Primal dual based ontology sparse vector learning for similarity measuring and ontology mapping

Abstract

From the mathematical point of view, the goal of ontology learning is to obtain the dimensionality function $f : ℝ^{p} \to ℝ$ , and the p-dimensional vector corresponding to the ontology vertex is mapped into one-dimensional real number. In the background of big data applications, the ontology concept corresponds to the high complexity of information, and thus sparse tricks are used in ontology learning algorithm. Through the ontology sparse vector learning, the ontology function f is obtained via ontology sparse vector β, and then applied to ontology similarity computation and ontology mapping. In this paper, the ontology optimization strategy is obtained by coordinate descent and dual optimization, and the optimal solution is obtained by iterative procedure. Furthermore, the greedy method and active sets are applied in the iterative process. Two experiments are presented where we will apply our algorithm to plant science for ontology similarity measuring and to mathematics ontologies for ontology mapping, respectively. The experimental data show that our primal dual based ontology sparse vector learning algorithm has high efficiency.

Keywords

1 Introduction

In computer science and information technology, ontology is a model for knowledge storage, representation, computation and analysis. It has been widely applied in query expansion, image retrieval, collaboration, information systems, and intelligent integration. Ontology, as a useful concept semantic model and an effective analysis tool, has been permeated in multidisciplinary, such as pharmacology science, medical science, education system, biology science, neuroscience, geographic information system (GIS) and nanotechnology (see Villalonga et al. [1], Gailly et al. [2], Benvenuti et al. [3], Podolskii [4] and Salas-Zarate et al. [5]).

Specifically, the structure of ontology can be expressed as a graph. Let O be an ontology and G be a graph corresponding to O. Each vertex in ontology graph represents to a concept in ontology and each edge in G denotes a directly relationship between two concepts. In the ontology engineering applications, it aimed to get the similarity between ontology vertices (concepts) and ontology mapping between multi-ontologies (which is aimed to bridge a similarity based connection between the vertices from different ontologies). In this way, the problem of ontology mapping can also be considered as the special kind of ontology similarity measuring.

Recent years, a variety of ontology approaches are applied in kinds of engineering applications. Bobillo and Straccia [6] raised a formal framework for type-2 fuzzy ontologies taking into account the needs of existing applications. Rani et al. [7] introduced two topic modeling algorithms called LSI & SVD and Mr.LDA for learning topic ontology. Pauwels et al. [8] focused on the optimization of geometric data in the semantic representation. They outlined and discussed the diverse available options in optimizing the data representations used, and also quantified the impact of these measures on the ifcOWL ontology. Tarus et al. [9] studied a hybrid knowledge-based recommender system based on ontology and sequential pattern mining for recommendation of e-learning resources to learners. Negri et al. [10] aimed at providing a representation of logistics aspects, which is to be used as an extension to the already-existing production systems ontologies that more focused on manufacturing processes.

In the background of big data applications, it faces the rigorous challenge of calculating massive amounts of data. Hence, the learning tricks are introduced in ontology algorithm. The popular ontology idea is to get an ontology function $f : V \to ℝ$ which maps each vertex to a real number. In this way, the similarity between two concepts is determined in light of the difference between their corresponding real numbers. In order to suit the ontology learning setting, all the information of one vertex is represented as a vector. Throughout our paper, to facilitate the representation, we slightly confuse the notations and denote v as both vertex and also its corresponding vector. Thus, the above-mentioned ontology learning trick can be summarized to obtain a dimensionality reduction function $f : ℝ^{p} \to ℝ$ .

