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A review on multitask metric learning
Big Data Analyticsvolume 3, Article number: 3 (2018)
Abstract
Distance metric plays an important role in machine learning which is crucial to the performance of a range of algorithms. Metric learning, which refers to learning a proper distance metric for a particular task, has attracted much attention in machine learning. In particular, multitask learning deals with the scenario where there are multiple related metric learning tasks. By jointly training these tasks, useful information is shared among the tasks, which significantly improves their performances. This paper reviews the literature on multitask metric learning. Various methods are investigated systematically and categorized into four families. The central ideas of these methods are introduced in detail, followed by some representative applications. Finally, we conclude the review and propose a number of future work directions.
Background
In the area of machine learning, pattern recognition, and data mining, the concept of distance metric usually plays an important role. For many algorithms, a proper distance metric is critical to their performances. For example, the nearest neighbor classification relies on the metric to identify the nearest neighbor and determine their class, whilst kmeans clustering uses the metric to determine which cluster a sample should belong to.
The metric is usually used as a measure of the similarity or dissimilarity, and there are various types of predefined distance metrics, such as Euclidean distance, cosine similarity, Hamming distance, etc. However, in practical applications, these generalpurpose metrics are insufficient to catch the sundry particular properties of various tasks. Therefore, researchers propose learning a metric from data for particular tasks, to improve algorithm performance. This is termed metric learning [1–7].
With the advent of data science, challenging and evolving problems have arisen. Obtaining training data is a costly process, hence complex models are being trained on small datasets, resulting in poor generalization. Alongside this the number of tasks to be learnt has increased significantly. To overcome these problems, multitask learning is proposed [8–13]. It aims to consider multiple tasks simultaneously at a higher level, whilst transferring useful information among different tasks to improve their performances.
After multitask learning was proposed by Caruana [8] in 1997, various strategies have been designed based on different assumptions. There are also some closely related topics, such as transfer learning [14, 15], domain adaptation [16], metalearning [17], lifelong learning [18], learning to learn [19], etc. In spite of some minor discrepancies among them, they share the same basic idea that the performance is improved by considering multiple learning tasks jointly and sharing information with other tasks.
Under such a background, it is natural to consider the problem of multitask metric learning. However, most multitask learning algorithms designed for traditional models are difficult for applying to metric learning algorithms due to the obvious differences between the two kinds of models. To resolve this problem, a series of multitask metric learning approaches are specifically designed for the metric learning models. By properly coupling multiple metric learning tasks, their performances are effectively improved.
Metric learning has the particularity that its effect on performance can be only given indirectly by the algorithm relying on the metric. This requires a different way to construct the multitask learning framework from traditional models. As far as we know, there is no review at present on the multitask metric learning, hence this paper will give a general overview of the existing works.
The rest of the paper is organized as follows. First we provide an overview of the basic concepts of metric learning and briefly introduce multitask metric learning. Next, various strategies of multitask metric learning approaches are reviewed. We then introduce some representative applications of multitask metric learning, and conclude with a discussion on potential future issues.
Overview
In this section, we first provide an overview of metric learning, including its concept and several representative algorithms. Then a general description about multitask metric learning is presented, leaving the details of the algorithms for the next section.
A brief review on metric learning
The notion of distance metric was originally a concept in mathematics, which refers to a function defined on \(\mathcal {X}\) as \(d:\mathcal {X}\times \mathcal {X}\rightarrow \mathbf {R}_{+}=[\!0,+\infty)\) satisfying positiveness, symmetry, and triangle inequality [20]. In the community of machine learning, metric is unnecessary to keep its original definition from mathematics, but usually refers to a general measure of dissimilarity or similarity. A lot of machine learning algorithms use it to measure the dissimilarity between samples without explicitly referring its definition, such as nearest neighbor classification, kmeans clustering, etc.
