LDL

BaseLDL

class pyldl.algorithms.base.BaseLDL(random_state: int | None = None)

Let \(\mathcal{X} = \mathbb{R}^{q}\) denote the input space and \(\mathcal{Y} = \lbrace y_i \rbrace_{i=1}^{l}\) denote the label space. The description degree of \(y \in \mathcal{Y}\) to \(\boldsymbol{x} \in \mathcal{X}\) is denoted by \(d_{\boldsymbol{x}}^{y}\). Then the label distribution of \(\boldsymbol{x}\) is defined as \(\boldsymbol{d} = \lbrace d_{\boldsymbol{x}}^{y} \rbrace_{y \in \mathcal{Y}}\). Note that \(\boldsymbol{d}\) is under the constraints of probability simplex, i.e., \(\boldsymbol{d} \in \Delta^{l-1}\), where \(\Delta^{l-1} = \lbrace \boldsymbol{d} \in \mathbb{R}^{l} \,|\, \boldsymbol{d} \geq 0,\, \boldsymbol{d}^{\text{T}} \boldsymbol{1} = 1 \rbrace\). Given a training set of \(n\) samples \(\mathcal{S} = \lbrace (\boldsymbol{x}_i,\, \boldsymbol{d}_i) \rbrace_{i=1}^{n}\), the goal of LDL is to learn a conditional probability mass function \(p(\boldsymbol{d} \,|\, \boldsymbol{x})\).

LRLDL

class pyldl.algorithms._LRLDL(mode='threshold', param=None, random_state=None)

Base class for pyldl.algorithms.TLRLDL and pyldl.algorithms.TKLRLDL.

ADMM is used as optimization algorithm.

_update_V()

Please note that Eq. (11) in paper [LDL-KWT+24] should be corrected to:

\[\boldsymbol{\Gamma}_1 \leftarrow \boldsymbol{\Gamma}_1 + \mu \left(\boldsymbol{W}\boldsymbol{X}^{\text{T}}\boldsymbol{O} - \boldsymbol{G}\right)\text{.}\]

_update_W()

Please note that Eq. (8) in paper [LDL-KWT+24] should be corrected to:

\[\begin{split}\begin{aligned} \boldsymbol{W} \leftarrow & \left(\left(\mu \boldsymbol{G} + \boldsymbol{\Gamma}_1 + \boldsymbol{L}\right) \boldsymbol{O}^{\text{T}} \boldsymbol{X} + \boldsymbol{D}\boldsymbol{X} \right) \\ & \left( \boldsymbol{X}^{\text{T}}\boldsymbol{X} + 2 \lambda \boldsymbol{I} + (1+\mu) \boldsymbol{X}^{\text{T}}\boldsymbol{O}\boldsymbol{O}^{\text{T}}\boldsymbol{X} \right)^{-1}\text{,} \end{aligned}\end{split}\]

where \(\boldsymbol{I}\) is the identity matrix.

And Eq. (10) should be corrected to:

\[\begin{split}\begin{aligned} \boldsymbol{O} \leftarrow & \left( (1+\mu) \boldsymbol{X}\boldsymbol{W}^{\text{T}} \left( \boldsymbol{X}\boldsymbol{W}^{\text{T}} \right)^{\text{T}} + 2 \lambda \boldsymbol{I} \right)^{-1} \\ & \boldsymbol{X}\boldsymbol{W}^{\text{T}} \left(\boldsymbol{L} + \mu \boldsymbol{G} - \boldsymbol{\Gamma}_1\right)\text{.} \end{aligned}\end{split}\]

TKLRLDL

class pyldl.algorithms.TKLRLDL(param=None, random_state=None)

TKLRLDL is proposed in paper [LDL-KWT+24].

A top-\(k\) binaryzation method is used to generate the logical label matrix.

TLRLDL

class pyldl.algorithms.TLRLDL(param=None, random_state=None)

TLRLDL is proposed in paper [LDL-KWT+24].

A threshold-based binaryzation method is used to generate the logical label matrix.

LDL-LRR

class pyldl.algorithms.LDL_LRR(*args, **kwargs)

LDL-LRR is proposed in paper [LDL-JSL+23].

