Ontology optimization tactics via distance calculating

Ontology originally comes from philosophy. It is used to describe the natural connection of things and their components’ inherently hidden connections. Ontology is set up as a model for knowledge storage and representation in information and computer science. It has been extensively applied in different fields such as knowledge management, machine learning, information systems, image retrieval, information retrieval search extension, collaboration and intelligent information integration. As a conceptually semantic model and an analysis tool, being quite effective, ontology has been favored by researchers from pharmacology science, biology science, medical science, geographic information system and social sciences since a few years ago (for instance, see Przydzial et al., [1], Koehler et al., [2], Ivanovic and Budimac [3], Hristoskova et al., [4], and Kabir [5]).

A simple graph is usually used by researchers to represent the structure of ontology. Every concept, objects and elements in ontology are made to correspond to a vertex. Each (directed or undirected) edge on an ontology graph represents a relationship (or potential link) between two concepts (objects or elements). Let O be an ontology and G be a simple graph corresponding to O. It can be attributed to getting. We use the similarity calculating function, the nature of ontology engineer application to compute the similarities between ontology vertices, which represent the intrinsic link between vertices in ontology graph. The ontology similarity measuring function is obtained by measuring the similarity between vertices from different ontologies. That is the goal of ontology mapping. The mapping serves as a bridge connecting different ontologies. Only through mapping, we gain a potential association between the objects or elements from different ontologies. The semi-positive score function Sim : V × V → ℝ⁺ ∪ {0} maps each pair of vertices to a non-negative real number.

Several effective methods exist for getting efficient ontology similarity measure or ontology mapping algorithm in terms of ontology function. The ontology similarity calculation in terms of ranking learning technology was considered by Wang et al., [12]. The fast ontology algorithm in order to cut the time complexity for ontology application was raised by Huang et al., [13]. An ontology optimizing model in which the ontology function is determined by virtue of NDCG measure was presented by Gao and Liang [14], which is successfully applied in physics education. More ontology applications on various engineering can be refered to Gao et al., [11].

In this article, we determine a new ontology learning method by means of distance calculating. Moreover, we give a theoretical analysis for proposed ontology algorithm.

Let S={(vi,vj,yij)}i,j=1N$\mathscr{S}=\{(v_{i},v_{j},y_{ij})\}_{i,j=1}^{N}$ be the ontology training data, where v_i,v_j ∊ ℝ^p are ontology vectors and y_ij = ±1 (if v_i and v_j are similar, then y_ij = 1; otherwise, y_ij = −1. We also fixed m relevant source ontology training sets Sq={(vqi,vqj,yqij)}i,j=1Nq$\begin{array}{} \displaystyle \mathscr{S}_{q}=\{(v_{qi}, v_{qj}, y_{qij})\}_{i,j=1}^{N_{q}} \end{array}$ (q = 1,...,m) if the number of target ontology training samples N is not large, and v_qi,v_qj ∊ ℝ^p belong to the certain ontology feature space as v_i, v_j in this setting.

We aim to learn a distance function d(v_i,v_j|W) = (v_i − v_j)^T W(v_i − v_j) which equals to learning a distance matrix W, and the similarity or dissimilarity between a ontology vertex pair v_i and v_j is obtained by comparing d(v_i,v_j|W) with a constant threshold parameter c. Specifically, our ontology optimization problem can be stated as

arg min1(N2)∑i<jg(yij[1−‖vi−vj‖W2])+η2‖W‖F2s.t.∑i=1mαq=1,αq≥0,q=1,⋯,m.$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {{\rm{arg }}\min \frac{1}{{\left( {\begin{array}{*{20}{c}} N\\ 2 \end{array}} \right)}}\mathop \sum\limits_{i < j} g({y_{ij}}[1 - \left\| {{v_i} - {v_j}} \right\|_{\bf{W}}^2]) + \frac{\eta }{2}\left\| {\bf{W}} \right\|_F^2}\\ {{\rm{s}}.{\rm{t}}.\quad \mathop \sum\limits^m_{i = 1} {\alpha _q} = 1,{\alpha _q} \ge 0,q = 1, \cdots ,m.} \end{array} \end{array}$$(1)

where ‖vi−vj‖W2=(vi−vj)TW(vi−vj),g(z)=max (0,b−z)$\begin{array}{} \displaystyle {v_i} - {v_j}_{\bf{W}}^2 = {({v_i} - {v_j})^T}{\bf{W}}({v_i} - {v_j}),g\left( z \right) = {\rm{max }}\left( {0,b - z} \right) \end{array}$ is an ontology hinge loss function, ||W||_F is the Frobenius norm of the metric W which is applied to control the model complexity, η is a balance parameter, and the constraint condition reveals that W is positive semi-definite.

