In the past decades, various social image retagging methods [1, 2, 3, 4, 5, 6] have been proposed to improve the tag quality of social images. At the early stage, image retagging methods refine tags of social images by utilizing semantic correlation among tags and visual similarity among images [7, 8]. A basic assumption of such methods is that two images with high visual similarity should have similar semantic tags, and vice versa. To well leverage the available image-tag inter-association, some researchers improved the matrix completion based on the low-rank decomposition to simultaneously recover the missing tags, and remove the noisy tags [9, 10]. Candes and Plan  have proven that the matrix completion model enables to complete the missing entries from a small number of observed entries in the original matrix between the dyadic data. However, the aforementioned methods achieve unsatisfied retagging results when there exists disambiguation between the visual content and label taxonomy (e.g., WordNet taxonomy ).
Intuitively, a set of images uploaded by one common user tends to have close relations. That is to say, the user information (e.g., user interests and backgrounds) bridges the inter-relationship between tags and social images , especially for some certain tags, e.g., geo-related tags, event tags, etc. Therefore, several methods were proposed to simultaneously leverage the visual information, tag information, and user information for social image retagging. Sang et al.  attempted to model the inter-association among users, images and tags, and presented a tensor completion based on the low-rank decomposition to refine the social image tags. However, since the process of tensor completion generates many large-scale temporary matrices and tensors, it requires extremely large computational cost. To handle this problem, Tang et al.  proposed Tri-clustered Tensor Completion (TTC) to divide the original tag-image-user tensor among all tags, images, and users into several sub-tensors, and implement the sub-tensor completion in a parallel way. However, the computational cost still increases accompanied with the number of images increasing.
To this end, we aim to develop an efficient image retagging framework to assign high-quality tags to social images, regardless of the data scale. We investigate that anchor graph can significantly accelerate large-scale graph-based learning by only exploring a small number of anchor points on the graph [14, 15]. Inspired by this, we propose a Social anchor-Unit GrAph Regularization (SUGAR) model on a tag-image-user graph by exploring a small number of representative anchor units, where one anchor unit consists of an anchor image and an anchor user, as shown in Figure 1. Accordingly, we propose a novel Social anchor-Unit GrAph Regularized Tensor Completion (SUGAR-TC) method to efficiently refine the tags of social images. The framework of SUGAR-TC is shown in Figure 2, which mainly consists of three modules. (a) Anchor-unit graph construction module. We employ the co-clustering algorithm to obtain the representative anchor units in multiple domains (i.e., image and user domains) rather than the traditional anchors in a single domain, and then construct a social anchor-unit graph. (b) SUGAR tensor completion module. We propose a tensor completion model with Social anchor-Unit GrAph Regularization (SUGAR). It completes the anchor tensor by decomposing the non-anchor tensor to refine the tags of anchor images. (c) Anchor-aware tag assignment module. By leveraging the potential relationships between non-anchor units and anchor units, we use the weighted average tags of anchor images to efficiently assign high-quality tags to the non-anchor images.
Overall, the main contributions of this work are two-fold. (1) We propose an efficient Social anchor-Unit GrAph Regularized Tensor Completion (SUGAR-TC) method for social image retagging, even when the data scale is very large. (2) To the best of our knowledge, it is the first time that social anchor unit across multiple domains rather than the traditional anchors in a single domain is presented. In the experiments, the proposed SUGAR-TC gains superior performance on both effectiveness and efficiency compared with the state-of-the-art methods.
Ii Related Work
Ii-a Social Image Retagging
The goal of social image retagging is to improve tag quality by recovering missing tags and removing noisy tags [16, 17]. It is an essential step for large-scale tag-based image retrieval. The image retagging task is closely related to tag refinement [18, 4], tag completion . Overall, social image retagging methods have gone through two stages, i.e., image-tag association based image retagging [19, 8, 20, 7, 1, 3, 21, 22, 23, 24, 25], and tag-image-user association based image retagging [2, 4, 26, 5].