Several effective ontology learning approaches for getting ontology score function and their theoretical analysis have been proposed. Gao et al. [11] studied the strong and weak stability of k-partite ranking based ontology algorithm. Gao and Xu [12] gave the stability analysis of learning algorithms for ontology similarity computation. Gao et al. [13] introduced an ontology sparse vector learning algorithm for ontology similarity measuring and ontology mapping by means of ADAL approach. Gao and Farahani [14] determined the generalization bounds and uniform bounds for multi-dividing ontology algorithms with convex ontology loss function. Gao et al. [15] proposed an ontology algorithm based on singular value decomposition. Gao and Wang [16] gave the analysis of k-partite ranking based ontology algorithm in area under the receiver operating characteristic curve criterion. More ontology learning algorithms and related theoretical results can refer to Simón et al. [17] and Pérez-Benito et al. [18].

In this paper, we focus on the ontology learning algorithm where the ontology function is obtained via ontology sparse vector. The new ontology sparse vector learning algorithm will be presented and the experiments will be depicted to show its effectiveness.

2 Mathematical setting of ontology algorithm

Assume that V is an ontology vertex space (instance space) and we use a p dimension vector to express all the information related to the ontology vertex v (which corresponding to a concept). Specifically, let v = {v¹, v², ⋯, v^p} ^T be a vector corresponding to ontology vertex v. An optimal ontology function $f : V \to ℝ$ can be regarded as a score function which is yielded via the ontology learning algorithms, and the similarity between v and v′ is determined by |f (v) - f (v′) |. Such ontology function is also can be considered as a dimensionality reduction function $f : ℝ^{p} \to ℝ$ .

Using ontology sparse vector, the ontology function can be formulated as $f (v) = v^{T} β + δ,$ (1) where $β \in ℝ^{p}$ is an ontology sparse vector and δ is a noise term. Thus, the learning purpose of ontology function f is to transfer to learn the ontology sparse vector β. The standard ontology sparse vector learning framework can be stated as $min_{β \in ℝ^{p}} \frac{1}{n} \sum_{i = 1}^{n} l_{i} (v_{i}^{T} β) + Ω (β),$ (2) where n is the number of ontology sample volume, $l_{i} : ℝ \to ℝ$ is a convex ontology loss function, $Ω : ℝ^{p} \to ℝ$ is a convex balance function which is used to control the sparseness of β.

Let {v₁, v₂, ⋯, v_n} be the ontology sample set, where $v_{i} = {v_{i}^{1}, v_{i}^{2}, \dots, v_{i}^{p}}^{T}$ for i ∈ {1, ⋯, n}. We use $V \in ℝ^{n \times p}$ denote the ontology data matrix with rows $V_{i} = v_{i}^{T}$ . Set V^j (j ∈ {1, ⋯, p}) be the j-th column of V. For the optimization of our learning algorithm, we give the following assumptions for the mathematical setting. Balance function Ω is supposed to be μ-strongly convex (i.e., for any subgradient ▽Ω (v) ∈ ∂Ω (v) and $v, v^{'} \in ℝ^{p}$ , we have $Ω (v^{'}) \geq Ω (v) + < ▽ Ω (v), v^{'} - v > + \frac{μ ∥ v^{'} - v ∥_{2}^{2}}{2}$ ) and has decomposable structure Ω (v) = ∑_iΩ_i (vⁱ). Ontology loss l_i is assumed to be $\frac{1}{γ}$ -smooth (i.e., for any $x, y \in ℝ$ and i ∈ {1, ⋯, n}, we infer $l_{i} (x) \leq l_{i} (y) + {l^{'}}_{i} (y) (x - y) + \frac{γ {(x - y)}^{2}}{2}$ ) and ${l^{'}}_{i}$ to be Lipschitz continuous (i.e., ${l^{'}}_{i} (y) - {l^{'}}_{i} (x) \leq \frac{| y - x |}{γ}$ ).