The general version of ontology distance learning approach is formulated by

argmin1(N2)∑i<jL(vi,vj,yij)+γ12‖W−WD‖F2+γ22‖α‖22+γ3‖θ‖1,s.t.∑i=1mαq=1,αq≥0,q=1,⋯,m.$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {{\rm{arg}}\min \frac{1}{{\left( {\begin{array}{*{20}{c}} N\\ 2 \end{array}} \right)}}\mathop \sum\limits_{i < j} L({v_i},{v_j},{y_{ij}}) + \frac{{{\gamma _1}}}{2}\left\| {{\bf{W}} - {{\bf{W}}_D}} \right\|_F^2 + \frac{{{\gamma _2}}}{2}\left\| \alpha \right\|_2^2 + {\gamma _3}{{\left\| \theta \right\|}_1},}\\ {{\rm{s}}.{\rm{t}}.\quad \mathop \sum\limits^m_{i = 1} {\alpha _q} = 1,{\alpha _q} \ge 0,q = 1, \cdots ,m.} \end{array} \end{array}$$(2)

where W=∑i=1nθiuiuiT$\begin{array}{} \displaystyle {\bf{W}} = \sum\nolimits_{i = 1}^n {{\theta _i}{{\bf{u}}_i}{\bf{u}}_i^T} \end{array}$ and WD=∑q=1mαqWq$\begin{array}{} \displaystyle {{\bf{W}}_D} = \sum\nolimits_{q = 1}^m {{\alpha _q}{{\bf{W}}_q}} \end{array}$. Both ‖α‖22$\begin{array}{} \displaystyle \|\alpha\|_{2}^{2} \end{array}$ and ||θ||₁ are employed to control the complexity of model. In what follows, γ₁, γ₂ and γ₃ are all positive balance parameters.

Select L(vi,vj,yij)=g(yij[1−‖vi−vj‖W2])$\begin{array}{} \displaystyle L(v_{i},v_{j},y_{ij})=g(y_{ij}[1-\|v_{i}-v_{j}\|_{{\bf W}}^{2}]) \end{array}$ and use the ontology hinge loss for g, that is to say, g(z) = max(0, b − z) and b is set to be 0. Thus, we deduce the following ontology optimization problem:

argmin1(N2)∑i<jg(yij[1−‖vi−vj‖W2])+γW2‖W−WD‖F2+γ22‖α‖22+γ3‖θ‖1,s.t.∑i=1nαq=1,αq≥0,q=1,⋯,m.$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {{\rm{arg}}\min \frac{1}{{\left( {\begin{array}{*{20}{c}} N\\ 2 \end{array}} \right)}}\mathop \sum\limits_{i < j} g({y_{ij}}[1 - \left\| {{v_i} - {v_j}} \right\|_{\bf{W}}^2]) + \frac{{{\gamma _{\bf{W}}}}}{2}\left\| {{\bf{W}} - {{\bf{W}}_D}} \right\|_F^2 + \frac{{{\gamma _2}}}{2}\left\| \alpha \right\|_2^2 + {\gamma _3}{{\left\| \theta \right\|}_1},}\\ {{\rm{s}}.{\rm{t}}.\quad \mathop \sum\limits^n_{i = 1} {\alpha _q} = 1,{\alpha _q} \ge 0,q = 1, \cdots ,m.} \end{array} \end{array}$$(3)

For short expressions, we use v_i, x_j and y_{i j} to denote vk1$\begin{array}{} \displaystyle v_{k}^{1} \end{array}$, vk2$\begin{array}{} \displaystyle v_{k}^{2} \end{array}$ and y_k with k=1,⋅⋅⋅,(N2)=N′$\begin{array}{} \displaystyle k = 1, \cdot \cdot \cdot ,\left( {\begin{array}{*{20}{c}} N\\ 2 \end{array}} \right) = N\prime \end{array}$. Let δk=vk1−vk2$\begin{array}{} \displaystyle \delta_{k}=v_{k}^{1}-v_{k}^{2} \end{array}$ with ‖vk1−vk2‖W2=∑i=1nθiδkTuiuiTδk=δTfk$\begin{array}{} \displaystyle \left\| {v_k^1 - v_k^2} \right\|_{\bf{W}}^2 = \sum\nolimits_{i = 1}^n {{\theta _i}\delta _k^T{{\bf{u}}_i}{\bf{u}}_i^T{\delta _k}} = {\delta ^T}{f_k} \end{array}$, fk=[fk1,⋯,fkn]T$\begin{array}{} \displaystyle f_{k}=[f_{k}^{1},\cdots, f_{k}^{n}]^{T} \end{array}$ and fki=δkTuiuiTδk$\begin{array}{} \displaystyle f_{k}^{i}=\delta_{k}^{T}{\bf u}_{i}{\bf u}_{i}^{T}\delta_{k} \end{array}$. Therefore, the ontology problem (3) can be re-expressed as

