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.” 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), Ant Colony Optimization (ACO), Genetic Algorithm (GA) and Fish School Search (FSS). 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
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).
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
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).
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