BFGS is used as the optimization algorithm.

LDL-DPA

class pyldl.algorithms.LDL_DPA(*args, **kwargs)

LDL-DPA is proposed in paper [LDL-JQLL24].

BFGS is used as the optimization algorithm.

CAD

class pyldl.algorithms.CAD(*args, **kwargs)

CAD is proposed in paper [LDL-WZYY23].

QFD2

class pyldl.algorithms.QFD2(*args, **kwargs)

QFD2 is proposed in paper [LDL-WZYY23].

CJS

class pyldl.algorithms.CJS(*args, **kwargs)

CJS is proposed in paper [LDL-WZYY23].

DF-LDL

class pyldl.algorithms.DF_LDL(estimator=None, random_state=None)

DF-LDL is proposed in paper [LDL-GGAT+21].

LDL-SCL

class pyldl.algorithms.LDL_SCL(*args, **kwargs)

LDL-SCL is proposed in paper [LDL-ZJL18].

Adam is used as optimizer.

See also:

[LDL-SCL-JLZ+21]

Xiuyi Jia, Zechao Li, Xiang Zheng, Weiwei Li, and Sheng-Jun Huang. Label distribution learning with label correlations on local samples. IEEE Transactions on Knowledge and Data Engineering, 33(4):1619–1631, 2021. URL: https://doi.org/10.1109/TKDE.2019.2943337.

LDL-LCLR

class pyldl.algorithms.LDL_LCLR(random_state: int | None = None)

LDL-LCLR is proposed in paper [LDL-RJLZ19].

ADMM is used as the optimization algorithm.

_update_W()

Please note that Eq. (9) in paper [LDL-RJLZ19] should be corrected to:

\[\begin{split}\begin{aligned} \nabla_\boldsymbol{W} = & \boldsymbol{X}^{\text{T}} \left(\hat{\boldsymbol{D}} - \boldsymbol{D}\right) + 2 \lambda_1 \boldsymbol{W} - \boldsymbol{X}^{\text{T}} \left(\left(\hat{\boldsymbol{D}} - \hat{\boldsymbol{D}}^2\right) \odot \boldsymbol{\Gamma}_1\right) \boldsymbol{S}^{\text{T}} \\ - & \rho \boldsymbol{X}^{\text{T}} \left(\left(\hat{\boldsymbol{D}} - \hat{\boldsymbol{D}}^2\right) \odot \left(\boldsymbol{D} - \hat{\boldsymbol{D}}\boldsymbol{S} - \boldsymbol{E}\right)\right) \boldsymbol{S}^{\text{T}}\text{,} \end{aligned}\end{split}\]

where \(\odot\) denotes element-wise multiplication.

LDLSF

class pyldl.algorithms.LDLSF(random_state: int | None = None)

LDLSF is proposed in paper [LDL-RJL+19].

ADMM is used as optimization algorithm.

LDLLC

class pyldl.algorithms.LDLLC(*args, **kwargs)

LDLLC is proposed in paper [LDL-JLLZ18].

BFGS is used as optimization algorithm.

BCPNN

class pyldl.algorithms.BCPNN(*args, **kwargs)

BCPNN is proposed in paper [LDL-YSS17].

BCPNN is based on CPNN. See also:

[BCPNN-GYZ13]

Xin Geng, Chao Yin, and Zhi-Hua Zhou. Facial age estimation by learning from label distributions. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(10):2401–2412, 2013. URL: https://doi.org/10.1109/TPAMI.2013.51.

ACPNN

class pyldl.algorithms.ACPNN(*args, **kwargs)

ACPNN is proposed in paper [LDL-YSS17].

ACPNN is based on CPNN. See also:

[ACPNN-GYZ13]

Xin Geng, Chao Yin, and Zhi-Hua Zhou. Facial age estimation by learning from label distributions. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(10):2401–2412, 2013. URL: https://doi.org/10.1109/TPAMI.2013.51.

LDLF

class pyldl.algorithms.LDLF(*args, **kwargs)

LDLF is proposed in paper [LDL-SZGY17].

