Collaboration based Multi-Label Learning

02/08/2019 ∙ by Lei Feng, et al. ∙ Nanyang Technological University Southwest University 0

It is well-known that exploiting label correlations is crucially important to multi-label learning. Most of the existing approaches take label correlations as prior knowledge, which may not correctly characterize the real relationships among labels. Besides, label correlations are normally used to regularize the hypothesis space, while the final predictions are not explicitly correlated. In this paper, we suggest that for each individual label, the final prediction involves the collaboration between its own prediction and the predictions of other labels. Based on this assumption, we first propose a novel method to learn the label correlations via sparse reconstruction in the label space. Then, by seamlessly integrating the learned label correlations into model training, we propose a novel multi-label learning approach that aims to explicitly account for the correlated predictions of labels while training the desired model simultaneously. Extensive experimental results show that our approach outperforms the state-of-the-art counterparts.

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Introduction

Multi-label learning deals with the problem where an instance can be associated with multiple labels simultaneously. Formally speaking, let be -dimensional feature space and be the label space with labels. Given the multi-label training set where

is a feature vector and

is the label vector, the goal of multi-label learning is to learn a model , which maps from the space of feature vectors to the space of label vectors. As a learning framework that handles objects with multiple semantics, multi-label learning has been widely applied in many real-world applications, such as image annotation  [Yang et al.2016], document categorization  [Li, Ouyang, and Zhou2015], bioinformatics  [Zhang and Zhou2006], and information retrieval  [Gopal and Yang2010].

The most straightforward multi-label learning approach [Boutell et al.2004] is to decompose the problem into a set of independent binary classification tasks, one for each label. Although this strategy is easy to implement, it may result in degraded performance, due to the ignorance of correlations among labels. To compensate for this deficiency, the exploitation of label correlations has been widely accepted as a key component of effective multi-label learning approaches [Gibaja and Ventura2015, Zhang and Zhou2014].

So far, many methods have been developed to improve the performance of multi-label learning by exploring various types of label correlations [Tsoumakas et al.2009, Cesa-Bianchi, Gentile, and Zaniboni2006, Petterson and Caetano2011, Huang, Zhou, and Zhou2012, Huang, Yu, and Zhou2012, Zhu, Kwok, and Zhou2018]. There has been increasing interest in exploiting the label correlations by taking the label correlation matrix as prior knowledge [Hariharan et al.2010, Cai et al.2013, Huang et al.2016, Huang et al.2018]. Concretely, these methods directly calculate the label correlation matrix by the similarity between label vectors using common similarity measures, and then incorporate the label correlation matrix into model training for further enhancing the predictions of multiple label assignments. However, the label correlations are simply obtained by common similarity measures, which may not be able to reflect complex relationships among labels. Besides, these methods exploit label correlations by manipulating the hypothesis space, while the final predictions are not explicitly correlated.

To address the above limitations, we make a key assumption that for each individual label, the final prediction involves the collaboration between its own prediction and the predictions of other labels

. Based on this assumption, a novel multi-label learning approach named CAMEL, i.e., CollAboration based Multi-labEl Learning, is proposed. Different from most of the existing approaches that calculate the label correlation matrix simply by common similarity measures, CAMEL presents a novel method to learn such matrix and show that it is equivalent to sparse reconstruction in the label space. The learned label correlation matrix is capable of reflecting the collaborative relationships among labels regarding the final predictions. Subsequently, CAMEL seamlessly incorporates the learned label correlations into the desired multi-label predictive model. Specifically, label-independent embedding is introduced, which aims to fit the final predictions with the learned label correlations while guiding the estimation of the model parameters simultaneously. The effectiveness of CAMEL is clearly demonstrated by experimental results on a number of datasets.

Related Work

In recent years, many algorithms have been proposed to deal with multi-label learning tasks. In terms of the order of label correlations being considered, these approaches can be roughly categorized into three strategies [Zhang and Zhou2014, Gibaja and Ventura2015].