argminα,θ1N′∑k=1N′g(yk[1−θTfk)+γW2‖W−WD‖F2+γ22‖α‖22+γ3‖θ‖1,s.t.∑k=1nαq=1,αq≥0,q=1,⋯,m.$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {{\bf{arg}}\mathop {\min }\limits_{_{\alpha ,\theta }} \frac{1}{{N'}}\sum\limits_{k = 1}^{N'} {g({y_k}[1 - {\theta ^T}{f_k})} + \frac{{{\gamma _{\bf{W}}}}}{2}\left\| {{\bf{W}} - {{\bf{W}}_D}} \right\|_F^2 + \frac{{{\gamma _2}}}{2}\left\| \alpha \right\|_2^2 + {\gamma _3}{{\left\| \theta \right\|}_1},} \\ {{\rm{s}}{\rm{.t}}.\quad \sum\limits_{k = 1}^n {{\alpha _q} = 1,{\alpha _q} \ge 0,q = 1, \cdots ,m} .} \\ \end{array} \end{array}$$(4)

The answer can be inferred by alternating between two sub ontology problems (minimization α = [α₁,··· ,α_m]^T and θ = [θ₁,··· ,θ_n]^T respectively) until its convergence.

Given α, the ontology optimization problem with respect to θ then it can be stated as

argminθF(θ)=Λ(θ)+Ω(θ)$$\begin{array}{} \displaystyle {\rm arg}\min\limits_{\theta}F(\theta)=\Lambda(\theta)+\Omega(\theta) \end{array}$$(5)

where Λ(θ)=1N′∑Nk=1g(yk[1−θTfk)+γ3‖θ‖1$\begin{array}{} \displaystyle \Lambda (\theta ) = \frac{1}{{{N^\prime }}}\sum\nolimits_N^{k = 1} {g({y_k}[1 - {\theta ^T}{f_k}) + {\gamma _3}{{\left\| \theta \right\|}_1}} \end{array}$, and Ω(θ)=γW2‖W−WD‖F2$\begin{array}{} \displaystyle \Omega(\theta) =\frac{\gamma_{{\bf W}}}{2}\|{\bf W}-{\bf W}_{D}\|_{F}^{2} \end{array}$. Since the ontology loss part Λ(θ) is non-differentiable, we should smooth the ontology loss and then solve (5) in terms of the gradient trick. Let Θ = {x : 0 ≤ x_k ≤ 1,x ∊ ℝ^N′} and σ be the smooth parameter. Then, the smoothed expression of the ontology hinge loss g(f_k, y_k, θ) = max{0, −y_k(1 − θ^T f_k)} can be formulated as

gσ=maxx∈Θxk(−yk(1−θTfk))−σ2∥fk∥∞xk2,$$\begin{array}{} \displaystyle g_{\sigma}=\max\limits_{x\in\Theta}x_{k}(-y_{k}(1-\theta^{T}f_{k}))-\frac{\sigma}{2}\|f_{k}\|_{\infty}x_{k}^{2}, \end{array}$$(6)

where ||f_k||_∞ term is used as a normalization. In view of setting the objective ontology function of (6) to 0 and projecting x_k on Θ, we infer the following solution: xk=median{−yk(1−θTfk)σ‖fk‖∞,0,1}$\begin{array}{} \displaystyle x_{k}={\rm median}\{\frac{-y_{k}(1-\theta^{T}f_{k})}{\sigma\|f_{k}\|_{\infty}},0,1\} \end{array}$. Furthermore, the piece-wise approximation of g can be expressed as