The algorithms employs deep neural decision forests. See also:

[LDLF-KFCB15]

Peter Kontschieder, Madalina Fiterau, Antonio Criminisi, and Samuel Rota Bulo. Deep neural decision forests. In Proceedings of the IEEE/CVF International Conference on Computer Vision, 1467–1475. 2015. URL: https://doi.org/10.1109/ICCV.2015.172.

Adam is used as the optimizer.

SA

class pyldl.algorithms._SA(random_state=None)

Base class for pyldl.algorithms.SA_IIS and pyldl.algorithms.SA_BFGS.

SA refers to specialized algorithms, where MaxEnt is employed as model.

SA-BFGS

class pyldl.algorithms.SA_BFGS(random_state=None)

SA-BFGS is proposed in paper [LDL-Gen16].

BFGS is used as optimization algorithm.

SA-IIS

class pyldl.algorithms.SA_IIS(random_state=None)

SA-IIS is proposed in paper [LDL-Gen16].

IIS is used as optimization algorithm.

IIS-LLD is the early version of SA-IIS. See also:

[SA-IIS-GYZ13]

Xin Geng, Chao Yin, and Zhi-Hua Zhou. Facial age estimation by learning from label distributions. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(10):2401–2412, 2013. URL: https://doi.org/10.1109/TPAMI.2013.51.

[SA-IIS-GSMZ10]

Xin Geng, Kate Smith-Miles, and Zhi-Hua Zhou. Facial age estimation by learning from label distributions. In Proceedings of the AAAI Conference on Artificial Intelligence, 451–456. 2010. URL: https://doi.org/10.1609/aaai.v24i1.7657.

[SA-IIS-ZWG15]

Zhaoxiang Zhang, Mo Wang, and Xin Geng. Crowd counting in public video surveillance by label distribution learning. Neurocomputing, 166:151–163, 2015. URL: https://doi.org/10.1016/j.neucom.2015.03.083.

AA-\(k\)NN

class pyldl.algorithms.AA_KNN(random_state: int | None = None)

AA-kNN is proposed in paper [LDL-Gen16].

AA-BP

class pyldl.algorithms.AA_BP(*args, **kwargs)

AA-BP is proposed in paper [LDL-Gen16].

PT

class pyldl.algorithms._PT(random_state: int | None = None)

Base class for pyldl.algorithms.PT_Bayes and pyldl.algorithms.PT_SVM.

PT refers to problem transformation.

PT-Bayes

class pyldl.algorithms.PT_Bayes(random_state: int | None = None)

PT-Bayes is proposed in paper [LDL-Gen16].

PT-SVM

class pyldl.algorithms.PT_SVM(random_state: int | None = None)

PT-SVM is proposed in paper [LDL-Gen16].

LDSVR

class pyldl.algorithms.LDSVR(random_state: int | None = None)

LDSVR is proposed in paper [LDL-GH15].

CPNN

class pyldl.algorithms.CPNN(*args, **kwargs)

CPNN is proposed in paper [LDL-GYZ13].

References

[LDL-KWT+24] (1,2,3,4)

Zhiqiang Kou, Jing Wang, Jiawei Tang, Yuheng Jia, Boyu Shi, and Xin Geng. Exploiting multi-label correlation in label distribution learning. In Proceedings of the International Joint Conference on Artificial Intelligence, 4326–4334. 2024. URL: https://doi.org/10.24963/ijcai.2024/478.

[LDL-JSL+23]

Xiuyi Jia, Xiaoxia Shen, Weiwei Li, Yunan Lu, and Jihua Zhu. Label distribution learning by maintaining label ranking relation. IEEE Transactions on Knowledge and Data Engineering, 35(2):1695–1707, 2023. URL: https://doi.org/10.1109/TKDE.2021.3099294.

[LDL-JQLL24]

Xiuyi Jia, Tian Qin, Yunan Lu, and Weiwei Li. Adaptive weighted ranking-oriented label distribution learning. IEEE Transactions on Neural Networks and Learning Systems, 35(8):11302–11316, 2024. URL: https://doi.org/10.1109/TNNLS.2023.3258976.