The image-tag association based methods focus on exploring inter- and intra- associations among images and tags. As one of the most classic works, Liu et al.  assumed that two images with higher visual similarity are more likely to have similar tags, or common tags. Yang et al.  proposed to mine the multi-tag intra-association for tag expansion and denoising. Chen et al. 
proposed co-regularized learning with two classifiers by jointly mapping visual features and text features into a common subspace.
Matrix completion based on the low-rank decomposition is popularly applied for image retagging. As an early study, Zhu et al.  solved the image retagging problem by factorizing the image-tag inter-association matrix into an approximately low-rank completed matrix and a sparse error matrix, where this low-rank completed matrix reveals the image-tag associations. Besides, Feng et al.  theoretically analyzed that the matrix completion method is able to simultaneously recover missing tags and remove noisy tags even with a limited number of observations. Motivated by the success of low-rank model in image tagging, Xu et al.  proposed Non-linear Matrix Completion (NMC) for social image tagging by constructing the original image-tag matrix with a non-linear kernel mapping. Li et al.  divided the original image-tag matrix into several sub-matrices, and proposed a Locality Sensitive Low-Rank (LSLR) model on each sub-matrix to recover the image-tag matrix via matrix factorization.
Tag-image-user association based image retagging methods construct a tag-image-user inter-association tensor instead of the image-tag inter-association matrix to explore the important user information . It can further improve the performance of image retagging with the help of user information. Tang et al.  proposed a Tri-clustered Tensor Completion (TTC) framework to first divide the original super-sparse tensor into several sub-tensors, and complete all sub-tensors regularized by a tensor kernel. However, to a certain degree, TTC will break the tag-image-user inter-association when dividing the original super-sparse tensor into several sub-tensors.
Ii-B Anchor Graph-based Learning
Graph-based learning methods have achieved impressive performance in various applications . However, such methods require much computational cost when the size of data rapidly increases. Therefore, several strategies have been proposed to reduce the computational cost of graph-based learning. To sum up, these strategies can be summarized into three categories. The first strategy uses neighborhood propagation from the approximate neighborhood graph to current graph [29, 30, 31, 32]. The second strategy utilizes hashing to improve the performance in terms of efficiency [33, 34, 35, 36]. The last strategy employs the anchor graph to simultaneously reduce the computational cost and memory cost [37, 38, 39, 15, 40]. On an anchor graph, anchors covering vast data point cloud can predict the label for each data point, even when the data size is large. As a result, the anchor graph model has been successfully applied to solve many practical tasks, including image retrieval 42], image classification , object tracking , and so on. In this work, we extend the anchor graph in a single domain to the anchor-unit graph across multiple domains, to simultaneously explore the association information among different domains.
Iii The Proposed Framework
This section introduces the whole framework of SUGAR-TC, as shown in Figure 2. It includes three main modules, i.e.
, Social anchor-unit graph construction module, SUGAR tensor completion module, and Anchor-aware tag assignment module. In this paper, tensors, matrices, vectors, variables and sets are denoted by calligraphic uppercase letters (e.g., ), bold uppercase letters (e.g., ), bold lowercase letters (e.g., ), letters (e.g., m, I) and blackboard bold letters (e.g., ), respectively. denotes the -th row and -th column entry of . denotes the -th row, -th column and -th tube entry of . denotes the number of data in . , and are the matrix form of by accumulating the entries of along the row, column, and tube axes, respectively. For convenience, some important notations are defined in Table I.
Iii-a Background and Problem Definition
In this work, three types of data are collected from photo sharing websites, namely the image set , the tag set and the user set , where and denote the i-th image, the j-th tag, and the k-th user, respectively. Since many tags of images freely given by users with various backgrounds and interests are ambiguous, noisy and incomplete, the original user-provided tags are weakly-supervised, which cannot represent the semantics of images well. If we directly implement the tag-based image retrieval on social images with original tags, the performance of retrieval will degrade. In this work, we aim to refine the tags of social images by utilizing the associations among tags, images and users. However, the numbers of images, tags and users are large, and even huge on photo sharing websites. This will dramatically increase the computational cost.