There have been various types of predefined metrics for general purposes. For example, assuming two points in the ddimensional space \(\mathbf {x}_{i},\mathbf {x}_{j}\in \mathcal {X}=\mathbb {R}^{d}\), the most frequently used Euclidean distance is defined as d(x_{ i },x_{ j })=∥x_{ i }−x_{ j }∥_{2}. Another example is the Mahalanobis metric [21] that is defined as \(d_{\mathbf {M}}(\mathbf {x}_{i},\mathbf {x}_{j})=\sqrt {(\mathbf {x}_{i}\mathbf {x}_{j})^{\top }\mathbf {M}(\mathbf {x}_{i}\mathbf {x}_{j})}\), where the symmetric positive semidefinite matrix M is the Mahalanobis matrix which determines the metric.
In spite of their widelyspread usage, the predefined metrics are incapable to capture the variety of real applications. Considering its importance to the performances of algorithms, researchers propose to learn a metric from the data instead of using the predefined metrics directly. By adapting the metric to the specific data for some algorithm, the performance is expected to be effectively improved. This is the central idea of the metric learning.
However, it is hardly practicable to learn a general metric by directly finding an optima in the functional space. A practical way is to define a family of metrics determined by some parameters, and transform the problem into the solving of the optimal parameters. Mahalanobis metric provides a perfect candidate for such a family of metrics, which has a simple formulation and is uniquely determined by the Mahalanobis matrix. In this case, the metric learning is equivalent to learning the Mahalanobis matrix.
Eric Xing et al. [1] proposes the idea of metric learning with the first algorithm in 2002. Since then, various metric learning methods have been proposed based on different strategies. Since metrics can be categorized into several families according to their properties, such as global vs. local, or linear vs. nonlinear, the metric learning approaches can also be categorized accordingly. Mahalanobis metric is a typical global linear metric. Because existing multitask learning approaches are almost based on global metrics, we focus on this type in this review, especially for the global linear metrics. Please refer to [22, 23] and their references for other types.
Most metric learning algorithms are formulated as a constrained optimization problem and the metric is obtained by solving this optimization. Since the distance is defined on two points, the supervised information to determine the metric, which is also called sideinformation in metric learning, is usually given by constraints on pairs or triplets as follows [22].

Mustlink / cannotlink constraints (positive/negative pairs)
$$ \begin{aligned} \mathcal{S}=&\{(\mathbf{x}_{i},\mathbf{x}_{j}): \mathbf{x}_{i}\ \text{and}\ \mathbf{x}_{j}\ \text{should be similar}\},\\ \mathcal{D}=&\{(\mathbf{x}_{i},\mathbf{x}_{j}): \mathbf{x}_{i}\ \text{and}\ \mathbf{x}_{j}\ \text{should be dissimilar}\}. \end{aligned} $$ 
Relative constraints (training triplets)
$$ \mathcal{R}=\{(\mathbf{x}_{i},\mathbf{x}_{j},\mathbf{x}_{k}): \mathbf{x}_{i}\ \text{should be more similar to}\ \mathbf{x}_{j}\ \text{than to}\ \mathbf{x}_{k}\}. $$
Using these constraints, we briefly introduce the strategies of some metric learning approaches. Xing’s method [1] aims to maximize the sum of distances between dissimilar pairs while keeping the sum of squared distances between similar pairs to be small. It is an example of learning with positive/negative pairs. Large Margin Nearest Neighbors (LMNN) [2, 24] requires the k nearest neighbors to belong to the same class and pushes out all the imposters (instances of other classes existing in the neighborhood). The sideinformation is provided by the relative constraints. Informationtheoretic metric learning (ITML) [3], which is also built with positive/negative pairs, models the problem with logdeterminant. Sparse Metric Learning [6] uses the mixed L_{2,1} norm to obtain a joint feature selection during metric learning, and Huang et al. [4, 5] proposes a unified framework for Generalized Sparse Metric Learning (GSML). Robust Metric Learning (RML) [25] deals with the noisy training constraints based on robust optimization.