gσ={0,yk(1−θTfk)>0−yk(1−θTfk)−σ2‖fk‖∞, yk(1−θTfk)<−σ‖fk‖∞(yk(1−θTfk))22σ‖fk‖∞,Otherwise.$$\begin{array}{} \displaystyle {g_\sigma } = \left\{ {\begin{array}{*{20}{c}} {0,} \hfill & {{y_k}(1 - {\theta ^T}{f_k}) \gt 0} \hfill \\ { - {y_k}(1 - {\theta ^T}{f_k}) - \frac{\sigma }{2}{{\left\| {{f_k}} \right\|}_\infty },} \hfill & {{y_k}(1 - {\theta ^T}{f_k}) < - \sigma {{\left\| {{f_k}} \right\|}_\infty }} \hfill \\ {\frac{{{{({y_k}(1 - {\theta ^T}{f_k}))}^2}}}{{2\sigma {{\left\| {{f_k}} \right\|}_\infty }}},} \hfill & {{\rm{Otherwise}}.} \hfill \\ \end{array}} \right. \end{array}$$(7)

By the computation and deduction, the gradient of the smoothed hinge ontology loss g_σ (θ ) is

∂gσ(fk,yk,θ)∂θ=ykfkxk.$$\begin{array}{} \displaystyle \frac{\partial g_{\sigma}(f_{k},y_{k},\theta)}{\partial\theta}=y_{k}f_{k}x_{k}. \end{array}$$(8)

Let H^Λ = [ f₁,··· , f_N′] and Y = diag(y). We get gσ(θ)∂θ=∑kykfkxk=HRYx$\begin{array}{} \displaystyle \frac{{{g_\sigma }(\theta )}}{{\partial \theta }} = \sum\nolimits_k {{y_k}{f_k}{x_k}} = {H^R}Yx \end{array}$, and Lg(θ)N′σmax‖fkfkT‖2‖fk‖∞$\begin{array}{} \displaystyle L^{g}(\theta)\frac{N'}{\sigma}\max\frac{\|f_{k}f_{k}^{T}\|_{2}}{\|f_{k}\|_{\infty}} \end{array}$ is the Lipschitz constant of g_σ (θ ).

By setting l(θ) = ||θ||₁, we infer the approximation of l with the smooth parameter σ^′ as

lσ'={−θr−σ′2,θr<−σ′θr−σ′2,θr>σ′θr22σ′,Otherwise.$$\begin{array}{} \displaystyle l_{\sigma}^{'}=\left\{\begin{array}{ll}-\theta_{r}-\frac{\sigma'}{2},& \hbox{$\theta_{r}<-\sigma'$} \\ \theta_{r}-\frac{\sigma'}{2},& \hbox{$\theta_{r}>\sigma'$}\\ \frac{\theta_{r}^{2}}{2\sigma'},& \hbox{Otherwise}. \end{array}\right. \end{array}$$

Furthermore, for each xr'=median{θrσ′,−1,1}$\begin{array}{} \displaystyle x_{r}^{'}={\rm median}\{\frac{\theta_{r}}{\sigma'},-1, 1\} \end{array}$, the gradient can be computed by ∂∑ni=1lσ′(θr)∂θ=x′$\begin{array}{} \displaystyle \frac{\partial\sum_{i=1}^{n}l_{\sigma'}(\theta_{r})}{\partial\theta}=x' \end{array}$ and the Lipschitz constant is denoted as Ll(θ)=1σ′$\begin{array}{} \displaystyle L^{l}(\theta)=\frac{1}{\sigma'} \end{array}$.

Moreover, set HstΩ=γ1Tr((ususT)(ututT))$\begin{array}{} \displaystyle H_{st}^{\Omega}=\gamma_{1}{\rm Tr}(({\bf u}_{s}{\bf u}_{s}^{T})({\bf u}_{t}{\bf u}_{t}^{T})) \end{array}$ and frΩ=γ1Tr(WST(ututT))$\begin{array}{} \displaystyle f_{r}^{\Omega}=\gamma_{1}{\rm Tr}({\bf W}_{S}^{T}({\bf u}_{t}{\bf u}_{t}^{T})) \end{array}$, we have ∂Ω(θ)∂θ=HΩθ−fΩ$\begin{array}{} \displaystyle \frac{\partial\Omega(\theta)}{\partial\theta}=H^{\Omega}\theta-f^{\Omega} \end{array}$, ∂Fσ(θ)∂θ=1N′HRYx+γcx′+HΩθ−fΩ$\begin{array}{} \displaystyle \frac{\partial F_{\sigma}(\theta)}{\partial\theta}=\frac{1}{N'}H^{R}Yx+\gamma cx'+H^{\Omega}\theta-f^{\Omega} \end{array}$ and Lσ=1σmaxk‖fkfkT‖2‖fk‖∞+γCσ′+‖HΩ‖2$\begin{array}{} \displaystyle L_{\sigma}=\frac{1}{\sigma}\max_{k}\frac{\|f_{k}f_{k}^{T}\|_{2}}{\|f_{k}\|_{\infty}}+\frac{\gamma C}{\sigma'}+\|H^{\Omega}\|_{2} \end{array}$ is the Lipschitz constant of F(θ ).