It has been proven that the anchor graph can significantly accelerate large-scale graph-based learning by exploring only a small number of anchor points. However, traditional anchor graph is constructed only in image domain. To leverage the user information, we extend the traditional anchor to anchor unit, which is a tuple including an anchor image and an anchor user. And the graph constructed on the anchor units is called anchor-unit graph.
Let , , , and indicate the non-anchor image set, the non-anchor user set, the anchor image set, and the anchor user set respectively, where and , we construct a tag-image-user non-anchor tensor among all the tags, non-anchor images and non-anchor users, and a tag-image-user anchor tensor among all the tags, anchor images and anchor users, in which some associations are incorrect or missing. Let denote the refined anchor tensor among the refined tags, anchor images and anchor users, this work first aims to learn with the help of the inter-association in and . After that, benefiting from the tag-image-user association in , we can efficiently retag the non-anchor images based on the inter-relationship between non-anchor units and anchor units on the anchor-unit graph.
Iii-B Social Anchor-Unit Graph Construction
To construct the anchor-unit graph, we need to obtain the anchor units from the image and user data. Generally speaking, the perfect anchor units should satisfy two conditions: 1) they can adequately represent the distribution of image and use data; 2) the number of the anchor units should be much smaller than the number of images 
. Existing methods employ the K-means clustering to obtain the traditional anchors[35, 38]. However, K-means algorithm performing in the single domain cannot cluster the image and user data simultaneously. Fortunately, co-clustering algorithm can construct the association between two types of data by an associated matrix, and then cluster rows and columns of this matrix simultaneously into several co-clusters . Therefore, we can build an associated image-user matrix by accumulating entries of tag-image-user tensor along the tag axis, and adopt the co-clustering algorithm instead of the K-means algorithm to find co-cluster centers. Subsequently, we select image-user units (a user uploads this image) that is most close to the -th co-cluster center as the anchor unit. In total, we can obtain image-user anchor units for co-cluster centers, where .
|Non-anchor image set.|
|Non-anchor user set.|
|Anchor image set.|
|Anchor user set.|
|Original tag-image-user tensor on , and .|
|Original tag-image-user non-anchor tensor on , and .|
|Original tag-image-user anchor tensor on , and .|
|Refined tag-image-user anchor tensor. (to be learned)|
|Inter-adjacency matrix between non-anchor image and anchor image.|
|Inter-adjacency matrix between non-anchor user and anchor user.|
|Intra-adjacency matrix between non-anchor image and non-anchor image.|
|Intra-adjacency matrix between non-anchor user and non-anchor user.|
|Intra-adjacency matrix between tag and tag.|
Based on the anchor units, we construct a social anchor-unit graph , where indicates the collection of the adjacency edges between the data points. To measure the weight of each edge in anchor-unit graph , we design two types of inter-adjacency matrices between non-anchor data and anchor data (i.e., image inter-adjacency matrix and user inter-adjacency matrix), and three types of intra-adjacency matrices among non-anchor images, non-anchor users and tags (i.e., image intra-adjacency matrix, user intra-adjacency matrix, and tag intra-adjacency matrix).
Inter-adjacency matrices. First, we design the image inter-adjacency matrix between non-anchor images and anchor images, i.e.,
where is a parameter of the RBF kernel, and indicate features (i.e., CNN feature ) of a non-anchor image and an anchor image , respectively. Here, if , we set to obtain a sparse matrix for reducing the memory cost.
Second, we also design user inter-adjacency matrix between the non-anchor users and anchor users. Similar to , we assume that two users with higher co-occurrence are more likely to be related with each other, and vice versa, namely
where denotes the number of groups that both a non-anchor user and an anchor user join, and is the number of groups that a non-anchor user joins, is the number of groups that an anchor user joins.
