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Semi-supervised regression with label-guided adaptive graph optimization

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Abstract

For the semi-supervised regression task, both the similarity of paired samples and the limited label information serve as core indicators. Nevertheless, most traditional semi-supervised regression methods cannot make full use of both simultaneously. To alleviate the above deficiency, this paper proposes a novel semi-supervised regression with label-guided adaptive graph optimization (LGAGO-SSR). Basically, LGAGO-SSR involves two phases: graph representation and label-guided adaptive graph construction. The first phase seeks two low-dimensional manifold spaces based on two similarity matrices. The second phase aims at adaptively learning these similarity matrices by integrating the data structure information in both the low-dimensional manifold spaces and the label spaces. Each phase has its optimization problems, and the final solution is obtained by iteratively solving problems in two phases. Additionally, the idea of decomposition optimization in twin support vector regression (TSVR) is used to accelerate the training of our LGAGO-SSR. Regression results on 12 benchmark datasets with different unlabeled rates demonstrate the effectiveness of LGAGO-SSR in semi-supervised regression tasks.

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Data availability and access

After obtaining a license to use the data, the data can be accessed by visiting the following websites: https://archive.ics.uci.edu/ml/index.php, https://hastie.su.domains/ElemStatLearn/data.html and https://tianchi.aliyun.com/dataset/159885. Users can use the data for study and research purposes, but not for commercial purposes.

Notes

  1. https://tianchi.aliyun.com/dataset/159885

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Acknowledgements

We would like to thank five anonymous reviewers and Editor for their valuable comments and suggestions, which have significantly improved this paper. This work was supported in part by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant No. 19KJA550002, by the Six Talent Peak Project of Jiangsu Province of China under Grant No. XYDXX-054, by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and by the Collaborative Innovation Center of Novel Software Technology and Industrialization.

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Authors and Affiliations

  1. School of Computer Science and Technology, Soochow University, 215006, Suzhou, Jiangsu, China

    Xiaohan Zheng, Li Zhang, Leilei Yan & Lei Zhao

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  1. Xiaohan Zheng
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  2. Li Zhang
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  3. Leilei Yan
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  4. Lei Zhao
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Contributions

Xiaohan Zheng: Methodology, Software, Validation, Writing - original draft, Writing - review & editing. Li Zhang: Conceptualization, Methodology, Software, Writing - review & editing, Validation, Project administration, Funding acquisition. Leilei Yan: Investigation, Software, Visualization. Lei Zhao: Investigation, Software, Visualization.

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Correspondence to Li Zhang.

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Appendices

Appendices

1.1 Appendix A: Derivation from (13) to (17)

Let \(v^{low}_{ij}={\Vert \varvec{w}_1^T \varvec{x}_i-\varvec{w}_1^T \varvec{x}_j\Vert }_2^2+\Vert y^{low}_i-y^{low}_j\Vert _2^2\), then the objective function in (13) can be rewritten as

$$\begin{aligned}&\sum _{i=1}^{n}\sum _{j=1}^{n}\left( v^{low}_{ij}s^{low}_{ij}+\lambda ^{low}_i \left( {s^{low}_{ij}}\right) ^2\right) \nonumber \\ =&\sum _{i=1}^{n}\sum _{j=1}^{n}\lambda ^{low}_i\left( \left( \frac{v^{low}_{ij}s^{low}_{ij}}{\lambda ^{low}_i}+\left( {s^{low}_{ij}}\right) ^2+\frac{({v^{low}_{ij}})^2}{4({\lambda _i^{low}})^2}\right) -\frac{({v^{low}_{ij}})^2}{4({\lambda _i^{low}})^2}\right) \nonumber \\ =&\sum _{i=1}^{n}\lambda ^{low}_i\sum _{j=1}^{n}\left( \left( \frac{v^{low}_{ij}s^{low}_{ij}}{\lambda ^{low}_i}+\left( {s^{low}_{ij}}\right) ^2+\frac{({v^{low}_{ij}})^2}{4({\lambda _i^{low}})^2}\right) -\frac{({v^{low}_{ij}})^2}{4({\lambda _i^{low}})^2}\right) \nonumber \\ =&\sum _{i=1}^{n}\lambda ^{low}_i\!\left( \left( \frac{(\varvec{v}^{low}_{i})^T{\varvec{s}^{low}_{i}}}{\lambda ^{low}_i}\!+\!\left( \varvec{s}^{low}_{i}\right) ^T\varvec{s}^{low}_{i}\!+\!\frac{({\varvec{v}^{low}_{i}})^T{\varvec{v}^{low}_{i}}}{4{(\lambda ^{low}_i)}^2}\right) \!-\frac{({\varvec{v}^{low}_{i}})^T{\varvec{v}^{low}_{i}}}{4({\lambda ^{low}_i})^2}\right) \nonumber \\ =&\sum _{i=1}^{n}\lambda ^{low}_i\left\| \varvec{s}^{low}_{i}+\frac{\varvec{v}^{low}_{i}}{2\lambda ^{low}_i}\right\| ^2_2-\frac{(\varvec{v}^{low}_{i})^T{\varvec{v}^{low}_{i}}}{4\lambda ^{low}_i}, \end{aligned}$$
(57)

where \(\varvec{v}^{low}_i=[v^{low}_{i1},v^{low}_{i2},\cdots ,v^{low}_{in}]^T\).