It is notable that learning a Mahalanobis matrix can also be regarded as learning a linear transformation. For any symmetric positive semidefinite Mahalanobis matrix M, there is a symmetric decomposition M=L^{⊤}L and the distance can be then reformulated as
By (1), the Mahalanobis metric defined by M is equivalent to the Euclidean distance after performing the linear transformation L, and thus metric learning can be also performed by learning such a linear transformation. Neighbourhood Component Analysis (NCA) [26] is an example of this class that optimizes the expected leaveoneout error of a stochastic nearest neighbor classifier by learning a linear transformation. Furthermore, the linear metric can be easily extended to the nonlinear metric by replacing the linear transformation L with a nonlinear transformation f, which is defined as
Then, the metric is obtained by learning an appropriate nonlinear transformation f. Since the deep learning has achieved remarkable successes in computer vision and machine learning [27], some researchers proposed the deep metric learning recently [28, 29]. These methods resort to the deep neural network to learn a nonlinear transformation, which are different from a traditional neural network in that their learning objective are given by constraints on distances.
There are a lot of metric learning methods because the metric plays an important role in many applications. We cannot introduce them in detail due to the limit of the space. Readers can refer to the paper [22] for a systematic review on metric learning.
An overview of multitask metric learning
Since the concept of multitask learning was proposed by Caruana [8] in 1997, this topic has attracted much attention from researchers in machine learning. Multiple different methods are proposed to construct a framework for simultaneously learning multiple tasks for conventional models, such as linear classifier or support vector machines. The performances of the original models are effectively improved by learning simultaneously.
However, these methods cannot be used directly for metric learning since there exist significant discrepancies between the conventional learning models and metric learning models. Taking the popular support vector machine (SVM) [30] as an example of conventional models, we can show the differences between it and metric learning. First, the training data of the two models are of different structures. For SVM, the training samples are given by points with a label for each one, while for metric learning they are given by pairs or triplets with a label for each one. Second, their models are of different types. The model of SVM is a singleinput singleoutput function parameterized by a weight vector and a bias, while the model of metric learning is a doubleinput singleoutput function parameterized by a symmetric positive semidefinite matrix. Third, the algorithms take effect on the performance in different ways. For SVM, the classification accuracy is given by the algorithm directly, while for metric learning, the performance has to be evaluated indirectly by other algorithms working with the learned metric.
Due to the reasons mentioned above, strategies have to be specially designed to construct a multitask metric learning model. They have to deal with two problem: (1) what type of useful information is shared among different metric learning tasks; (2) how such information is shared by the proposed model and algorithm. Parameswaran et al. [31] proposes the first multitask metric learning approach in 2010, and in the following years a variety of strategies have been proposed for multitask metric learning. We generally categorize them into the following families according to the way how the information is shared:

1
Assume that the Mahalanobis matrix of each metric is composed of several components and share some composition.

2
Predefine the relationship among tasks or learn such relationship from data, and constrain the learning process with this relationship.

3
Use a common metric with proper regularization to couple the metrics.

4
Consider metric learning from the perspective of learning a transformation and share some parts of transformation.
There are some representative works in each family and we will introduce them in detail in next section. Figure 1 gives a summary of the multitask metric learning approaches mentioned in this paper.
Review on multitask metric learning approaches
In this section, we investigate the multitask metric learning approaches published todate and provide a detailed review on them. The methods are organized according to the type of strategies. We focus on only the models and algorithms in this section without mentioning their application backgrounds, which are left for the next section. The discussion about the relation between some closely related methods is also included.
Before diving into the details, we summarize the main features of these multitask metric learning methods in Table 1. Besides, in this section, we always use M to represent the Mahalanobis matrix to keep the notations uniform, which may be different from the original papers.
Sharing composition of Mahalanobis matrices
Since the Mahalanobis metric is uniquely determined by the Mahalanobis matrix, a natural way to couple multiple related metrics is to share some composition of their Mahalanobis matrices. Specifically, the Mahalanobis matrix of each task is assumed to be composed of both common composition shared by all tasks and taskspecific composition preserving its specific properties. This strategy is the most popular way to construct a multitask metric learning model and we introduce some representative ones below.