Intra-adjacency matrices. We first design image intra-adjacency matrix among the non-anchor images. One strategy is that we can measure the association between two non-anchor images by Eq. (1). However, the number of non-anchor images is much larger than the number of anchor images, which will increase the computational cost. Alternately, since anchor images are very representative in the whole image set, we can measure the association between non-anchor images by exploring links among common anchor images, which has also been claimed in [15, 14]. For two non-anchor images, the more number of common anchor images the two non-anchor images share, the stronger association such two non-anchor images have. Thus, can be computed as follows,
where the diagonal matrix is defined as (). From Eq (3), if , it means two non-anchor images share at least one anchor user, and otherwise .
Second, we use the same idea to design user intra-adjacency matrix among the non-anchor users, as follows,
where the diagonal matrix is defined as .
where and denote the weight coefficients (); denotes the occurrence count of tag in a dataset; denotes the co-occurrence count for tags and in a dataset; is the information content of tag ;
is the probability of tag; and is the least common sub-sumer of tags and in the WordNet taxonomy . The least common sub-sumer of two tags in WordNet is the sumer that does not have any children that are also the sub-sumer of two tags.
Iii-C SUGAR Tensor Completion
It has been proved that anchor graph can efficiently deal with the standard multi-class semi-supervised learning problem. Motivated by this, we extend the existing Anchor Graph Regularization  to a novel Social anchor-Unit GrAph Regularization (SUGAR). Specifically, it can be assumed that two images linked by the same anchor image should have closely similar tags and closely related users, two users linked by the same anchor user prefer to upload images with closely similar tags. Then, we can define the regularization as follows,
where and are nonnegative coefficients to control the penalty of the corresponding regularization. The matrix denotes the -mode product111More details about -mode product can be found in . of and , and describes the inter-association between the tags and the anchor users corresponding to the -th non-anchor image. The matrix describes the inter-association between the tags and the anchor images corresponding to the -th non-anchor user.
The great number of original tags provided by users on websites involve the important supervised information, which can guide us to mine the inter-association among images, tags, and users . Accordingly, we construct the tag-image-user non-anchor tensor among tags, non-anchor images and non-anchor users. Specifically, if the i-th non-anchor image uploaded by the k-th non-anchor user is annotated with the j-th tag, we set , otherwise , where , , and
. For a tensor with a few available entries, the tensor completion algorithm can estimate missing entries and remove the noisy ones by reconstructing an approximately low-rank tensor, as follows,
The Tucker decomposition  of tensor provides a factorization way to solve the low-rank tensor by the following objective function, i.e.,
where . Here, denotes the core tensor, , , and denote the factor matrices. Since , and describe the associations between tags and tags, non-anchor images and anchor images, as well as non-anchor users and anchor users, respectively, by setting , and , the learned core tensor can be regarded as the refined tensor . To well leverage the weakly supervised tag-image-user association, we introduce a new regularization term , which constrains the learned to be consistent with the original tag-image-user anchor tensor . Finally, we can obtain the refined tensor by minimizing the following tensor completion function , i.e.,
where and are nonnegative parameters to control the penalty of the regularization. In Eq. 9, the learned is a low-rank and compact tensor, which can reveal the association among anchor-images, anchor-users and tags.
In this work, we learn in an iteratively updating way. Specifically, we compute the partial derivatives of the objective function with respect to , i.e.,
where , , , and . Therefore, the multiplicative updating rule  of is
The details of algorithm are described in Algorithm 1. Here, is initialized by , where is a random small-disturbed tensor with mean 0. The convergence criterion is that the iterations stop when the relative cost of the objective function is smaller than a predefined threshold . The proposed SUGAR-TC convergences after about iterations in the experiments.