For the first-order strategy, the multi-label learning problem is tackled in a label-by-label manner where label correlations are ignored. Intuitively, one can easily decompose the multi-label learning problem into a series of independent binary classification problems (one for each label) [Boutell et al.2004]. The second-order strategy takes into consideration pairwise relationships between labels, such as the ranking between relevant labels and irrelevant labels [Elisseeff and Weston2002] or the interaction of paired labels [Zhu et al.2005]. For the third-order strategy, high-order relationships among labels are considered. Following this strategy, numerous multi-label algorithms are proposed. For example, by modeling all other labels’ influences on each label, a shared subspace [Ji et al.2008]

is extracted for model training. By addressing connections among random subsets of labels, a chain of binary classifiers  

[Read et al.2011] are sequentially trained.

Recently, there has been increasing interest in second-order approaches [Hariharan et al.2010, Cai et al.2013, Huang et al.2016, Huang et al.2018]

that take the label correlation matrix as prior knowledge for model training. These approaches normally directly calculate the label correlation matrix by the similarity between label vectors using common similarity measures, and then incorporate the label correlation matrix into model training for further enhancing the predictions of multiple label assignments. For instance, cosine similarity is widely used to calculate the label correlation matrix 

[Cai et al.2013, Huang et al.2016, Huang et al.2018]. Such label correlation matrix is further incorporated into a structured sparsity-inducing norm regularization [Cai et al.2013]

for regularizing the learning hypotheses, or performing joint label-specific feature selection and model training 

[Huang et al.2016, Huang et al.2018]. In addition, there are also some high-order approaches that exploit label correlations on the hypothesis space, while they do not rely on the label correlation matrix. For example, a boosting approach [Huang, Yu, and Zhou2012] is proposed to exploit label correlations with a hypothesis reuse mechanism.

Note that most of the existing approaches using label correlation matrix are second-order and focus on the hypothesis space. Such simple label correlations exploited in the hypothesis space may not correctly depict the real relationships among labels, and final predictions are not explicitly correlated. In the next section, a novel high-order approach with crafted label correlation matrix that focus on the label space will be introduced.

The CAMEL Approach

Following the notations used in Introduction, the training set can be alternatively represented by where denotes the instance matrix, and denotes the label matrix. In addition, we denote by the -th column vector of the matrix (versus for the -th row vector of ), and represents the matrix that excludes the -th column vector of .

Label Correlation Learning

To characterize the collaborative relationships among labels regarding the final predictions, CAMEL works by learning a label correlation matrix where reflects the contribution of the -label to the -label. Guided by the assumption that for each individual label, the final prediction involves the collaboration between its own prediction and the predictions of other labels, we thus take the given label matrix as the final prediction, and propose to learn the label correlation matrix in the following way:

(1)

where is the tradeoff parameter that controls the collaboration degree. In other words, is used to balance the -th label’s own prediction and the predictions of other labels. Since each label is normally correlated with only a few labels, the collaborative relationships between one label and other labels could be sparse. With a slight abuse of notation, we denote by the -th column vector of excluding (). Under canonical sparse representation, the coefficient vector is learned by solving the following optimization problem:

(2)

where controls the sparsity of the coefficient vector . By properly rewriting the above problem and setting , it is easy to derive the following equivalent optimization problem:

(3)

Here, this problem aims to estimate the collaborative relationships between the -th label and the other labels via sparse reconstruction. The first term corresponds to the linear reconstruction error via norm, and the second term controls the sparsity of the reconstruction coefficients by using norm. The relative importance of each term is balanced by the tradeoff parameter , which is empirically set to in the experiments. To solve problem (3), the popular Alternating Direction Method of Multiplier (ADMM) [Boyd et al.2011] is employed, and detailed information is given in Appendix A. After solving problem (3) for each label, the weight matrix can be accordingly constructed with all diagonal elements set to 0. Note that for most of the existing second-order approaches using label correlation matrix [Hariharan et al.2010, Cai et al.2013, Huang et al.2016, Huang et al.2018], only pairwise relationships are considered, and the relationships between one label and the other labels are separated. While for CAMEL, since the final prediction of each label is determined by all the predictions of other labels and itself, the relationships among all labels are exploited in a collaborative manner. Which means, the relationships between one label and the other labels are coordinated (influenced by each other). Therefore, CAMEL is a high-order approach.

