International Journal of Education and Information Technology, Vol. 1, No. 2, June 2015 Publish Date: Jun. 2, 2015 Pages: 48-54

Ontology Similarity Measuring and Ontology Mapping Algorithms Based on Majorization-Minimization Method

Yun Gao1, *, Wei Gao2

1Department of Editorial, Yunnan Normal University, Kunming, China

2School of Information Science and Technology, Yunnan Normal University, Kunming, China


Ontology similarity calculation is important research topics in information retrieval and widely used in science and engineering. In information retrieval, ontology is aiming to find the highly semantic similarity information of the original query concept, and then return the results to the user. Ontology mapping is aiming toconstruct the relationship between two or more ontologies. The core trick of ontology applications is to calculate the similarity between the vertices in the ontology graph. By analyzing the technology of majorization-minimization, we propose the new algorithm for ontology similarity measure and ontology mapping. Via the ontology sparse vector learning, the ontology graph is mapped into a line consists of real numbers. The similarity between two concepts then can be measured by comparing the difference between their corresponding real numbers. The experiment results show that the proposed new algorithm has high accuracy and efficiency on ontology similarity calculation and ontology mapping in biology and physics applications.


Ontology, Similarity Measure, Ontology Mapping, Sparse Vector, Majorization-Minimization

1. Introduction

Asa conceptual shared and knowledge representation model, ontologyhas been used in knowledge management, image retrieval and information retrieval search extension. Furthermore, acted as an effective concept semantic model, ontology is employed in other fields except computer science, including medical science, social science,pharmacology science, geography science and biology scienceRef. input by wrong way?(see Przydzial et al., [1], Koehler et al., [2], Ivanovic and Budimac[3], Hristoskova et al., [4], and Kabir et al., [5] for more detail).

The ontology model is a graph G=(V,E) such that each vertex vexpresses a concept and each directed edge e=vivjdenote a relationship between concepts vi and vj. Theaim of ontology similarity measure is to get a similarity function Sim: V×V {0} such that each pair of vertices is mapped to a non-negative real number.Moreover, the aim of ontology mapping is to obtain the link between two or more ontologies. In more applications, the keyof ontology mapping is to get a similarity function Sto determine the similarity between vertices from different ontologies.

In recent years, ontology similarity-based technologies were employed in many applications. By virtue of technology for stable semantic measurement, a graph derivation representation based trick for stable semantic measurement is presented by Ma et al., [6]. Li et al., [7] determined an ontology representation method which can be used in online shopping customer knowledge with enterprise information. A creative ontology matching system is proposed by Santodomingo et al., [8] such thatthe complex correspondences are deduced by processing expert knowledge with external domain ontologies and in view of novel matching technologies. The main features of the food ontology and several examples of application for traceability aims is reported by Pizzuti et al., [9]. Lasierra et al., [10]pointed out that ontologies can be employed in designing an architecture for taking care of patients at home. More ontology learning algorithms can refer to [11-22].

In this paper, we present the new ontology similarity computation and ontology mapping algorithms relied on sparse vector learning andmajorization-minimizationmethod or Technology. In terms of thesparse vector, the ontology graph is mapped into a real line and vertices are mapped into real numbers. Then the similarity between vertices is measured by the difference between their corresponding real numbers.

2. Basic Idea

Let V be an instance space and  be a n dimensional vector space.For any vertex in ontology graph G, its information (including its attribute, instance, structure, name and semantic information of the concept which iscorresponding to the vertex and that is contained in its vector)is denoted by a vector with p dimension. Let v= be a vector which is corresponding to a vertex v. For facilitating theexpression, we slightly confuse the notations and denotev both the ontology vertex and its corresponding vector.The purpose of ontology learning algorithmsis toget an optimal ontology function f: V, then the similarity between two vertices is determined by the difference between two real numbers which they correspond to.Theessence of such algorithmis dimensionality reduction, that is to say, use one dimension vector to represent p dimension vector. From this point of view, an ontology function f can be regarded as a dimensionality reduction map as f: .

In the real implement, thesparse ontology functioncan be denotedas

=,              (1)

where = is a sparse vector. For determining the ontology function f, we should get the sparse vector  first.

3. Main Ontology Algorithms

In this section, it willpresent our main ontology sparse vector learning algorithms for ontology similarity measuring and ontology mapping by virtue ofmajorization-minimization trick.

The ontology sparse vector learning model using fused lasso regression (FLR)was defined by

,       (2)

where y is the response vector, and V is the ontology matrix; n is the sample number,  and are non-negative regularization parameters,and E is the unordered index set of the adjacent pairs of variables specified in the model.

In thispaper,it will use an MM trick to solve the ontology problem (2). Given the current estimate  of the optimal solution to ontology problem (2), the majorization step finds a majorizing functionsuch that= and for. The minimizationstep updates the estimate with


It has a descent character that. Furthermore, if meets some regularity conditions, the MM produce converges to the optimal solution of.

The MM technology has been applied to the classical lasso problem such that it propose to minimize a perturbed version of the objective function of theclassical lasso

instead of

to avoid division by zero in the learning. The majorizing function forat is obtained by


The sequence with =converges to the minimizer of, and converges touniformly as tends to 0.