Under the above assumption, the ontology optimization problem (2) is equal to the following dual version: $max_{w \in ℝ^{n}} - Ω^{*} (\frac{V^{T} w}{n}) - \frac{1}{n} \sum_{i = 1}^{n} l_{i}^{*} (w_{i}),$ (3) or equal to the following convex-concave saddle point problem: $max_{w \in ℝ^{n}} min_{β \in ℝ^{p}} Ω (β) + \frac{w^{T} V β}{n} - \frac{1}{n} \sum_{i = 1}^{n} l_{i}^{*} (w_{i}) .$ (4)

3 Ontology sparse vector learning algorithm

In this section, we present the detailed learning trick for our main ontology sparse vector optimization problem.

3.1 Primal dual ontology algorithm and its efficient implementation trick

Now, we give the ontology algorithm to solve (4) using standard learning approaches in the special setting that Ω is separable, i.e., Ω (β) term in (2) can be formulated as $\sum_{j = 1}^{p} Ω_{j} (β^{j})$ .

Algorithm A: Primal dual ontology algorithm A1: Input $β^{(- 1)} = β^{(0)} = {\bar{β}}^{(0)} \in ℝ^{p}$ , $w^{(- 1)} = w^{(0)} = {\bar{w}}^{(0)} \in ℝ^{n}$ , and positive parameters (θ, τ, σ). A2: For t = 0, 1, ⋯, T - 1: randomly select two index sets I ⊂ {1, ⋯, n}, J ⊂ {1, ⋯, p} with |I| = n₁ and |J| = n₂. $w_{i}^{(t + 1)} \leftarrow \underset{m \in ℝ}{arg max} {\frac{1}{n} < V_{i}, {\bar{β}}^{(t)} > m - \frac{l_{i}^{*} (m)}{n} - \frac{(m - w_{i}^{(t)})^{2}}{2 σ}}$ if i ∈ I; otherwise $w_{i}^{(t + 1)} \leftarrow w_{i}^{(t)}$ ; ${\bar{w}}^{(t + 1)} \leftarrow w^{(t)} + \frac{n}{n_{1}} (w^{(t + 1)} - w^{(t)})$ ; $β_{j}^{(t + 1)} \leftarrow \underset{q \in ℝ}{arg min} {\frac{< V^{j}, {\bar{w}}^{(t + 1)} > q}{n} + Ω_{j} (q) + \frac{(q - β_{j}^{(t)})^{2}}{2 τ}}$ if j ∈ J; otherwise $β_{j}^{(t + 1)} \leftarrow β_{j}^{(t)}$ ; ${\bar{β}}^{(t + 1)} \leftarrow β^{(t)} + (θ + 1) (β^{(t + 1)} - β^{(t)})$ . A3: Return β^(T) and w^(T).

Set V = UX, then the efficient implementation of above ontology algorithm is listed as follows. Algorithm B: Efficient implementation of ontology algorithm A: B1: Input $β^{(- 1)} = β^{(0)} = {\bar{β}}^{(0)} \in ℝ^{p}$ , $w^{(- 1)} = w^{(0)} = {\bar{w}}^{(0)} \in ℝ^{n}$ , positive parameters (θ, τ, σ), u⁽⁰⁾ = U^Tw⁽⁰⁾, x⁽⁰⁾ = Xβ⁽⁰⁾, ${\bar{u}}^{(0)} = U^{T} {\bar{w}}^{(0)}$ , ${\bar{x}}^{(0)} = X {\bar{x}}^{(0)}$ . B2: For t = 0, 1, ⋯, T - 1: randomly select two index sets I ⊂ {1, ⋯, n}, J ⊂ {1, ⋯, p} with |I| = n₁ and |J| = n₂. $w_{i}^{(t + 1)} \leftarrow \underset{m \in ℝ}{arg max} {\frac{1}{n} < U_{i}, {\bar{x}}^{(t)} > m - \frac{l_{i}^{*} (m)}{n} - \frac{(m - w_{i}^{(t)})^{2}}{2 σ}}$ if i ∈ I; otherwise $w_{i}^{(t + 1)} \leftarrow w_{i}^{(t)}$ ; u^(t+1) ← u^(t) + U^T (w^(t+1) - w^(t)); ${\bar{u}}^{(t + 1)} \leftarrow u^{(t)} + \frac{n}{n_{1}} U^{T} (w^{(t + 1)} - w^{(t)})$ ; $β_{j}^{(t + 1)} \leftarrow \underset{q \in ℝ}{arg min} {\frac{< X^{j}, {\bar{u}}^{(t + 1)} > q}{n} + Ω_{j} (q) + \frac{(q - β_{j}^{(t)})^{2}}{2 τ}}$ if j ∈ J; otherwise $β_{j}^{(t + 1)} \leftarrow β_{j}^{(t)}$ ; x^(t+1) ← x^(t) + X^T (x^(t+1) - x^(t)); ${\bar{x}}^{(t + 1)} \leftarrow x^{(t)} + (1 + θ) X^{T} (x^{(t + 1)} - x^{(t)})$ . B3: Return β^(T) and w^(T).