Multi-Label Classifier Training

In this section, we propose a novel multi-label learning approach by seamlessly integrating the learned label correlations into the desired predictive model. Suppose the ordinary prediction matrix of is denoted by where denotes the individual label predictors respectively. In the ordinary setting, each label predictor is only in charge of a single label, while label correlations are fully lost. To absorb the learned label correlations into predictions, we reuse the assumption that for each individual label, the final prediction involves the collaboration between its own prediction and the predictions of other labels, and propose to compute the final prediction of the -th label as follows:

(4)

where is consistent with problem (1), which controls the collaboration degree of label predictions. By considering all the label predictions simultaneously, we thus obtain the following compact representation:

(5)

Here, the whole multi-label learning problem could be considered as two parallel subproblems, i.e., training the ordinary model and fitting the final predictions by the modeling outputs with label correlations. Thus, we propose to learn label-independent embedding denoted by , which works as a bridge between model training and prediction fitting. This brings several advantages: First, the two subproblems can be solved via alternation, which encourages the mutual adaption of model training and prediction fitting; Second, the relative importance of the two subproblems can be controlled by a tradeoff parameter; Third, closed-form solutions and kernel extension can be easily derived. Let , the proposed formulation is given as follows:

(6)

where controls the complexity of the model , and are the tradeoff parameters determining the relative importance of the above three terms. To instantiate the above formulation, we choose to train the widely-used model where and are the model parameters, denotes the column vector with all elements equal to 1, and is a feature mapping that maps the feature space to some higher (maybe infinite) dimensional Hilbert space. For the regularization term to control the model complexity, we adopt the widely-used squared Frobenius norm, i.e., . To further facilitate a kernel extension for the general nonlinear case, we finally present the formulation as a constrained optimization problem:

(7)
0:  : the multi-label training set

: the hyperparameters

: the unseen test instance
0:  : the predicted label for the test instance
0:  
1:  learn the label correlation matrix by solving problem (3) for each label via ADMM procedure;
2:  set ;
3:  initialize ;
4:  construct the kernel matrix by Gaussian kernel function;
5:  repeat
6:     update and according to (9);
7:     update ;
8:     update in terms of (11);
9:  until convergence.
10:  return the predicted label vector according to (12).
Algorithm 1 The CAMEL Algorithm

Optimization

Problem (7) is convex with respect to and with fixed, and also convex with respect to with and fixed. Therefore, it is a biconvex problem [Gorski, Pfeuffer, and Klamroth2007], and can be solved by an alternating approach.

Updating and with fixed

With fixed, problem (7) reduces to

(8)

By deriving the Lagrangian of the above constrained problem and setting the gradient with respect to to 0, it is easy to show where is the matrix that stores the Lagrangian multipliers. Let be the kernel matrix with its element , where represents the kernel function. For CAMEL, Gaussian kernel function is employed with set to the average Euclidean distance of all pairs of training instances. In this way, we choose to optimize with respect to and instead, and the close-form solutions are reported as follows:

(9)

where . The detailed information is provided in Appendix B.

Updating with and fixed

When and are fixed, the modeling output matrix is calculated by . By inserting , problem (7) reduces to:

(10)

Setting the gradient with respect to to 0, we can obtain the following closed-form solution:

(11)

Once the iterative process converges, the predicted label vector of the test instance is given as:

(12)

The pseudo code of CAMEL is presented in Algorithm 1. Since the proposed formulation is biconvex, this alternating optimization process is guaranteed to converge [Gorski, Pfeuffer, and Klamroth2007].

Experiments

In this section, we conduct extensive experiments on various datasets to validate the effectiveness of CAMEL.

Experimental Setup

Datasets

For comprehensive performance evaluation, we collect sixteen benchmark multi-label datasets. For each dataset , we denote by , , , , and the number of examples, the number of features (dimensions), the number of distinct class labels, the average number of labels associated with each example, and feature type, respectively. Table 1 summarizes the detailed characteristics of these datasets, which are organized in ascending order of . According to , we further roughly divide these datasets into regular-size datasets () and large-size datasets (

). For performance evaluation, 10-fold cross-validation is conducted on these datasets, where mean metric values with standard deviations are recorded.

