) \times pdw + i - 1]

On entry:

if $seed \leq 0$ , w should be set to an initial iterate for the non-negative matrix factor, $W$ .
If $seed \geq 1$ , w need not be set. f01sac will generate a random initial iterate.

On exit: the non-negative matrix factor,

W

7: $pdw$ – Integer Input

On entry: the stride separating matrix row elements in the array w.

Constraint:

pdw \geq m

8: $h [\dim]$ – double Input/Output

Note: the dimension, dim, of the array h must be at least

pdh \times n

The

(i, j)

th element of the matrix

H

is stored in

h [(j - 1) \times pdh + i - 1]

On entry:

if $seed \leq 0$ , h should be set to an initial iterate for the non-negative matrix factor, $H$ .
If $seed \geq 1$ , h need not be set. f01sac will generate a random initial iterate.

On exit: the non-negative matrix factor,

H

9: $pdh$ – Integer Input

On entry: the stride separating matrix row elements in the array h.

Constraint:

pdh \geq k

10: $seed$ – Integer Input

On entry:

if $seed \leq 0$ , the supplied values of $W$ and $H$ are used for the initial iterate.
If $seed \geq 1$ , 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.

11: $errtol$ – double Input

On entry: the convergence tolerance for when the Hierarchical Alternating Least Squares iteration has reached a stationary point. If

errtol \leq 0.0

\max (m, n) \times \sqrt{machine precision}

is used.

12: $maxit$ – Integer Input

On entry: specifies the maximum number of iterations to be used. If

maxit \leq 0

200

is used.

13: $fail$ – 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 $⟨ value ⟩$ had an illegal value.
NE_CONVERGENCE: The function has failed to converge after $⟨ value ⟩$ iterations. The factorization given by w and h may still be a good enough approximation to be useful. Alternatively an improved factorization may be obtained by increasing maxit or using different initial choices of w and h.
NE_INIT_ESTIMATE: An internal error occurred when generating initial values for w and h. Please contact NAG.
NE_INT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 2$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 2$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pda \geq m$ .

On entry, $pdh = ⟨ value ⟩$ and $k = ⟨ value ⟩$ .
Constraint: $pdh \geq k$ .

On entry, $pdw = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pdw \geq m$ .
NE_INT_3: On entry, $k = ⟨ value ⟩$ , $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $1 \leq k < \min (m, n)$ .
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 a, 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 f01sac is locally convergent; it is guaranteed to converge to a stationary point of

f (W, H)

, but this may not be the global minimum. The iteration is deemed to have converged if the gradient of

f (W, H)

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 f01sac 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 for each function call.

Note that even if f01sac exits with

fail . code =

NE_CONVERGENCE, the factorization given by

W

and

H

may still be a good enough approximation to be useful.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f01sac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f01sac 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.

9 Further Comments

Each iteration of the Hierarchical Alternating Least Squares algorithm requires

O (m n k)

floating-point operations.

The real allocatable memory required is

m \times n + k (m + n)

A

is large and sparse, then f01sbc should be used to compute a non-negative matrix factorization.

9.1 Uniqueness

Note that non-negative matrix factorization is not unique. For a factorization given by the matrices

W

and

H

, an equally good solution is given by

W D

and

D^{−1} H

, where

D

is any real non-negative

k \times k

matrix whose inverse is also non-negative. In f01sac,

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 non-negative matrix factorizations for several different values of

k

(typically with

k ≪ \min (m, n)

) 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 f08kbc.

9.3 Generating Random Initial Iterates

seed \geq 1

on entry, then f01sac 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.

10 Example

This example finds a non-negative matrix factorization for the matrix

A = (\begin{matrix} 8 & 10 & 5 & 10 & 5 \\ 9 & 21 & 7 & 17 & 10 \\ 12 & 17 & 8 & 14 & 6 \\ 14 & 18 & 9 & 16 & 7 \\ 13 & 29 & 10 & 23 & 13 \\ 10 & 17 & 7 & 14 & 7 \end{matrix}) .

f01sa: FL CL CPP AD PY MB

NAG CL Interfacef01sac (real_​nmf)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

9.1 Uniqueness

9.2 Choice of k

9.3 Generating Random Initial Iterates

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f01sac (real_nmf)

9.2 Choice of $k$