Introducing a perturbed version of the objective (2) and then the majorizing function for the ontology problem (2):

,       (3)




Since  is a smooth convex function on, we have that  majorizes at  and has a unique global minimum on the rank of V.

Now, let be the current estimate in the r-th iteration. The majorizing functionis minimized over if =0, and can be expressed as a linearsystem of equations

=           (5)

where =with = for  and = is a symmetric and positive semidefinite matrix with


==, .

The procedure presented as Algorithm 1 below.

Algorithm 1. FLR based on MM method.

Input: y, V, , , convergence tolerance , perturbation constant .

Step 1: for j=1,…,p do

Step 2:

Step 3: end for

Step 4: repeat r=0,1,2,…

Step 5:

Step 6:

Step 7: until

Output: =.

If is tridiagonal or blocktridiagonal, we can use the preconditioned conjugategradient (PCG) trick by using the matrix as the preconditioner to efficiently determine the linear system (5). The PCG technology is an iterative trick for solving a linear system=c, where Q is a symmetric positive definite matrix and Mis a preconditioner matrix. For instance, we can use M= for Q=. The PCG methoddeduces the solution via iteratively updating four sequences , , , and with





where=, =, , =c, =,and =. For the standard FLR ontology model, we can solve the linear system = using LAPACK functions dpbtrf and dpbtrs. Algorithm 2 manifests the proposedMM based ontology algorithm using the PCG method under the standard FLR framework.

Algorithm 2.MM based ontology algorithm using the PCG method for the standard FLR and the two-dimensional FLR framework

Input: y, V, ,, convergence tolerance , perturbation constant .

Step 1: for j=1,…,p do

Step 2:

Step 3: end for

Step 4: repeat r=0,1,2,…

Step 5:

Step 6:

Step 7:

Step 8:

Step 9:

Step 10: repeat j=1,2,…

Step 11:

Step 12:

Step 13:

Step 14: Solve

Step 15:

Step 16:

Step 17: until

Step 18:

Step 19: until

Output: =.

4. Experiments

Two simulation experiments on ontology similarity measure and ontology mapping are designed in this section.For detail implement, onefirst obtain the optimal sparse vector using the algorithm raised in our paper, and then the ontology function is yielded by (1).

4.1. Experiment on Biology Data

The "Go" ontology O1 was constructed byhttp: //www. geneontology. org. (Fig. 1 presents the graph structure of O1). We use P@N (defined by Craswell and Hawking [23]) to determine the equality of the experiment result.

Beside our ontology algorithm, ontology algorithms in Huang et al., [12], Gao and Liang [13] and Gao and Gao[14] are also acted to "Go" ontology. Then, we compare the precision ratio which we have deducedfrom the four tricks. Some parts of experiment results can be seen in Table 1.

Figure 1. "Go" ontology

Table 1. The experiment data for ontology similarity measure

  P@3 average precision ratio P@5 average precision ratio P@10 average precision ratio P@20 average precision ratio
Simulation or experiment Algorithm 47.08% 53.92% 63.78% 81.69%
Algorithmin [12] 46.38% 53.48% 62.34% 74.59%
Algorithm in [13] 43.56% 49.38% 56.47% 71.94%
Algorithm in[14] 42.13% 51.83% 60.19% 72.39%

Figure 2. "Physical Education" Ontology O2

Figure 3. "Physical Education" Ontology O3.

As we can see in the Figure 1, when N= 3, 5, 10 or 20, the precision ratio in view of our algorithmis higher than that got by trickswhich has been determinedin Huang et al., [12], Gao and Liang [13] and Gao and Gao[14].

4.2. Experiment on Physical Education Data

For the second experiment, we use physical education ontologies O2 and O3 (the graph structures of O2 and O3 are raised in Fig. 2 and Fig. 3 respectively). The purpose of this experiment is to construct the ontology mapping between O2 and O3. Again, P@N criterion is applied to measure the equality of the experiment results.

Furthermore, ontology technologiesrefer to Huang et al., [12], Gao and Liang [13] and Gao et al., [16] are employed to "physical education" ontology. At last, we compare the precision ratio that we have obtained from four tricks. Table 2 present several resultsfor this experiment.

From what we have obtained in Table 2, we find itmore efficient to use our Algorithmthan algorithms determinedbyHuang et al., [12], Gao and Liang [13] and Gao et al., [16],especially whereN is sufficiently large.

Table 2. The experiment data for ontology mapping

  P@1 average precision ratio P@3 average precision ratio P@5 average precision ratio
Algorithm 69.13% 80.65% 92.26%
Algorithm in [12] 61.29% 73.12% 79.35%
Algorithm in [13] 69.13% 75.56% 84.52%
Algorithm in [16] 67.74% 77.42% 89.68%


5. Conclusions

In this article, a new algorithmfor ontology similarity measure and ontology mapping application is presented by virtue ofmajorization-minimization method. Furthermore, experiment resultsreveal that our new algorithm has high efficiency in both biology and physics education. The ontology algorithmpresented in our paper illustrates the promising application prospects forontology use.


We thank the reviewers for their constructive comments in improving the quality of this paper.The research is financed by: NSFC (No.11401519).


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