Evaluation Metrics

For performance evaluation, we use seven widely-used evaluation metrics, including

One-error, Hamming loss, Coverage, Ranking loss, Average precision, Macro-averaging F1, and Micro-averaging F1. Note that for all the employed multi-label evaluation metrics, their values vary within the interval [0,1]. In addition, for the last three metrics, the larger values indicate the better performance, and we use the symbol to present such positive logic. While for the first five metrics, the smaller values indicate the better performance, which is represented by . More detailed information about these evaluation metrics can be found in [Zhang and Zhou2014].

Comparing Approaches

CAMEL is compared with three well-established and two state-of-the-art multi-label learning algorithms, including the first-order approach BR [Boutell et al.2004], the second-order approaches LLSF [Huang et al.2016] and JFSC [Huang et al.2018], and the high-order approaches ECC [Read et al.2011], and RAKEL [Tsoumakas, Katakis, and Vlahavas2011]. Here, LLSF and JFSC are the state-of-the-art counterparts using label correlation matrix.

BR, ECC, and RAKEL are implemented under the MULAN multi-label learning package [Tsoumakas et al.2011]

by using the logistic regression model as the base classifier. Furthermore, parameters suggested in the corresponding literatures are used, i.e., ECC: ensemble size 30; RAKEL: ensemble size

with . For LLSF, parameters are chosen from , and chosen from . For JFSC, parameters , and are chosen from , and is chosen from . For the proposed approach CAMEL, is empirically set to 1, is chosen from , and is chosen from . All of these parameters are decided by conducting 5-fold cross-validation on training set.

Dataset
cal500 502 68 174 26.04 numeric
emotions 593 72 6 1.87 numeric
genbase 662 1185 27 1.25 nominal
medical 978 1449 45 1.25 nominal
enron 1702 1001 53 3.38 nominal
image 2000 294 5 1.24 numeric
scene 2407 294 5 1.07 numeric
yeast 2417 103 14 4.24 numeric
science 5000 743 40 1.45 numeric
arts 5000 462 26 1.64 numeric
business 5000 438 30 1.59 numeric
rcv1-s1 6000 944 101 2.88 nominal
rcv1-s2 6000 944 101 2.63 nominal
rcv1-s3 6000 944 101 2.61 nominal
rcv1-s4 6000 944 101 2.48 nominal
rcv1-s5 6000 944 101 2.64 nominal
Table 1: Characteristics of the benchmark multi-label datasets.