Because \(\lambda ^{low}_i\) and \(\varvec{v}^{low}_{i}\) are unrelated to \(\varvec{s}_i^{low}\), the last term in (57) is a constant. Therefore, the objective function in (13) is equivalent to

$$\begin{aligned} \sum _{i=1}^{n}\left\| \varvec{s}^{low}_{i}+\frac{\varvec{v}^{low}_{i}}{2\lambda ^{low}_i}\right\| ^2_2. \end{aligned}$$

In a summary, we have completed the derivation from (13) to (17).

$$\begin{aligned} \min _{\varvec{w}_1,b_1,\varvec{\xi }_1,\varvec{S}^{low},\varvec{y}^{low}}~~~~&G_1\left( \varvec{w}_1,b_1,\varvec{\xi }_1,\varvec{S}^{low},\varvec{y}^{low}\right) \nonumber \\ =&\frac{1}{2}\Vert \varvec{w}_1\Vert ^2_2+\sum _{i=1}^{n}\Big ({C_1}\Vert y^{low}_i-\epsilon _1-f_1(\varvec{x}_i)\Vert _2^2\nonumber \\&+{C_2}{\xi _1}_i\Big )\frac{1}{2}\sum _{i=1}^{n}\sum _{j=1}^{n}\Big (\Vert f_1(\varvec{x}_i)-f_1(\varvec{x}_j)\Vert _2^2s^{low}_{ij}\nonumber \\&+\Vert y^{low}_i-y^{low}_j\Vert ^2_2s^{low}_{ij}+\lambda ^{low}_i \left( {s^{low}_{ij}}\Big )^2\right) \nonumber \\ s.t.~~~~~~~~~~~&y^{low}_i-\epsilon _1-f_1(\varvec{x}_i)+{\xi _1}_i\ge 0,\nonumber \\&{\varvec{y}}^{low}_u ={\left( \varvec{I}- {\varvec{S}}^{low}_{uu}\right) }^{-1} {\varvec{S}}^{low}_{ul}\varvec{y}_l,\nonumber \\&{\xi _1}_i\ge 0,~s^{low}_{ij}\ge 0,~ \varvec{1}^T{{\varvec{s}}^{low}_{i}} = 1,\nonumber \\&i,j=1,\cdots ,n, \end{aligned}$$
(58)

1.2 Appendix B: Proof of Theorem 1

The sequences \(\left\{ \left( \varvec{w}_1^{(p)},b_1^{(p)},\varvec{\xi }_1^{(p)},{\varvec{S}^{low}}^{(p)},{\varvec{y}^{low}}^{(p)}\right) \right\} \) and \(\left\{ \!(\varvec{w}_2^{(p)},b_2^{(p)},{\varvec{\xi }_2}^{(p)},{\varvec{S}^{up}}^{(p)},{\varvec{y}^{up}}^{(p)})\!\right\} \) generated by Algorithm 1 can guarantee that \(\left\{ G_1(\varvec{w}_1^{(p)},b_1^{(p)},\varvec{\xi }_1^{(p)},{\varvec{S}^{low}}^{(p)},{\varvec{y}^{low}}^{(p)})\right\} \) and \(\left\{ G_2(\varvec{w}_2^{(p)},b_2^{(p)},{\varvec{\xi }_2}^{(p)},{\varvec{S}^{up}}^{(p)},{\varvec{y}^{up}}^{(p)})\right\} \) are monotonic decreasing and bounded, respectively, where p refers to the current number of iterations.

Assume that p is the current iteration. First, we prove that the sequence \(\left\{ G_1\left( \varvec{w}_1^{(p)},b_1^{(p)},\varvec{\xi }_1^{(p)},{\varvec{S}^{low}}^{(p)},{\varvec{y}^{low}}^{(p)}\right) \right\} \) decreases monotonically. Given \(\left( \varvec{w}_1^{(p)},b_1^{(p)},\varvec{\xi }_1^{(p)}\right) \), we optimize (13) to obtain \(\left( {\varvec{S}^{low}}^{(p+1)},{\varvec{y}^{low}}^{(p+1)}\right) \), thus we have

$$\begin{aligned}&G_1\left( \varvec{w}_1^{(p)},b_1^{(p)},\varvec{\xi }_1^{(p)},{\varvec{S}^{low}}^{(p)},{\varvec{y}^{low}}^{(p)}\right) \ge G_1\Big (\varvec{w}_1^{(p)},b_1^{(p)},\nonumber \\&\quad \varvec{\xi }_1^{(p)},{\varvec{S}^{low}}^{(p+1)},{\varvec{y}^{low}}^{(p+1)}\Big ). \end{aligned}$$
(59)