Note that the precondition of Algorithm B is that the ontology information matrix V can be factorized.

3.2 Primal dual ontology algorithm in normal setting

In this part, we manifest the ontology sparse vector learning algorithm using standard trick for the ontology optimization problem presented in the last section, but the setting discussed in this part doesn’t need the separable of Ω. Algorithm C: Primal dual ontology sparse vector learning algorithm in normal setting: C1: Input positive parameters (θ, τ, σ), iteration number T, β⁽⁰⁾, w⁽⁰⁾, ${\bar{β}}^{(0)} = β^{(0)}$ , $u^{(0)} = \frac{\sum_{i = 1}^{n} w_{i}^{(0)} v_{i}}{n}$ . C2: For t = 0, 1, ⋯, T - 1: randomly select index set I ⊂ {1, ⋯, n}, with |I| = n₁. $w_{i}^{(t + 1)} \leftarrow \underset{m \in ℝ}{arg max} {< v_{i}, {\bar{β}}^{(t)} > m - l_{i}^{*} (m) - \frac{(m - w_{i}^{(t)})^{2}}{2 σ}}$ if i ∈ I; otherwise $w_{i}^{(t + 1)} \leftarrow w_{i}^{(t)}$ ; $β_{j}^{(t + 1)} \leftarrow \underset{β \in ℝ^{p}}{arg min} {< β, u^{(t)} + \frac{\sum_{i \in I} (w_{i}^{(t + 1)} - w_{i}^{(t)}) v_{i}}{n_{1}} > + Ω (β) + \frac{∥ β - β^{(t)} ∥_{2}^{2}}{2 τ}}$ ; $u^{(t + 1)} = u^{(t)} + \frac{\sum_{i \in I} (w_{i}^{(t + 1)} - w_{i}^{(t)}) v_{i}}{n}$ ; ${\bar{β}}^{(t + 1)} = β^{(t + 1)} + θ (β^{(t + 1)} - β^{(t)})$ . C3: Return β^(T) and w^(T).

Next, we show the Algorithm C in the nonuniform ontology sample setting.

Algorithm D: Primal dual ontology sparse vector learning algorithm in with nonuniform ontology sample:

D1: Input positive parameter (θ, τ, σ), iteration number T, β⁽⁰⁾, w⁽⁰⁾, ${\bar{β}}^{(0)} = β^{(0)}$ , $u^{(0)} = \frac{\sum_{i = 1}^{n} w_{i}^{(0)} v_{i}}{n}$ .