Experimental Results

Comparing
algorithms
One-error
cal500 emotions genbase medical enron image scene yeast
CAMEL 0.1290.053 0.2920.052 0.0010.001 0.1100.021 0.2070.038 0.2420.033 0.1750.027 0.2180.027
BR 0.8930.038 0.2840.077 0.0170.016 0.3220.055 0.6460.023 0.3870.027 0.3610.036 0.2440.028
ECC 0.2950.036 0.2960.074 0.0100.013 0.1560.037 0.4210.034 0.4060.023 0.3060.020 0.2380.030
RAKEL 0.6340.039 0.3000.070 0.0090.007 0.2430.055 0.5320.007 0.4020.024 0.2800.031 0.2440.027
LLSF 0.1380.050 0.4120.051 0.0020.005 0.1200.020 0.2500.042 0.3270.030 0.2590.020 0.3940.029
JFSC 0.1160.051 0.4380.086 0.0020.005 0.1280.024 0.2780.041 0.3460.023 0.2660.022 0.2420.021
Comparing
algorithms
Hamming loss
cal500 emotions genbase medical enron image scene yeast
CAMEL 0.1360.005 0.2030.021 0.0010.001 0.0110.001 0.0450.003 0.1440.012 0.0720.009 0.1900.005
BR 0.1890.005 0.2160.028 0.0020.002 0.0260.003 0.1110.006 0.2100.014 0.1390.009 0.2050.007
ECC 0.1540.005 0.2140.027 0.0090.004 0.0110.002 0.0670.002 0.2100.016 0.1120.006 0.2040.010
RAKEL 0.1950.004 0.2380.025 0.0020.001 0.0200.002 0.0920.004 0.2230.013 0.1390.008 0.2240.009
LLSF 0.1380.006 0.2670.022 0.0010.001 0.0100.002 0.0480.002 0.1800.010 0.1090.003 0.2780.009
JFSC 0.1910.004 0.2950.019 0.0010.001 0.0100.001 0.0510.003 0.1880.012 0.1100.007 0.2060.006
Comparing
algorithms
Coverage
cal500 emotions genbase medical enron image scene yeast
CAMEL 0.7520.019 0.3120.031 0.0120.005 0.0280.012 0.2390.028 0.1560.016 0.0620.006 0.4460.010
BR 0.7860.015 0.3190.026 0.0140.006 0.1130.030 0.5800.023 0.2160.018 0.1680.015 0.4630.011
ECC 0.7960.019 0.3100.029 0.0130.003 0.0340.012 0.2910.020 0.2330.022 0.1350.010 0.4600.010
RAKEL 0.9620.016 0.3620.027 0.0140.005 0.0950.018 0.5130.019 0.2530.017 0.1690.013 0.5440.013
LLSF 0.7780.025 0.3620.032 0.0210.006 0.0310.014 0.2830.023 0.1920.007 0.0920.006 0.6010.020
JFSC 0.7300.026 0.3920.046 0.0140.007 0.0300.012 0.3140.024 0.2000.009 0.1020.007 0.4550.011
Comparing
algorithms
Ranking loss
cal500 emotions genbase medical enron image scene yeast
CAMEL 0.1770.009 0.1800.032 0.0010.001 0.0160.008 0.079 0.028 0.1280.013 0.0580.005 0.1620.007
BR 0.2330.007 0.1820.030 0.0030.004 0.0910.027 0.3040.014 0.2040.017 0.1510.015 0.1760.008
ECC 0.2190.007 0.1720.031 0.0020.002 0.0220.010 0.1180.008 0.2250.023 0.1170.010 0.1790.009
RAKEL 0.3660.008 0.2250.029 0.0020.001 0.0730.018 0.2440.017 0.2210.018 0.1310.014 0.2400.009
LLSF 0.1840.012 0.2450.033 0.0020.003 0.0190.010 0.1070.009 0.1740.006 0.0930.005 0.3460.017
JFSC 0.1880.010 0.2710.041 0.0030.003 0.0170.008 0.1180.013 0.1830.007 0.1050.007 0.1790.009
Comparing
algorithms
Average precision
cal500 emotions genbase medical enron image scene yeast
CAMEL 0.5150.018 0.7880.035 0.9970.003 0.9170.017 0.7180.025 0.8430.018 0.8970.012 0.7750.013
BR 0.3450.018 0.7830.040 0.9880.008 0.7500.036 0.3880.016 0.7530.016 0.7710.021 0.7540.013
ECC 0.4420.014 0.7890.036 0.9910.008 0.8840.023 0.5570.015 0.7380.020 0.8110.012 0.7560.014
RAKEL 0.3290.016 0.7630.039 0.9930.006 0.8000.032 0.4560.019 0.7350.017 0.8040.022 0.7200.014
LLSF 0.5070.021 0.7160.035 0.9970.005 0.9120.015 0.6820.028 0.7900.014 0.8430.008 0.6010.015