[LDL-WZYY23] (1,2,3)

Changsong Wen, Xin Zhang, Xingxu Yao, and Jufeng Yang. Ordinal label distribution learning. In Proceedings of the IEEE/CVF International Conference on Computer Vision, 23481–23491. 2023. URL: https://doi.org/10.1109/ICCV51070.2023.02146.

[LDL-GGAT+21]

Manuel González, Germán González-Almagro, Isaac Triguero, José-Ramón Cano, and Salvador García. Decomposition-fusion for label distribution learning. Information Fusion, 66:64–75, 2021. URL: https://doi.org/10.1016/j.inffus.2020.08.024.

[LDL-ZJL18]

Xiang Zheng, Xiuyi Jia, and Weiwei Li. Label distribution learning by exploiting sample correlations locally. In Proceedings of the AAAI Conference on Artificial Intelligence, 4556–4563. 2018. URL: https://doi.org/10.1609/aaai.v32i1.11693.

[LDL-RJLZ19] (1,2)

Tingting Ren, Xiuyi Jia, Weiwei Li, and Shu Zhao. Label distribution learning with label correlations via low-rank approximation. In Proceedings of the International Joint Conference on Artificial Intelligence, 3325–3331. 2019. URL: https://doi.org/10.24963/ijcai.2019/461.

[LDL-RJL+19]

Tingting Ren, Xiuyi Jia, Weiwei Li, Lei Chen, and Zechao Li. Label distribution learning with label-specific features. In Proceedings of the International Joint Conference on Artificial Intelligence, 3318–3324. 2019. URL: https://doi.org/10.24963/ijcai.2019/460.

[LDL-JLLZ18]

Xiuyi Jia, Weiwei Li, Junyu Liu, and Yu Zhang. Label distribution learning by exploiting label correlations. In Proceedings of the AAAI Conference on Artificial Intelligence, 3310–3317. 2018. URL: https://doi.org/10.1609/aaai.v32i1.11664.

[LDL-YSS17] (1,2)

Jufeng Yang, Ming Sun, and Xiaoxiao Sun. Learning visual sentiment distributions via augmented conditional probability neural network. In Proceedings of the AAAI Conference on Artificial Intelligence, 224–230. 2017. URL: https://doi.org/10.1609/aaai.v31i1.10485.

[LDL-SZGY17]

Wei Shen, Kai Zhao, Yilu Guo, and Alan Yuille. Label distribution learning forests. In Advances in Neural Information Processing Systems, 834–843. 2017.

[LDL-Gen16] (1,2,3,4,5,6)

Xin Geng. Label distribution learning. IEEE Transactions on Knowledge and Data Engineering, 28(7):1734–1748, 2016. URL: https://doi.org/10.1109/TKDE.2016.2545658.

[LDL-GH15]

Xin Geng and Peng Hou. Pre-release prediction of crowd opinion on movies by label distribution learning. In Proceedings of the International Joint Conference on Artificial Intelligence, 3511–3517. 2015.

[LDL-GYZ13]

Xin Geng, Chao Yin, and Zhi-Hua Zhou. Facial age estimation by learning from label distributions. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(10):2401–2412, 2013. URL: https://doi.org/10.1109/TPAMI.2013.51.

Further Reading

[LDL-YJ24]

Peiqiu Yu and Xiuyi Jia. Exploiting indirect linear correlation for label distribution learning. Neurocomputing, pages 128022, 2024. URL: https://doi.org/10.1016/j.neucom.2024.128022.

[LDL-TCGJ23]

Chao Tan, Sheng Chen, Xin Geng, and Genlin Ji. A label distribution manifold learning algorithm. Pattern Recognition, 135:109112, 2023. URL: https://doi.org/10.1016/j.patcog.2022.109112.

[LDL-WG23]

Jing Wang and Xin Geng. Label distribution learning by exploiting label distribution manifold. IEEE Transactions on Neural Networks and Learning Systems, 34(2):839–852, 2023. URL: https://doi.org/10.1109/TNNLS.2021.3103178.