D2: For t = 0, 1, ⋯, T - 1: randomly select k ∈ {1, ⋯, n} with possibility $p_{k} = \frac{1}{2 n} + \frac{∥ v_{k} ∥_{2}}{2 \sum_{i = 1}^{n} ∥ v_{i} ∥_{2}}$ . $w_{i}^{(t + 1)} \leftarrow \underset{m \in ℝ}{arg max} {< v_{i}, {\bar{β}}^{(t)} > m - l_{i}^{*} (m) - \frac{p_{i} n (m - w_{i}^{(t)})^{2}}{2 σ}}$ if i = k; otherwise $w_{i}^{(t + 1)} \leftarrow w_{i}^{(t)}$ ; $β_{j}^{(t + 1)} \leftarrow \underset{β \in ℝ^{p}}{arg min} {< β, u^{(t)} + \frac{\sum_{i \in I} (w_{i}^{(t + 1)} - w_{i}^{(t)}) v_{i}}{p_{k} n} > + Ω (β) + \frac{∥ β - β^{(t)} ∥_{2}^{2}}{2 τ}}$ ; $u^{(t + 1)} = u^{(t)} + \frac{(w_{k}^{(t + 1)} - w_{k}^{(t)}) v_{i}}{n}$ ; ${\bar{β}}^{(t + 1)} = β^{(t + 1)} + θ (β^{(t + 1)} - β^{(t)})$ . D3: Return β^(T) and w^(T).

3.3 Greedy procedure and corresponding algorithm with active set

In the subsection, we present our finally ontology sparse vector learning algorithm in view of greedy techniques. Also, we assume the balance function Ω is separable.

Algorithm E: Primal dual ontology sparse vector learning algorithm in with greedy trick:

E1: Input ontology training data v₁, ⋯, v_n with ontology information matrix $V \in ℝ^{n \times p}$ , positive parameter η, $β^{(0)} \leftarrow 0 \in ℝ^{p}$ , $w^{(0)} \leftarrow 0 \in ℝ^{n}$ , $y^{(0)} = V β = 0 \in ℝ^{n}$ , $z^{(0)} = V^{T} w = 0 \in ℝ^{p}$ .

E2: For t = 1, ⋯, T: ${\bar{β}}_{k}^{(t)} \leftarrow \underset{m}{arg min} {\frac{z_{k}^{(t - 1)} m}{n} + Ω_{k} (m)}$ , for any k ∈ {1, ⋯, p}; $j^{(t)} \leftarrow \underset{k \in {1, \dots, p}}{arg min} {\frac{z_{k}^{(t - 1)} ({\bar{β}}_{k}^{(t)} - {\bar{β}}_{k}^{(t - 1)})}{n} + Ω_{k} ({\bar{β}}_{k}^{(t)}) - Ω_{k} ({\bar{β}}_{k}^{(t - 1)})}$ ; $β_{k}^{(t)} \leftarrow {\bar{β}}_{k}^{(t)}$ if k = j^(t); otherwise $β_{k}^{(t)} \leftarrow β_{k}^{(t - 1)}$ ; $y^{(t)} \leftarrow y^{(t - 1)} + (β_{j}^{(t)} - β_{j}^{(t - 1)}) V^{j}$ ; $i^{(t)} \leftarrow \underset{k \in {1, \dots, n}}{arg max} | y_{k}^{(t - 1)} - \frac{1}{n} (l_{k}^{*})^{'} (w_{y}^{(t - 1)}) |$ ; $w_{k}^{(t)} \leftarrow \underset{q}{arg max} {\frac{y_{k}^{(t)} q}{n} - ł_{k}^{*} (q) - \frac{(q - w_{k}^{(t - 1)})^{2}}{2 η}}$ if k = i^(t); otherwise, $w_{k}^{(t)} \leftarrow w_{k}^{(t - 1)}$ ; $z^{(t)} \leftarrow z^{(t - 1)} + (w_{i^{(t)}}^{(t)} - w_{i^{(t)}}^{(t - 1)}) V_{i^{(t)}}$ .

E3: Return β^(T) and w^(T).