JFSC 0.4920.020 0.6910.040 0.9970.004 0.9080.016 0.6550.025 0.7790.011 0.8350.010 0.7460.012
Comparing
algorithms
Macro-averaging F1
cal500 emotions genbase medical enron image scene yeast
CAMEL 0.1800.032 0.6250.052 0.9710.030 0.7790.043 0.3250.044 0.6600.030 0.7870.023 0.4110.018
BR 0.1670.019 0.6200.044 0.9510.029 0.6400.060 0.2360.016 0.5530.027 0.6230.026 0.3910.021
ECC 0.2360.027 0.6220.043 0.9280.037 0.7550.054 0.3030.030 0.5400.030 0.6620.026 0.3950.015
RAKEL 0.1870.020 0.6140.044 0.9580.030 0.6890.051 0.2560.017 0.5400.028 0.6440.024 0.3810.020
LLSF 0.1800.031 0.6150.056 0.9710.031 0.7690.057 0.2920.043 0.5540.031 0.6150.007 0.2350.016
JFSC 0.2390.031 0.3450.023 0.9710.031 0.7720.043 0.3390.048 0.5590.035 0.7050.019 0.3000.007
Comparing
algorithms
Micro-averaging F1
cal500 emotions genbase medical enron image scene yeast
CAMEL 0.3370.017 0.6490.041 0.9880.012 0.8350.019 0.5800.023 0.6590.031 0.7800.026 0.6550.010
BR 0.3390.016 0.6390.050 0.9780.014 0.6110.032 0.3590.015 0.5580.028 0.6190.023 0.6330.013
ECC 0.3640.015 0.6420.046 0.9070.035 0.7960.023 0.4520.015 0.5410.030 0.6530.023 0.6430.017
RAKEL 0.3510.018 0.6290.049 0.9830.011 0.6780.042 0.3920.014 0.5410.031 0.6290.026 0.6320.016
LLSF 0.3250.015 0.6370.049 0.9920.003 0.8230.027 0.5340.025 0.5570.032 0.6180.008 0.2800.018
JFSC 0.4730.013 0.4060.022 0.9950.006 0.8180.018 0.5550.026 0.5650.033 0.6950.022 0.6090.012
Table 2: Predictive performance of each algorithm (meanstd. deviation) on the regular-scale datasets.
Comparing
algorithms
One-error
science arts rcv1-s1 rcv1-s2 rcv1-s3 rcv1-s4 rcv1-s5 business
CAMEL 0.4570.021 0.4620.024 0.4040.019 0.4030.018 0.4130.019 0.3310.016 0.4040.010 0.1010.009
BR 0.7600.015 0.6420.022 0.7420.019 0.7230.024 0.7180.021 0.6620.021 0.7150.015 0.4170.016
ECC 0.5740.022 0.5260.023 0.4710.020 0.4410.021 0.4480.021 0.3780.019 0.4250.016 0.1530.008
RAKEL 0.6230.014 0.5430.024 0.6130.019 0.5920.022 0.5780.020 0.5520.020 0.5750.014 0.2010.009
LLSF 0.4860.013 0.4540.027 0.4090.015 0.4060.016 0.4150.021 0.3330.016 0.3990.018 0.1220.016
JFSC 0.4890.027 0.4470.027 0.4180.016 0.4070.014 0.4180.025 0.3370.015 0.4070.023 0.1220.019
Comparing
algorithms
Hamming loss
science arts rcv1-s1 rcv1-s2 rcv1-s3 rcv1-s4 rcv1-s5 business
CAMEL 0.0300.001 0.0550.002 0.0260.008 0.0230.001 0.0230.001 0.0180.001 0.0220.001 0.0240.001
BR 0.0720.002 0.0790.003 0.0560.001 0.0530.001 0.0530.001 0.0410.001 0.0510.002 0.0490.001
ECC 0.0360.002 0.0630.002 0.0280.001 0.0240.001 0.0240.001 0.0190.001 0.0240.001 0.0300.001
RAKEL 0.0420.002 0.0750.002 0.0460.001 0.0390.001 0.0350.001 0.0350.001 0.0360.003 0.0350.002
LLSF 0.0360.002 0.0540.002 0.0270.001 0.0250.001 0.0250.001 0.0190.001 0.0230.001 0.0480.007
JFSC 0.0350.002 0.0570.002 0.0290.001 0.0250.001 0.0250.001 0.0190.001 0.0250.001 0.0270.002
Comparing
algorithms
Coverage
science arts rcv1-s1 rcv1-s2 rcv1-s3 rcv1-s4 rcv1-s5 business
CAMEL 0.1890.010 0.2050.009 0.1510.008 0.1420.012 0.1310.006 0.1430.003 0.1320.005 0.0820.006
BR 0.3030.011 0.2040.009 0.3930.011 0.3410.013 0.3510.018 0.2940.015 0.3360.013 0.1410.002
ECC 0.1960.009 0.2290.009 0.1660.011 0.1540.007 0.1540.008 0.1080.003 0.1450.001 0.1040.001