[LDL-LWY+22]

Qiang Li, Jingjing Wang, Zhaoliang Yao, Yachun Li, Pengju Yang, Jingwei Yan, Chunmao Wang, and Shiliang Pu. Unimodal-concentrated loss: fully adaptive label distribution learning for ordinal regression. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 20513–20522. 2022. URL: https://doi.org/10.1109/CVPR52688.2022.01986.

[LDL-ZMSZ22]

Tianyue Zhang, Yingke Mao, Furao Shen, and Jian Zhao. Label distribution learning through exploring nonnegative components. Neurocomputing, 501:212–221, 2022. URL: https://doi.org/10.1016/j.neucom.2022.06.017.

[LDL-XJS+20]

Suping Xu, Hengrong Ju, Lin Shang, Witold Pedrycz, Xibei Yang, and Chun Li. Label distribution learning: a local collaborative mechanism. International Journal of Approximate Reasoning, 121:59–84, 2020. URL: https://doi.org/10.1016/j.ijar.2020.02.003.

[LDL-JZL+19]

Xiuyi Jia, Xiang Zheng, Weiwei Li, Changqing Zhang, and Zechao Li. Facial emotion distribution learning by exploiting low-rank label correlations locally. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 9841–9850. 2019. URL: https://doi.org/10.1109/CVPR.2019.01007.

[LDL-WG19a]

Jing Wang and Xin Geng. Theoretical analysis of label distribution learning. In Proceedings of the AAAI Conference on Artificial Intelligence, 5256–5263. 2019. URL: https://doi.org/10.1609/aaai.v33i01.33015256.

[LDL-WG19b]

Ke Wang and Xin Geng. Discrete binary coding based label distribution learning. In Proceedings of the International Joint Conference on Artificial Intelligence, 3733–3739. 2019. URL: https://doi.org/10.24963/ijcai.2019/518.

[LDL-XSS19]

Suping Xu, Lin Shang, and Furao Shen. Latent semantics encoding for label distribution learning. In Proceedings of the International Joint Conference on Artificial Intelligence, 3982–3988. 2019. URL: https://doi.org/10.24963/ijcai.2019/553.

[LDL-CWFL18]

Mengting Chen, Xinggang Wang, Bin Feng, and Wenyu Liu. Structured random forest for label distribution learning. Neurocomputing, 320:171–182, 2018. URL: https://doi.org/10.1016/j.neucom.2018.09.002.

[LDL-WG18]

Ke Wang and Xin Geng. Discrete binary coding based label distribution learning. In Proceedings of the International Joint Conference on Artificial Intelligence, 3733–3739. 2018. URL: https://doi.org/10.24963/ijcai.2018/386.

[LDL-ZZ18]

Peng Zhao and Zhi-Hua Zhou. Label distribution learning by optimal transport. In Proceedings of the AAAI Conference on Artificial Intelligence, 4506–4513. 2018. URL: https://doi.org/10.1609/aaai.v32i1.11609.

[LDL-HGHL17]

Peng Hou, Xin Geng, Zeng-Wei Huo, and Jia-Qi Lv. Semi-supervised adaptive label distribution learning for facial age estimation. In Proceedings of the AAAI Conference on Artificial Intelligence, 2015–2021. 2017. URL: https://doi.org/10.1609/aaai.v31i1.10822.

[LDL-GXX+17]

Bin-Bin Gao, Chao Xing, Chen-Wei Xie, Jianxin Wu, and Xin Geng. Deep label distribution learning with label ambiguity. IEEE Transactions on Image Processing, 26(6):2825–2838, 2017. URL: https://doi.org/10.1109/TIP.2017.2689998.

[LDL-XGX16]

Chao Xing, Xin Geng, and Hui Xue. Logistic boosting regression for label distribution learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 4489–4497. 2016. URL: https://doi.org/10.1109/CVPR.2016.486.

[LDL-YGZ16]

Xu Yang, Xin Geng, and Deyu Zhou. Sparsity conditional energy label distribution learning for age estimation. In Proceedings of the International Joint Conference on Artificial Intelligence, 2259–2265. 2016. URL: https://www.ijcai.org/Abstract/16/322.