If we define ontology active sets V_β and V_w for both primal and dual variables, then the ontology Algorithm E can be re-stated as follows:

Algorithm F: Primal dual ontology sparse vector learning algorithm in with greedy trick and active sets:

F1: Input ontology training data v₁, ⋯, v_n with ontology information matrix $V \in ℝ^{n \times p}$ , positive parameter η, $β^{(0)} \leftarrow 0 \in ℝ^{p}$ , $w^{(0)} \leftarrow 0 \in ℝ^{n}$ , $V_{β}^{(0)} \leftarrow \emptyset$ , $V_{w}^{(0)} \leftarrow \emptyset$ . F2: For t = 1, ⋯, T: ${\bar{β}}_{k}^{(t)} \leftarrow \underset{m}{arg min} {\frac{< V^{k}, w^{(t - 1)} > m}{n} + Ω_{k} (m)}$ , for any k ∈ {1, ⋯, p}; $j^{(t)} \leftarrow \underset{k \in {1, \dots, p}}{arg min} | {\bar{β}}_{k}^{(t - 1)} |$ ; $V_{β}^{(t)} \leftarrow V_{β}^{(t - 1)} \cup {j^{(t)}}$ ; $β_{j}^{(t)} \leftarrow {\bar{β}}_{j}^{(t)}$ if $j \in V_{β}^{(t)}$ ; otherwise $β_{j}^{(t)} \leftarrow β_{j}^{(t - 1)}$ ; $i^{(t)} \leftarrow \underset{k \in {1, \dots, n} - V_{w}^{(t - 1)}}{arg max} | < V_{k}, β^{(t)} > - \frac{1}{n} (l_{k}^{*})^{'} (w_{k}^{(t - 1)}) |$ ; $V_{w}^{(t)} \leftarrow V_{w}^{(t - 1)} \cup {i^{(t)}}$ ; $w_{k}^{(t)} \leftarrow \underset{q}{arg max} {\frac{< V_{i}, β^{(t)} > q}{n} - ł_{i}^{*} (q) - \frac{(q - w_{k}^{t - 1})^{2}}{2 η}}$ if $i \in V_{w}^{(t)}$ ; otherwise, $w_{k}^{(t)} \leftarrow w_{k}^{(t - 1)}$ ; $V_{w}^{(t)} \leftarrow V_{w}^{(t)} - \cup_{i : w_{i}^{(t)} = 0} {i}$ ; $V_{β}^{(t)} \leftarrow V_{β}^{(t)} - \cup_{j : β_{j}^{(t)} = 0} {j}$ .

F3: Return β^(T) and w^(T).

Finally, after getting the ontology sparse vector β, the ontology function is obtained by f = v^Tβ.

4 Experiments

To implement ontology learning algorithm with mathematical setting, for each vertex in each ontology, we use fix dimensional vector to represent all the information of vertex, which contains it’s name, structure, instance, and attribute, etc. In this section, two experiments are manifested to test the effectiveness of our ontology sparse vector learning algorithm.

4.1 Ontology similarity measuring test on plant data

In plant science, “PO” ontology O₁ (see http: //www.plantontology.org, and its basic structure is depicted in Fig. 1) is used to present plant anatomy and morphology, and development stages of plants. We use P @ N (Precision Ratio) to measure the equality of the experiment data. The detailed procedures can be described as follows:

Fig.1

The Structure of “PO” Ontology O₁.

first step: the top similarity N vertices for every vertex are listed in light of field experts;

second step: in terms of our ontology learning algorithm, other top similarity N vertices for every vertex are determined;

third step: the P @ N precision ratios for all ontology vertices are obtained;

last step: the average precision ratio for the whole ontology graph is computed.

To compare the results, ontology learning approaches introduced in Gao and Zhu [19], Gao et al. [20] and [21] are acted on the “PO” ontology data, and their corresponding precision ratios are yielded respectively. Parts of the compared results can be referred to Table 1.