Denote θ^t, y^t and z^t as the solutions in the t-th iteration round, and use θ^$\begin{array}{} \displaystyle \widehat{\theta} \end{array}$ as a guessed solution of θ. We obtain that L_σ is the Lipschitz constant of F_σ (θ ) and the two attached ontology optimizations are stated as

miny<▽Fσ(θt),y−θt>+Lσ2∥y−θt∥22$$\begin{array}{} \displaystyle \min\limits_{y} \lt \bigtriangledown F_{\sigma}(\theta^{t}),y-\theta^{t} \gt +\frac{L_{\sigma}}{2}\|y-\theta^{t}\|_{2}^{2} \end{array}$$

and

minz∑i=0ti+12[Fσ(θi)+<▽Fσ(θi),y−θi>]+Lσ2∥y−θ^∥22,$$\begin{array}{} \displaystyle \min\limits_{z}\sum_{i=0}^{t}\frac{i+1}{2}[F_{\sigma}(\theta^{i})+ \lt \bigtriangledown F_{\sigma}(\theta^{i}),y-\theta^{i} \gt ]+\frac{L_{\sigma}}{2}\|y-\widehat{\theta}\|_{2}^{2}, \end{array}$$

respectively. Set the gradients of the two objective ontology functions in the above two attached ontology problems to be zeros, we yield yt=θt−1Lσ▿Fσ(θt)$\begin{array}{} \displaystyle y^{t}=\theta^{t}-\frac{1}{L_{\sigma}}\bigtriangledown F_{\sigma}(\theta^{t}) \end{array}$ and zt=θ^−1Lσ∑i=0ti+12▿Fσ(θi)$\begin{array}{} \displaystyle z^{t}=\widehat{\theta}-\frac{1}{L_{\sigma}}\sum_{i=0}^{t}\frac{i+1}{2}\bigtriangledown F_{\sigma}(\theta^{i}) \end{array}$. Hence, we deduce θt+1=2t+3zt+t+1t+3yt$\begin{array}{} \displaystyle \theta^{t+1}=\frac{2}{t+3}z^{t}+\frac{t+1}{t+3}y^{t} \end{array}$ and the stop criterion is given by |F_σ (θ^t+1) − F_σ (θ^t)| < ε.

Given θ , the optimization ontology problem on parameter α can be stated as

argminαγ12W−∑p=1mαpWpF2+γ22α22s.t.∑q=1mαq=1,αq≥0,q=1,⋯,m.$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {{\rm{arg}}\;{\min\limits_\alpha }\frac{{{\gamma _1}}}{2}\left\| {{\bf{W}} - \mathop \sum\limits^m_{p = 1} {\alpha _p}{{\bf{W}}_p}} \right\|_F^2 + \frac{{{\gamma _2}}}{2}\left\| \alpha \right\|_2^2}\\ {{\rm{s}}.{\rm{t}}.\quad \mathop \sum\limits^m_{q = 1} {\alpha _q} = 1,{\alpha _q} \ge 0,q = 1, \cdots ,m.} \end{array} \end{array}$$(9)

And, the ontology problem (9) can be expressed in compact form which is stated by

argminα12αTHαT+γ22α22s.t.∑q=1mαq=1,αq≥0,q=1,⋯,m.$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {{\rm{arg}}\;{\min\limits_\alpha }\frac{1}{2}{\alpha ^T}{\bf{H}}{\alpha ^T} + \frac{{{\gamma _2}}}{2}\left\| \alpha \right\|_2^2}\\ {{\rm{s}}.{\rm{t}}.\quad \mathop \sum\limits^m_{q = 1} {\alpha _q} = 1,{\alpha _q} \ge 0,q = 1, \cdots ,m.} \end{array} \end{array}$$(10)