Given \(\left( {\varvec{S}^{low}}^{(p+1)},{\varvec{y}^{low}}^{(p+1)}\right) \), we optimize (9) to obtain \(\left( \varvec{w}_1^{(p+1)},b_1^{(p+1)},\varvec{\xi }_1^{(p+1)}\right) \), thus we have

$$\begin{aligned}&G_1\left( \varvec{w}_1^{(p)},b_1^{(p)},\varvec{\xi }_1^{(p)},{\varvec{S}^{low}}^{(p+1)},{\varvec{y}^{low}}^{(p+1)}\right) \ge G_1\Big (\varvec{w}_1^{(p+1)},\nonumber \\&\quad b_1^{(p+1)},\varvec{\xi }_1^{(p+1)},{\varvec{S}^{low}}^{(p+1)},{\varvec{y}^{low}}^{(p+1)}\Big ). \end{aligned}$$
(60)

By combining (59) and (60), we have

$$\begin{aligned}&G_1\left( \varvec{w}_1^{(p)},b_1^{(p)},\varvec{\xi }_1^{(p)},{\varvec{S}^{low}}^{(p)},{\varvec{y}^{low}}^{(p)}\right) \ge G_1\Big (\varvec{w}_1^{(p+1)},\nonumber \\&\quad b_1^{(p+1)},\varvec{\xi }_1^{(p+1)},{\varvec{S}^{low}}^{(p+1)},{\varvec{y}^{low}}^{(p+1)}\Big ), \end{aligned}$$
(61)

which indicates that the sequence \(\Big \{G_1\Big (\varvec{w}_1^{(p)},b_1^{(p)},\varvec{\xi }_1^{(p)},\) \({\varvec{S}^{low}}^{(p)},{\varvec{y}^{low}}^{(p)}\Big )\Big \}\) for \(p=1, 2, \cdots \) is monotonic decreasing. In the same way, we can prove that the sequence \(\left\{ G_2\left( \varvec{w}_2^{(p)},b_2^{(p)},{\varvec{\xi }_2}^{(p)},{\varvec{S}^{up}}^{(p)},{\varvec{y}^{up}}^{(p)}\right) \right\} \) is monotonic dec-reasing when \(p=1,2,\cdots \).

Next, we prove that both the sequences \(\Big \{G_1\Big (\varvec{w}_1^{(p)},b_1^{(p)},\) \(\varvec{\xi }_1^{(p)},{\varvec{S}^{low}}^{(p)},{\varvec{y}^{low}}^{(p)}\Big )\Big \}\) and \(\Big \{G_2\Big (\varvec{w}_2^{(p)},b_2^{(p)},{\varvec{\xi }_2}^{(p)},{\varvec{S}^{up}}^{(p)},\) \({\varvec{y}^{up}}^{(p)}\Big )\Big \}\) have the infimum. In (15) and (16), we know that \(s^{low}_{ij}\), \(s^{up}_{ij}\), \(\lambda ^{low}_i\), \(\lambda ^{up}_i\), and \(C_i~(i=1,2,3,4)\) are all great than zero. Thus, it is easy to infer that

$$\begin{aligned} G_1\left( \varvec{w}_1^{(p)},b_1^{(p)},\varvec{\xi }_1^{(p)},{\varvec{S}^{low}}^{(p)},{\varvec{y}^{low}}^{(p)}\right) \ge 0, \end{aligned}$$
(62)

and

$$\begin{aligned} G_2\left( \varvec{w}_2^{(p)},b_2^{(p)},{\varvec{\xi }_2}^{(p)},{\varvec{S}^{up}}^{(p)},{\varvec{y}^{up}}^{(p)}\right) \ge 0. \end{aligned}$$
(63)

In the other word, \(\left\{ G_1\!\left( \varvec{w}_1^{(p)},b_1^{(p)},\varvec{\xi }_1^{(p)},\!{\varvec{S}^{low}}^{(p)},{\varvec{y}^{low}}^{(p)}\right) \!\right\} \) and \(\left\{ G_2\left( \varvec{w}_2^{(p)},b_2^{(p)},{\varvec{\xi }_2}^{(p)},{\varvec{S}^{up}}^{(p)},{\varvec{y}^{up}}^{(p)}\right) \right\} \) have the infimum that equals 0.

This completes the proof of Theorem 1.

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Zheng, X., Zhang, L., Yan, L. et al. Semi-supervised regression with label-guided adaptive graph optimization. Appl Intell 54, 10671–10694 (2024). https://doi.org/10.1007/s10489-024-05766-7

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