Table 1

The experiment results of ontology similarity measure on “PO” data

	P@3 average precision ratio	P@5 average precision ratio	P@10 average precision ratio	P@20 average precision ratio
Algorithm in our paper	0.5371	0.6782	0.9107	0.9719
Algorithm in Gao and Zhu [19]	0.5042	0.6216	0.7853	0.9034
Algorithm in Gao et al. [20]	0.4921	6152	0.8113	0.9174
Algorithm in Gao et al. [21]	0.5360	0.6664	0.9004	0.9673

The compared data depicted in Table 1 reveals that the precision ratio using our ontology sparse vector learning algorithm is obviously higher than the precision ratio determined by algorithms in Gao and Zhu [19], Gao et al. [20, 21] when N=3, 5, 10 or 20. Therefore, we conclude that our proposed ontology algorithm is superior to the learning approaches that Gao and Zhu [19], Gao et al. [20, 21] introduced.

4.2 Ontology mapping experiment on mathematical data

Mathematical ontologies O₂ and O₃ (the basic structures of O₂ and O₃ are extracted and presented in Figs. 2 and 3, respectively) are used in our second experiment to show the effectiveness of our ontology sparse vector learning algorithm in ontology mapping construction. The aim is to building the similarity based ontology mapping between O₂ and O₃, and P @ N criterion is also applied as criterion to test the quality of experiment results. Moreover, ontology learning approaches introduced in Gao and Zhu [19], Wu et al. [22] and Gao et al. [23] are implemented in mathematical ontologies as well. Parts of compared experiment results are listed inTable 2.

Fig.2

“Mathematical” ontology O₂.

Fig.3

“Mathematical” ontology O₃.

Table 2

The experiment results of ontology mapping

	P@1 average precision ratio	P@3 average precision ratio	P@5 average precision ratio
Algorithm in our paper	0.3846	0.5128	0.6923
Algorithm in Gao and Zhu [19]	0.3077	0.4359	0.5615
Algorithm in Wu et al.[22]	0.3846	0.5000	0.6769
Algorithm in Gao et al. [23]	0.3462	0.3974	0.5231

As presented in Table 2, the experiment compared data implies that our ontology sparse vector leanring algorithm performances much more efficient than ontology learning algorithms raised in Gao and Zhu [19], Wu et al. [22] and Gao et al. [23]. With the increase in the value of N, this advantage is more obvious.

5 Conclusions

Ontology, as a widely applied semantic model, has raised much attention from the scientists and engineers. One popular ontology learning method in recent years is mapping each concept (ontology vertex) to a real number by means of ontology function, and the similarities between ontology vertices v and v′ can be calculated in view of |f (v) - f (v′) |.

In our article, we focus on the ontology function learning algorithm via ontology sparse vector computation. We manifest a new learning trick for ontology similarity measuring and ontology mapping by virtue of coordinate descent and dual optimization. Furthermore, the greedy method and active sets are used in the iterative programme. Finally, our proposed ontology learning algorithm is applied in two classes of engineering applications for similarity computation and ontology mapping construction, respectively. The compared data proved the effectiveness of our ontology sparse vector learningalgorithm.

Conflict of Interests

The authors hereby declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the reviewers for their constructive comments in improving the quality of this paper. This work is partly supported by the 2016 scientific research project of the Guangdong university of science and technology – A Study on the design of the campus intelligent space situational awareness system based on Ontology [GKY-2016KYYB-10].

References

Villalonga

, Pomares

, Rojas

and Banos

, MIMU-wear: Ontology-based sensor selection for real-world wearable activity recognition, Neurocomputing250 (2017), 76–100.

Gailly

, Alkhaldi

, Casteleyn

and Verbeke

, Recommendation-based conceptual modeling and ontology evolution framework (CMOE plus), Bus Inform Syst Eng59 (2017), 235–250.

Benvenuti

, Diamantini

, Potena

and Storti

, An ontology-based framework to support performance monitoring in public transport systems, Transport Res C-Emer81 (2017), 188–208.

Podolskii

V.V.