Iii-D Anchor-Aware Tag Assignment
After obtaining the completed tensor , we accumulate its entries along the user axis and image axis to acquire the desired tag-image association matrix and tag-user association matrix , respectively. Then, we can employ the completed tags associated with anchor units to predict tags for non-anchor images. For one non-anchor image () with an available user (), we use the weighted average of tags of nearest-neighbor anchor units to estimate its tag vector , as follows,
where is an index set of closest anchor-images of , is an index set of users corresponding to these closest anchor-images, is a parameter to control the degrees of tag-image and tag-user associations. Finally, we rank the elements of based on the values in the descending order, and select top tags as the final tags of .
Iv-a Experimental Settings
We conduct experiments on a real-world social image dataset NUS-WIDE-128  to evaluate the performance of the proposed SUGAR-TC method. It is extended from the widely-used NUS-WIDE dataset , and contains images, user-provides tags crawled from the Internet, as well as the manually labeled ground-truth of 128 predefined concepts for evaluation. In this dataset, some user IDs are invalid or unavailable, thus we delete the corresponding images. Finally, we obtain images with user IDs and the information of user groups.
. Besides, we also set a baseline method called Original Tagging (OT), where the labels used for evaluation are the user-provided tags crawled from Internet. To evaluate the performance, the widely-used F-score is adopted. For each concept, F-score is calculated as F-score. And then the average F-score over all the concepts is reported. For fair comparison, we use the CNN features  and tune the hyper-parameters by using the grid search strategy for all methods in the experiments. The best results are reported for comparison. All methods in the experiments are implemented on a server with an 8-core 2.67 GHz CPU and 32 GB memory.
Iv-B Results and Analysis
In this section, we conduct experiments to evaluate the effectiveness of the proposed SUGAR-TC. The compared results in terms of the average F-score are presented in Table II. We find that the proposed SUGAR-TC achieves the highest F-score compared with the other related methods. MRTF, TTC1 and TTC2 utilize the inter- and intra- associations among images, tags and users, thus they perform better than TRVSC, LR, NMC and LSLR. In particular, since TTC1 and TTC2 address the super-sparse problem existing in the original tensor, they gain higher F-scores than MRTF that directly completes the entries in the original tensor. However, to a certain degree, both TTC1 and TTC2 break the latent inter-association among images, tags and users, when dividing the original tensor into several sub-tensors. In turn, the proposed SUGAR-TC did not break the inherent tag-image-user inter-association. Thus, the F-score obtained by SUGAR-TC is , which is about higher than of TTC2.
We further detail the F-scores obtained by different methods on all the 128 predefined concepts, as shown in Figure 3. We can see that SUGAR-TC achieves the highest F-scores on most of these concepts. We can also see that all image retagging methods improve the quality of almost all concepts compared with OT. By adding user information, MRTF, TTC1, TTC2 and SUGAR-TC perform better than LR, NMC, LSLR and TRVSC, especially for some summarized or complex tags (e.g., “military”, “nighttime”, and “cityscape”). In particular, for the frequently-used tags (e.g., “bus”, “flowers”, and “sunset”, etc) in Figure 3, SUGAR-TC shows remarkable improvement than TTC1 and TTC2, one reason may be that TTC1 and TTC2 break the tag-image-user inter-association for such common tags when decomposing the original tensor into several sub-tensors.
We also show some tag assignment results for some specific cases, as shown in Figure 4. We can see that SUGAR-TC can effectively assign tags of images, even though there are some globally complex images (e.g., Figure 4(c)) and locally abstract images (e.g., Figure 4(d)), since SUGAR-TC can simultaneously mine the context information between users and images, and the semantic relation between images and tags. Besides, although some tags, i.e., geo-related tags, event tags, and time-related tags, are hardly inferred by only using visual information, SUGAR-TC considering extra user information can correctly infer them. For example, as shown in Figure 4(a), the train is a landmark of Japan, thus this image should be tagged with the geo-related tags “Japan”. It is very difficult to reveal the relation between the tag “Japan” and this image by only mining the visual and tag information, but SUGAR-TC can do it well based on the user background information. Similarly in Figure 4(b), this image shows the Obama’s speaking of the United States presidential election, it is reasonable to assign the event tag “election” to this image.