where f = [ f₁,··· , f_m] with f_q = γ₁Tr(W^TW_q), and H is a symmetric positive semi-definite matrix such that Hst=γ1Tr(WsTWt)$\begin{array}{} \displaystyle {\bf H}_{st}=\gamma_{1}{\rm Tr}({\bf W}_{s}^{T}{\bf W}_{t}) \end{array}$. We only choose two elements α_i and α_j to update for each iteration. In order to meet the restraint ∑q=1mαq=1$\begin{array}{} \sum\nolimits_{q = 1}^m {{\alpha _q}} = 1 \end{array}$, we get αi*+αj*=αi+αj$\begin{array}{} \displaystyle \alpha_{i}^{*}+\alpha_{j}^{*}=\alpha_{i}+\alpha_{j} \end{array}$, where αi*$\begin{array}{} \displaystyle \alpha_{i}^{*} \end{array}$ and αj*$\begin{array}{} \displaystyle \alpha_{j}^{*} \end{array}$ are the solutions of the current iteration. Then, according to (10) and set εij=(Hii−Hij−Hji+Hjj)αi−∑k(Hik−Hjk)αk$\begin{array}{} \displaystyle {\varepsilon _{ij}} = ({H_{ii}} - {H_{ij}} - {H_{ji}} + {H_{jj}}){\alpha _i} - \sum\nolimits_k {({H_{ik}} - {H_{jk}}){\alpha _k}} \end{array}$, we designed the updating rule as follows: αi*=γ2(αi+αj)+(fi−fj)+εij(Hii−Hij−Hji+Hjj)+2γ2$\begin{array}{} \displaystyle \alpha_{i}^{*}=\frac{\gamma_{2}(\alpha_{i}+\alpha_{j})+(f_{i}-f_{j})+\varepsilon_{ij}}{(H_{ii}-H_{ij}-H_{ji}+H_{jj})+2\gamma_{2}} \end{array}$ and αj*=αi+αj−αi*$\begin{array}{} \displaystyle \alpha_{j}^{*}=\alpha_{i}+\alpha_{j}-\alpha_{i}^{*} \end{array}$. In case the obtained αi*$\begin{array}{} \displaystyle \alpha_{i}^{*} \end{array}$ and αj*$\begin{array}{} \displaystyle \alpha_{j}^{*} \end{array}$ don’t meet the restraint α_q ≥ 0, we further set

{αi*=0,αj*=αi+αjifγ2(αi+αj)+(hi−hj)+εij≤0αj*=0,αi*=αi+αj,ifγ2(αi+αj)+(hj−hi)+εij≤0.$$\begin{array}{} \displaystyle \left\{ {\begin{array}{*{20}{l}} {\alpha _i^* = 0,\alpha _j^* = {\alpha _i} + {\alpha _j}}&{if{\gamma _2}({\alpha _i} + {\alpha _j}) + ({h_i} - {h_j}) + {\varepsilon _{ij}} \le 0}\\ {\alpha _j^* = 0,\alpha _i^* = {\alpha _i} + {\alpha _j},}&{if{\gamma _2}({\alpha _i} + {\alpha _j}) + ({h_j} - {h_i}) + {\varepsilon _{ij}} \le 0.} \end{array}} \right. \end{array}$$

The whole ontology algorithm is stated as follows:

Initialize: α⁽⁰⁾, θ⁽⁰⁾, γ2(0)$\begin{array}{} \displaystyle \gamma_{2}^{(0)} \end{array}$ and γ3(0)$\begin{array}{} \displaystyle \gamma_{3}^{(0)} \end{array}$. Set t = 0, construct W0=∑r=0nθr(0)ururT$\begin{array}{} \displaystyle {{\bf{W}}^0} = \sum\nolimits_{r = 0}^n {\theta _r^{(0)}{{\bf{u}}_r}{\bf{u}}_r^T} \end{array}$ and WS0=∑q=1mαq(0)Wq$\begin{array}{} \displaystyle {\bf{W}}_S^0 = \sum\nolimits_{q = 1}^m {\alpha _q^{(0)}{{\bf{W}}_q}} \end{array}$.