, Bounds in ontology-based data access via circuit complexity, Theor Comput Syst61 (2017), 464–493.

Salas-Zarate

M.D.

, Valencia-Garcia

, Ruiz-Martinez

and Colomo-Palacios

, Feature-based opinion mining in financial news: An ontology-driven approach, J Inf Sci43 (2017), 458–479.

Bobillo

and Straccia

, Generalizing type-2 fuzzy ontologies and type-2 fuzzy description logics, Int J Approx Reason87 (2017), 40–66.

Rani

, Dhar

A.K.

and Vyas

O.P.

, Semi-automatic terminology ontology learning based on topic modeling, Eng Appl Artif Intel63 (2017), 108–125.

Pauwels

, Krijnen

, Terkaj

and Beetz

, Enhancing the ifcOWL ontology with an alternative representation for geometric data, Automat Constr80 (2017), 77–94.

Tarus

J.K.

, Niu

Z.D.

and Yousif

, A hybrid knowledge-based recommender system for e-learning based on ontology and sequential pattern mining, Future Gener Comp Sy72 (2017), 37–48.

10.

Negri

, Perotti

, Fumagalli

, Marchet

and Garetti

, Modelling internal logistics systems through ontologies, Comput Ind88 (2017), 19–34.

11.

Gao

, Gao

and Zhang

Y.G.

, Strong and weak stability of k-partite ranking algorithms, Information15 (2012), 4585–4590.

12.

Gao

and Xu

T.W.

, Stability analysis of learning algorithms for ontology similarity computation, Abstr Appl Anal (2013). 10.1155/2013-174802 .

13.

Gao

, Zhu

L.L.

and Wang

K.Y.

, Ontology sparse vector learning algorithm for ontology similarity measuring and ontology mapping via ADAL technology, Int J Bifurcat Chaos25 (2015). DOI: 10.1142/S0218127415400349.

14.

Gao

and Farahani

M.R.

, Generalization bounds and uniform bounds for multi-dividing ontology algorithms with convex ontology loss function, Comput J (2017). DOI: 10.1093/comjnl/bxw107.

15.

Gao

, Guo

and Wang

K.Y.

, Ontology algorithm using singular value decomposition and applied in multidisciplinary, Cluster Comput19 (2016), 2201–2210.

16.

Gao

and Wang

W.F.

, Analysis of k-partite ranking algorithm in area under the receiver operating characteristic curve criterion, Int J Comput Math (2017). 10.1080/00207160.2017.1322688 .

17.

Simón

C.B.

, López

J.C.C.

, Altúzar

L.V.

and Micó

R.J.V.

, Mean square calculus and random linear fractional differential equations: Theory and applications, Appl Math Nonlinear Sci2 (2017), 317–328.

18.

Pérez-Benito

, Morillas

, Jordán

and Conejero

J.A.

, Smoothing vs. sharpening of color images-together or separated, Appl Math Nonlinear Sci2 (2017), 299–316.

19.

Gao

and Zhu

L.L.

, Gradient learning algorithms for ontology computing, Comput Intel Neurosc (2014). 10.1155/2014-438291.

20.

Gao

, Farahani

M.R.

, Aslam

and Hosamani

, Distance learning techniques for ontology similarity measuring and ontology mapping, Cluster Comput20 (2017), 959–968.

21.

Gao

, Baig

A.Q.

, Ali

, Sajjad

and Farahani

M.R.

, Margin based ontology sparse vector learning algorithm and applied in biology science, Saudi J Biol Sci24 (2017), 132–138.

22.

J.Z.

, Yu

and Gao

, Disequilibrium multi-dividing ontology learning algorithm, Commun Stat-Theor M46 (2017), 8925–8942.

23.

Gao

, Zhu

L.L.

and Wang

K.Y.

, Ranking based ontology scheming using eigenpair computation, J Intell Fuzzy Syst31 (2016), 2411–2419.