*Swarm intelligence *

Optimization
Problems have being one of major problems and can be found everywhere,
especially in the academy ﬁeld and industrial world. To solve these kind problems,
many methods have been proposed. Recently Swarm Intelligence (SI) optimization
has been a very hot subject and method for optimization problems and showed its
great success. James Kennedy deﬁned it as follows : ” Swarm intelligence refers
to a kind of problem-solving ability that emerges in the interactions of simple
information-processing units.[3]” The units that build up the swarm can be
living creature and also can be lifeless bodies. In the years since the
introduction of Swarm Intelligence, SI has a number of branches with a number
of researchers exploring the concepts, behavior, cognition underlie the nature
and made great contribution to these ﬁelds, such as, Particle Swarm
Optimization (PSO)[4], Ant Colony Optimization (ACO)[5], Genetic Algorithm
(GA)[6] and Fish School Search (FSS)[7]. In PSO algorithms, it simulates the
behavior of ” searching food among ﬂocks” who share the social information,
previous experiences among the swarm and individuals.

*Fig1. **From
Biological phenomenon or process to Artificial Computational Approach*

*Motivation of FWA*

In the
same way, the inspiration of Inspired by observing ﬁreworks explosion, a novel
swarm intelligence algorithm, called Fireworks Algorithm (FA), is proposed for
global optimization of complex functions. The FA is presented and implemented
by simulating the explosion process of ﬁreworks. In the FA, two explosion
(search) processes are employed and mechanisms for keeping diversity of sparks
are also well designed.

*Fig2. **Mathematical
Modeling Process of Fireworks Algorithm*

*Fireworks Algorithm *

When a ﬁrework is set oﬀ, a shower of sparks will ﬁll
the local space around the ﬁrework. In our opinion, the explosion process of a ﬁrework
can be viewed as a search in the local space around a speciﬁc point where the ﬁrework
is set oﬀ
through the sparks generated in the explosion. Mimicking the process of setting
ﬁreworks, a rough framework of the FA is depicted in Fig. 3.

*Fig.3.
**Framework of ﬁreworks algorithm*

To validate the convergence speed of the FA, we
conducted more thorough experiments. Fig. 4 depicts the convergence curves of
the FA, the CPSO and the SPSO on eight benchmark functions averaged over 20
independent runs. From these results, we can arrive at a conclusion that the
proposed FA has a much faster speed than the CPSO and the SPSO. From Table 3,
we can ﬁnd that the FA can ﬁnd excellent solutions with only 10000 times of
function evaluations. This also reﬂects the fast convergence speed of the
proposed FA.

*Fig.4.**Convergence curves of the FA, the CPSO and the SPSO
on eight benchmark functions.*

*Table. 1. **Statistical
mean and standard deviation of solutions found by the FA, the CPSO and the SPSO
on nine benchmark functions over 20 independent runs of 10000 function
evaluations*

*Application based on FWA*

Low-rank approximations are utilized in several
content based retrieval and data mining applications, such as text and
multimedia mining, web search, etc. and achieve a more compact representation
of the data with only limited loss in information. They reduce storage and
runtime requirements, and also reduce redundancy and noise in the data
representation while capturing the essential associations. The Non-negative
Matrix Factorization (NMF, (Lee
and Seung 1999))
leads to a low-rank approximation which satisfies non-negativity constraints.
NMF approximates a data matrix by where and are the NMF
factors. NMF requires all entries in, andto be zero or positive.

*Figure
*5* **- Scheme of very coarse NMF
approximation with very low rank k. Although k is significantly smaller than m
and n, the typical structure of the original data matrix can be retained (note
the three different groups of data objects in the left, middle, and right part
of A).*

**Optimization Strategy 1 –
Initialization**

*Algorithm **1** **– Pseudo code for the initialization
procedure for NMF factors W and H. The two for-loops in lines 4 and 10 can be
executed concurrently. SIO = Swarm Intelligence Optimization*

**Evaluation of Optimization Strategy
1**

Initialization: Before evaluating the improvement of
the NMF approximation quality as such, we first measure the initial error after
initializing and (before running the NMF algorithm). Figure 3 and Figure
4 show the average approximation error (i.e. Frobenius
norm / fitness) per row (left) and per column (right) for data set DS-RAND.

*Figure
**3** **–
Left hand-side: average approximation error per row (after initializing rows of
W). Right hand-side: average approximation error per column (after initializing
of H). NMF rank k = 5. Legends are ordered according to approximation error
(top = worst, bottom = best).*

**For more details about this paper, please see [8][9].**

*Reference *

[1] Y. Tan and Y. Zhu
(2010). Fireworks algorithm for optimization. ICSI 2010, Part I, LNCS 6145, pp.
355-364

[2] D. Bratton and J.
Kennedy (2007). Deﬁning a Standard for Particle Swarm Optimization. Proceedings
of the 2007 IEEE Swarm Intelligence Symposium (SIS 2007), pp. 120-127

[3] J. Kennedy (2006).
Handbook of Nature-Inspired and Innovative, Springer, Chapter 6, pp. 187-219

[4] J. Kennedy and R.
Eberhart(1995), Particle Swarm Optimization. Proceedings of the 1995 IEEE
International Conference on Neural Networks(Perth, Australia): IEEE Service
Center, Piscataway, NJ, IV, pp. 1942-1948.

[5] M. Dorigo, V. Maniezzo
and A. Colorni (1996). Ant system: optimization by a colony of cooperating
agents. Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions
on, Volume: 26, Issue: 1, pp. 29-41.

[6] J. Holland (1975).
Adaptation in natural and artiﬁcial systems. University of Michigan Press, Ann Arbor.

[7] C. J. A. B. Filho, F. B.
de Lima Neto, A. J. C. C. Lins, A. I. S. Nascimento and M. P. Lima (2008). A
novel search algorithm based on ﬁsh school behavior. Proceedings of the IEEE
International Conference on Systems, Man and Cybernetics. pp. 2646-2651

[8] Andreas Janecek* and
Ying Tan, Swarm Intelligence for Non-negative Matrix Factorization, IJSIR 2011

[9] Andreas Janecek* and
Ying Tan, Using population
based algorithms for initializing nonnegative matrix factorization,
ICSI 2011, Part II, LNCS 6729, pp. 307–316, 2011.