RAKEL 0.2090.012 0.2140.008 0.2730.011 0.3290.012 0.2930.017 0.2730.012 0.2460.012 0.1070.003
LLSF 0.1970.014 0.1950.011 0.1410.009 0.1460.008 0.1330.008 0.1090.006 0.1330.006 0.0860.013
JFSC 0.1960.011 0.2330.018 0.1400.006 0.1430.009 0.1360.010 0.1060.005 0.1390.006 0.0860.011
Comparing
algorithms
Ranking loss
science arts rcv1-s1 rcv1-s2 rcv1-s3 rcv1-s4 rcv1-s5 business
CAMEL 0.1390.007 0.1350.008 0.0580.003 0.0770.005 0.0470.003 0.0570.002 0.0730.002 0.0400.004
BR 0.2450.009 0.1450.006 0.1970.006 0.1900.008 0.1980.010 0.1730.009 0.1810.006 0.0880.006
ECC 0.1510.006 0.1640.007 0.0740.005 0.0690.003 0.0700.002 0.0470.004 0.0630.003 0.0550.002
RAKEL 0.1950.007 0.1560.008 0.1830.006 0.1530.008 0.1780.010 0.1120.009 0.1230.006 0.0670.005
LLSF 0.1490.009 0.1410.009 0.0600.003 0.0600.004 0.0480.003 0.0340.003 0.0450.003 0.0450.009
JFSC 0.1470.008 0.1590.009 0.0610.003 0.0620.006 0.0610.004 0.0470.003 0.0600.003 0.0450.008
Comparing
algorithms
Average precision
science arts rcv1-s1 rcv1-s2 rcv1-s3 rcv1-s4 rcv1-s5 business
CAMEL 0.6240.016 0.6070.018 0.6150.009 0.6440.012 0.6350.010 0.7170.008 0.6260.009 0.8910.009
BR 0.3830.011 0.5140.013 0.3530.011 0.3820.015 0.3820.015 0.4430.013 0.3900.009 0.7090.008
ECC 0.5160.020 0.5530.018 0.5450.016 0.5870.015 0.5850.016 0.6770.017 0.6000.009 0.8440.005
RAKEL 0.4870.012 0.5260.015 0.4240.012 0.4890.016 0.4590.014 0.4790.012 0.4320.009 0.8580.007
LLSF 0.5940.021 0.6310.016 0.6270.009 0.6370.008 0.6320.013 0.7140.010 0.6250.013 0.8670.013
JFSC 0.5950.020 0.6210.020 0.6060.008 0.6300.009 0.6240.014 0.7000.012 0.6240.013 0.8740.018
Comparing
algorithms
Macro-averaging F1
science arts rcv1-s1 rcv1-s2 rcv1-s3 rcv1-s4 rcv1-s5 business
CAMEL 0.3100.038 0.3120.029 0.2500.023 0.2580.022 0.2470.025 0.3400.031 0.2530.016 0.3260.046
BR 0.2150.048 0.2570.020 0.2320.018 0.2100.017 0.2210.019 0.3130.016 0.2360.019 0.2490.017
ECC 0.2850.024 0.2820.021 0.2710.023 0.2570.022 0.2660.012 0.3340.018 0.2850.014 0.3260.032
RAKEL 0.2670.028 0.2750.019 0.2660.019 0.2370.023 0.2430.017 0.3220.017 0.2550.018 0.3070.024
LLSF 0.3120.038 0.2190.032 0.2610.022 0.2570.025 0.2700.027 0.3340.031 0.2170.018 0.3250.028
JFSC 0.3080.039 0.3050.032 0.3080.026 0.2490.019 0.2580.024 0.3370.032 0.2540.019 0.3180.036
Comparing
algorithms
Micro-averaging F1
science arts rcv1-s1 rcv1-s2 rcv1-s3 rcv1-s4 rcv1-s5 business
CAMEL 0.4280.018 0.4150.015 0.4010.015 0.4370.017 0.4310.025 0.4910.017 0.4410.015 0.7460.011
BR 0.2770.013 0.3490.018 0.3010.009 0.3100.009 0.3070.013 0.3560.009 0.3210.009 0.5950.003
ECC 0.3430.028 0.3770.018 0.3850.016 0.4100.022 0.4140.013 0.4820.024 0.4400.011 0.6900.007
RAKEL 0.3370.014 0.3680.017 0.3410.010 0.3370.008 0.3350.014 0.3690.008 0.3500.008 0.7010.014
LLSF 0.4460.025 0.3680.018 0.4630.016 0.4320.018 0.4280.023 0.4780.017 0.4380.019 0.6930.035
JFSC 0.4490.026 0.4420.017 0.4560.008 0.4220.011 0.4240.012 0.4820.013 0.4380.011 0.7120.021
Table 3: Predictive performance of each algorithm (meanstd. deviation) on the large-scale datasets.
(a)
(b)
(c)
(d) Convergence Curve
Figure 1: Parameter sensitivity and convergence analysis of CAMEL on the enron dataset.