Iterate:

Optimize θ(t+1)←argminθ1N′∑k=1N′g(yk(1−θTfk))+γ12W−WStF2+γ3(t)θt$\begin{array}{l} {\theta ^{(t + 1)}} \leftarrow {\rm{arg mi}}{{\rm{n}}_\theta }\frac{1}{{{N^\prime }}}\sum\nolimits^{{N^\prime }}_{k = 1} {g({y_k}(1 - {\theta ^T}{f_k})) + \frac{{{\gamma _1}}}{2}\left\| {{\bf{W}} - {\bf{W}}_S^t} \right\|_F^2 + \gamma _3^{(t)}{{\left\| \theta \right\|}_t}} \end{array}$ and update W(t+1)=∑r=1nθr(t+1)ururT$\begin{array}{} \displaystyle {{\bf{W}}^{(t + 1)}} = \sum\nolimits_{r = 1}^n {\theta _r^{(t + 1)}{{\bf{u}}_r}{\bf{u}}_r^T} \end{array}$;

Optimize α(t+1)←argminαγ12‖W(t+1)−WS‖F2+γ2(t)2‖α‖22$\begin{array}{} \displaystyle \alpha^{(t+1)}\gets{\rm arg}\min\limits_{\alpha}\frac{\gamma_{1}}{2}\|{\bf W}^{(t+1)}-{\bf W}_{S}\|_{F}^{2}+\frac{\gamma_{2}^{(t)}}{2}\|\alpha\|_{2}^{2} \end{array}$

and update WSt+1=∑q=1mαq(t+1)Wq$\begin{array}{l} {\bf W}_{S}^{t+1}=\sum\nolimits_{q=1}^{m}\alpha_{q}^{(t+1)}{\bf W}_{q} \end{array}$;

Determine γ3t+1=|ρC|(1N′∑k=1N′g(yk(1−(θt+1)Tfk))+γ12∥W(t+1)−WS(t)∥F2)∥θ(t+1)∥1$\begin{array}{} \displaystyle \gamma_{3}^{t+1}=\frac{|\rho_{C}|(\frac{1}{N'}\sum_{k=1}^{N'}g(y_{k}(1-(\theta^{t+1})^{T}f_{k}))+\frac{\gamma_{1}}{2}\|{\bf W}^{(t+1)}-{\bf W}_{S}^{(t)}\|_{F}^{2})}{\|\theta^{(t+1)}\|_{1}} \end{array}$;

Obtain γ2(t+1)=|ρB|(γ1‖W(t+1)−WS(t+1)‖F2)‖α(t+1)‖22$\begin{array}{} \displaystyle \gamma_{2}^{(t+1)}=\frac{|\rho_{B}|(\gamma_{1}\|{\bf W}^{(t+1)}-{\bf W}_{S}^{(t+1)}\|_{F}^{2})}{\|\alpha^{(t+1)}\|_{2}^{2}} \end{array}$;

t ← t + 1.

Until convergence.

In this section, we give the theoretical analysis of our ontology algorithm via stability assumption.

In this section, we design five simulation experiments respectively concerning ontology measure and ontology mapping. In our experiment, we select the ontology loss function as the square loss. To make sure the accuracy of the comparison, we ran our algorithm in C++ through available LAPACK and BLAS libraries for linear algebra and operation computations. We implement five experiments on a double-core CPU with a memory of 8GB.

4.1

Ontology similarity measure experiment on plant data

We use O₁, a plant “PO” ontology in the first experiment. It was constructed in www.plantontology.org. We use the structure of O₁ presented in Fig. 1. P@N (Precision Ratio see Craswell and Hawking [5]) to measure the quality of the experiment data. At first, the closest N concepts for every vertex on the ontology graph in plant field was given by experts. Then we gain the first N concepts for every vertex on ontology graph by our algorithm, and compute the precision ratio.

Fig. 1

Meanwhile, we apply ontology methods in [12], [13] and [14] to the “PO” ontology. Then after getting the average precision ratio by means of these three algorithms, the results with our algorithm are compared. Parts of the data can be referred to Table 1.

Table 1

Tab. 1.The Experiment Results of Ontology Similarity measure

	P@3 average precision ratio	P@5 average precision ratio	P@10 average precision ratio
Our Algorithm	0.5358	0.6517	0.8821
Algorithm in [12]	0.4549	0.5117	0.5859
Algorithm in [13]	0.4282	0.4849	0.5632
Algorithm in [14]	0.4831	0.5635	0.6871

When N =3, 5 or 10, the precision ratio gained from our algorithms are a little bit higher than the precision ratio determined by algorithms proposed in [12], [13] and [14]. Furthermore, the precision ratios show it tends to increase apparently as N increases. As a result, our algorithms is proved to be better and more effective than those raised by [12], [13] and [14].