Large margin multitask metric learning (mtLMNN) Parameswaran et al. [31] proposes a multitask learning model based on the idea to share a common composition of the Mahalanobis matrices. It is motivated by the regularized multitask learning (RMTL) [9], and obtained by adapting RMTL to the largemargin nearest neighbor metric learning (LMNN) [2, 24]. To couple multiple tasks, each Mahalanobis matrix is decomposed into a common part M_{0} and a taskspecific part M_{ t }. Thus the distance between two points \(\mathbf {x}_{i},\mathbf {x}_{j}\in \mathcal {X}\) defined by the metric of the tth task is defined as
By restricting that M_{0}≽0 and M_{ t }≽0,∀t, the Mahalanobis matrix for each task is ensured to be positive semidefinite, which induces a positive semidefinite metric. In this model, M_{0} picks up the general trends across all tasks while M_{ t } gathers the individual information for each task. The obtained regularization of mtLMNN is
In (3), the sideinformation is incorporated by constraints generated from triplets as LMNN [2]. The regularization on taskspecific matrices M_{ t }’s represses the specialty of each task and encourages the shared part of all tasks, while the regularization on M_{0} restricts the common part to be close to the identity. They further make the learnt metric of each task not far from the Euclidean metric.
The hyperparameters γ_{t>0}’s control the balance between the commonness and speciality, while γ_{0} controls the regularization of the common part. As the increasing of γ_{t>0}, the taskspecific parts become small and the learnt metrics of all tasks tend to be similar. When γ_{t>0}→∞, the algorithm learns a unique metric M_{0} for all tasks, while when γ_{t>0}→0, all tasks tend to be learnt individually. On the other hand, when γ_{0}→∞, the common part M_{0} becomes identity and Euclidean metric is obtained. When γ_{0}→0, there tends to be no regularization on the common part. This model is convex and can be solved effectively.
This is the first attempt to apply a multitask approach to metric learning problem. It provides a simple yet effective way to improve the performance of metric learning by jointly learning multiple tasks. However, the idea of splitting each Mahalanobis matrix into a common part and an individual part is not easy to explain from the perspective of distance metric and can only deal with some simple cases.
Multitask multifeature similarity learning learning (M^{2}SL) Wang et al. [32] proposes a multitask multifeature metric learning approach to adapt the metric learning to large scale visual applications. For each sample, M types of features are extracted and the metrics are learnt individually for each feature. For each feature channel, there are T tasks and each task learns a distance metric. To make the information shared among tasks, the Mahalanobis matrix of the tth task in the mth feature channel is defined to be a combination of a common part \(\mathbf {M}_{0}^{(m)}\) and an individual part \(\mathbf {M}_{t}^{(m)}\). Then the authors incorporate such a formulation into the idealized kernel learning [33] and obtain the multifeature multitask metric learning model as
where the distance is defined as
The variable \(\delta _{t}^{ij}\) denotes the label of similar/dissimilar labeled pairs, and \(\sigma _{t}^{ij}\) is a predefined threshold for hinge loss. The parameters b_{0} and b_{ t } represent weights for the sharing part and discriminating parts respectively, and the last term is the regularization on these weights.
Using this approach, the information contained in different tasks is shared among them and the multiple features are used in a more effective way. It uses the same strategy as mtLMNN to construct the multitask metric learning model and thus has the similar advantages/disadvantages.
Multitask sparse compositional metric learning (mtSCML) Shi et al. [34] proposes a multitask metric learning framework from the perspective of sparse combination. The authors first propose a sparse compositional metric learning (SCML) approach which regards a Mahalanobis matrix as a nonnegative weighted sum of K rank1 positive semidefinite matrices:
where the b_{ i }’s are Ddimensional column vectors. Noting that the distance between any two points (x,y) determined by M is calculated by
the vectors b_{ i }’s span the common lowdimensional subspace in which the metric is defined.