Table 2 and 3 report the detailed experimental results on the regular-scale and large-scale datasets respectively, where the best performance among all the algorithms is shown in boldface. From the two result tables, we can see that CAMEL outperforms other comparing algorithms in most cases. Specifically, on the regular-size datasets (Table 2), across all the evaluation metrics, CAMEL ranks first in 80.4% (45/56) cases, and on the large-scale datasets (Table 3), across all the evaluation metrics, CAMEL ranks first in 69.6% (39/56) cases. Compared with the three well-established algorithms BR, ECC, and RAKEL, CAMEL introduces a new type of label correlations, i.e., collaborative relationships among labels, and achieves superior performance in 93.8% (315/336) cases. Compared with the two state-of-the-art algorithms LLSF and JFSC, instead of employing simple similarity measures to regularize the hypothesis space, CAMEL introduces a novel method to learn label correlations for explicitly correlating the final predictions, and achieves superior performance in 80.4% (180/224) cases. These comparative results clearly demonstrate the effectiveness of the collaboration based multi-label learning approach.

Sensitivity Analysis

In this section, we first investigate the sensitivity of CAMEL with respect to the two tradeoff parameters and , and the parameter that controls the degree of collaboration, then illustrate the convergence of CAMEL. Due to page limit, we only report the experimental results on the enron dataset using the Coverage () metric. Concretely, we study the performance of CAMEL when we vary one parameter while keeping other parameters fixed at their best setting. Figure 1(a), 1(b), and 1(c) show the sensitivity curve of CAMEL with respect to , , and respectively. It can be seen that and have an important influence on the final performance, because and control the collaboration degree and the model complexity. Figure 1(d) illustrates the convergence of CAMEL by using the difference of the optimization variable between two successive iterations, i.e., . From Figure 1(d), we can observe that quickly decreases to 0 within a few number of iterations. Hence the convergence of CAMEL is demonstrated.

Conclusion

In this paper, we make a key assumption for multi-label learning that for each individual label, the final prediction involves the collaboration between its own prediction and the predictions of other labels. Guided by this assumption, we propose a novel method to learn the high-order label correlations via sparse reconstruction in the label space. Besides, by seamlessly integrating the learned label correlations into model training, we propose a novel multi-label learning approach that aims to explicitly account for the correlated predictions of labels while training the desired model simultaneously. Extensive experimental results show that our approach outperforms the state-of-the-art counterparts.

Despite the demonstrated effectiveness of CAMEL, it only considers the global collaborative relationships between labels, by assuming that such collaborative relationships are shared by all the instances. However, as different instances have different characteristics, such collaborative relationships may be shared by only a subset of instances rather than all the instances. Therefore, our further work is to explore different collaborative relationships between labels for different subsets of instances.

Acknowledgements

This work was supported by MOE, NRF, and NTU.

Appendix A. The ADMM Procedure

To solve problem (3) by ADMM, we first reformulate problem (3) into the following equivalent form:

(13)

Following the ADMM procedure, the above constrained optimization problem (13) can be solved as a series of unconstrained minimization problems using augmented Lagrangian function, which is presented as:

(14)

Here, is the penalty parameter and is the Lagrange multiplier. By introducing the scaled dual variable , a sequential minimization of the scaled ADMM iterations can be conducted by updating the three variables , and sequentially:

(15)

where is the proximity operator of the norm, which is defined as .

Appendix B. Model Parameter Optimization

The Lagrangian of problem (8) is expressed as:

(16)

where is the trace operator, and is the introduced matrix that stores the Lagrangian multipliers. Besides, we have used the property of trace operator that . By Setting the gradient w.r.t. to 0 respectively, the following equations will be induced:

(17)

The above linear equations can be simplified by the following steps:

(18)

Here, we define , then we can obtain:

(19)

In this way, can be calculated by .

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