Iv-C Computational Cost Analysis
To illustrate the efficiency of the proposed SUGAR-TC, we also compare the computational cost of SUGAR-TC and the closely related methods, including MRTF , TTC1 , and TTC2 . Table III lists the computing time and memory cost of these methods in details. We can see that: 1) SUGAR-TC with total computing time 4.2h runs much faster than MRTF, TTC1 and TT2; 2) the memory cost of SUGAR-TC is more than TTC1 and TTC2, but less than MRTF. By jointly considering the accuracy, computing time and memory cost, SUGAR-TC is more applicable for image retagging.
|Method||MRTF ||TTC1 ||TTC2 ||SUGAR-TC|
Iv-D Parameter Sensitivity Analysis
In the proposed SUGAR-TC method, there are several hyper-parameters to be set in advance. Following , the weight coefficients and in Eq. (5) are set to and respectively. The radius parameter in Eq. (1) is set to . The number of anchor units is set as for each co-cluster center. In the following, we will discuss the sensitiveness of other parameters.
For the number of image centers (denoted by ) and the number of user clusters (denoted by ), we tune them by 10, 20, 40, 60, 80, 100, 200, 500 and 3, 6, 12, 18, 24, 30, 50, 200, respectively. The corresponding F-scores are shown in Figure 5(a) and Figure 5(b), respectively. From the results, we can see that the best result is obtained with and . When these two parameters are set within and respectively, the performance changes slightly. In the experiments, the default number of image clusters and user clusters are set to 40 and 12, respectively.
For , , , , and , we tune them by 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.05, 0.1, 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.05, 0.1, 0, 0.01, 0.05, 0.1, 0.25, 0.75, 1, 10, 0, 0.01, 0.05, 0.1, 0.25, 0.75, 1, 10, and 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1. The corresponding results are shown in Figure 5(c-e). SUGAR-TC achieves the best result with , , , , and , which are set as the default values of these parameters in the experiments.
Iv-E Application: Image Retrieval
In this work, for each image, image retagging task can not only add and remove tags, but also re-rank the completed tag list by assigning tags with different confidence scores. For each image, we can obtain a tag ranking list, which can improve the performance of tag-based image retrieval. Therefore, we conduct experiment of tag-based image retrieval to illustrate the effectiveness of SUGAR by comparing with other related methods, including TRVSC , LR , NMC , LSLR , MRTF , TTC1  and TTC2 .
Following the experimental setting in , we perform tag-based social image retrieval with ten queries on NUS-WIDE-128, i.e., birds, building, butterfly, dog, fish, flowers, horses, plane, sunset and zoo
. Average Precision (AP) is used as the evaluation metric. Specifically, given a ranked list with length, AP is defined as . Here, is the number of relevant images in this ranked list, is the precision at cut-off top , is an indicator function: if the -th image is relevant with respect to the ground-truth concept, otherwise . Finally, we use MAP over all queries to evaluate the overall performance. Figure 6 shows MAP values with different ranking length T. We can see that the proposed SUGAR-TC outperforms the other four image retagging methods. This demonstrates SUGAR-TC is more effective than other related image reatgging methods in terms of tag-based social image retrieval.
In this work, we propose a novel Social anchor-Unit GrAph Regularized Tensor Completion (SUGAR-TC) method to efficiently refine the tags of social images, regardless of the data scale. First, we utilize the co-clustering algorithm to obtain the representative anchor units (anchor image and anchor user) across image and user domains rather than traditional anchors in a single domain, and then construct an anchor-unit graph with multiple intra/inter adjacency edges. Second, we present a SUGAR tensor completion to refine tags of anchor images. Finally, we efficiently assign high-quality tags to all non-anchor images by leveraging the potential relationships between non-anchor units and anchor units. Experimental results on a real-world social image database demonstrate the effectiveness and efficiency of SUGAR-TC compared with the state-of-the-art methods.
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