4.2

Ontology mapping experiment on humanoid robotics data

“Humanoid robotics” ontologies O₂ and O₃ are used in the second experiment. The structure of O₂ and O₃ are respectively presented in Fig. 2 and Fig. 3. The leg joint structure of bionic walking device for six-legged robot is presented by the ontology O₂. The exoskeleton frame of a robot with wearable and power assisted lower extremities is presented by the ontology O₃.

Fig. 2

Fig. 3

Fig. 4

We set the experiment, aiming to get ontology mapping between O₂ and O₃. P@N Precision Ratio is taken as a measure for the quality of experiment. After applying ontology algorithms in [24], [13] and [14] on “humanoid robotics” ontology and getting the average precision ratio, the precision ratios gained from these three methods are compared. Some results can refer to Table 2.

Table 2

Tab. 2. The Experiment Results of Ontology Mapping

	P@1 average precision ratio	P@3 average precision ratio	P@5 average precision ratio
Our Algorithm	0.2778	0.5000	0.7667
Algorithm in [24]	0.2778	0.4815	0.5444
Algorithm in [13]	0.2222	0.4074	0.4889
Algorithm in [14]	0.2778	0.4630	0.5333

When N = 1, 3 or 5, the precision ratios gained from our new ontology algorithm are higher than the precision ratios determined by algorithms proposed in [24], [13] and [14]. Furthermore, the precision ratios show they tend to increase apparently as N increases. As a result, our algorithms shows much more efficiency than those raised by [24], [13] and [14].

4.4

Ontology mapping experiment on physics education data

“Physics education” ontologies O₅ and O₆ are used in the fourth experiment. We respectively present the structures of O₅ and O₆ in Fig. 5 and Fig. 6.

Fig. 5

Fig. 6

We set the experiment, aiming to give ontology mapping between O₅ and O₆. P@N precision ratio is taken as a measure for the quality of the experiment. Ontology algorithms are applied in [13], [14] and [26] on “physics education” ontology. The precision ratio gotten from the three methods is compared. Some results can be referred to Table 4.

Table 4

Tab. 4. The Experiment Results of Ontology Mapping

	P@1 average precision ratio	P@3 average precision ratio	P@5 average precision ratio
Our Algorithm	0.6913	0.7556	0.9161
Algorithm in [13]	0.6129	0.7312	0.7935
Algorithm in [14]	0.6913	0.7556	0.8452
Algorithm in [26]	0.6774	0.7742	0.8968

When N = 1, 3 or 5, the precision ratio in terms of our new ontology mapping algorithms are much higher than the precision ratio determined by algorithms proposed in [13], [14] and [26]. Furthermore, the precision ratios show they tend to increase apparently as N increases. As a result, our algorithms shows more effectiveness than those raised by [13], [14] and [26].

4.5

Ontology mapping experiment on university data

“University” ontologies O₇ and O₈ are applied in the last experiment. We present the structures of O₇ and O₈ in Fig. 7 and Fig. 8.

Fig. 7

Fig. 8

We set the experiment, aiming to give ontology mapping between O₇ and O₈. P@N precision ratio is taken as a criterion to measure the quality of the experiment. Ontology algorithms are applied in [12], [13] and [14] on “University” ontology. The precision ratios gotten from the three methods are compared. Some results can be referred to Table 5.

Table 5

Tab. 5. The Experiment Results of Ontology Mapping

	P@1 average precision ratio	P@3 average precision ratio	P@5 average precision ratio
Our Algorithm	0.5714	0.6786	0.7714
Algorithm in [12]	0.5000	0.5952	0.6857
Algorithm in [13]	0.4286	0.5238	0.6071
Algorithm in [14]	0.5714	0.6429	0.6500

When N = 1, 3 or 5, the precision ratios in terms of our new ontology mapping algorithms are much higher than the precision ratios determined by algorithms proposed in [12], [13] and [14]. Furthermore, the precision ratios show they tend to increase apparently as N increases. As a result, our algorithms turn out to have more effectiveness than those raised by [12], [13] and [14].

In this paper, the new ontology learning framework and its optimal approaches are manifested for ontology similarity calculating and ontology mapping. The new ontology algorithm is based on distance function learning tricks. Also, the stability analysis and generalized bounding computation of ontology learning algorithm are presented. Finally, simulation data in five experiments show that our new ontology learning algorithm has high efficiency in these engineering applications. The distance learning based ontology algorithm proposed in our paper illustrates the promising application prospects for multiple disciplines.