Using such a formulation, each rank1 matrix is a basis and the metric can be reformulated as a sparse combination of these bases. Then the metric learning is a process of learning such weights, which is shown as
where L defines the loss from sideinformation as
with [ ·]_{+}= max(0,·), and the ℓ_{1} regularization encourages a sparse solution of w.
When there are T tasks to be learned together, the multitask learning can be easily obtained by applying a structure regularization on these weights. To be specific, the authors assume that the different tasks share a common lowdimensional subspace for the reconstruction weights, and use a mixed norm to obtain the structure sparsity. The formulation of mtSCML is shown as
where W is a T×K nonnegative matrix where each row w_{ t } defines the reconstruct weight vector for the tth task, and ∥W∥_{2,1} is the ℓ_{2}/ℓ_{1} mixed norm. It equals to the ℓ_{1} norm applied to the ℓ_{2} norm of the columns of W, which induces the group sparsity at the column level, i.e., it encourages some columns to be zero together and thus make the different tasks share the same reconstruction bases.
This method naturally introduces the idea of group sparse to construct multitask metric learning, and the proposed approach is not difficult to be realized. However, this algorithm requires the set of rankone metrics to be pretrained and thus cannot be optimized simultaneously with the weights.
Twolevel multitask metric learning (TMTL) Liu, et al. [35] proposes a twolevel multitask metric learning approach that combines multiple metrics directly without an explicit optimization procedure. It is developed based on KISSME [36], which is a metric learning approach motivated by a statistical inference and defines the Mahalanobis matrix as
This model is extended to twolevel multitask learning paradigm in a rather simple way. The authors first learn a Mahalanobis matrix for each task respectively and a common metric for all samples. Then the final individual Mahalanobis matrix is given by a direct weighted composition
This method is so simple that no optimization procedure is needed. To be strict, it is not a typical metric learning and can deal with only some special problems.
Online semisupervised multitask distance metric learning (onlineSMDM) Li et al. [37] proposes a semisupervised metric learning approach that is capable to utilize the unlabeled data to learn the metric. The method is designed based on the regularized distance metric learning [38] and extended to a multitask metric learning model called online semisupervised multitask distance metric learning. It assumes each Mahalanobis matrix to be composed of a common part M_{0} and a taskspecific part M_{ t } as [31] does, and proposes an online algorithm to solve the model effectively.
To utilize the unlabeled data during training process, the authors assign labels for the unlabeled pairs:
where N(x_{ i }) indicates the nearest neighbor set of x_{ i } calculated by Euclidean distance. The Eq. (5) indeed assumes that if a point is one of the nearest neighbors of the other point, they should have the same label. Then the model of unlabeled model can be formulated as
where D_{ tl } and D_{ tu } represent the sets of labeled data pairs and unlabeled data pairs respectively, N_{ tl } and N_{ tu } are the numbers of the labeled and unlabeled training data, λ and γ are both hyperparameters to control the regularization on the individual parts and the common part, and M represents all the M_{ t }’s and M_{0} for brevity.
This method utilizes the unlabeled data by assigning labels for them according to the original distances. The strategy of constructing the multitask learning is same as the previous ones.
Hierarchical multitask metric learning (HMML) Zheng et al. [39] proposes an approach to learn a hierarchical tree of multiple sparse metrics hierarchically over a visual tree. In this work, a visual tree is first constructed to organize the categories in a coarsetofine fashion. Then a topdown approach is used to learn multiple metrics along the visual tree, where the model is expected to benefit from leveraging both the internode visual correlation and the interlevel visual correlations.
Construction of the visual tree is composed of two key steps: (a) Active Sampling for Category Representation, which utilizes active sampling to find multiple representative samples for each image category. (b) Hierarchical Affinity Propagation Clustering for Node Partitioning, which is a topdown approach to hierarchical affinity propagation (AP) clustering. It starts from the root node containing all the image categories and ends at the leaf nodes containing only one single image category. Figure 2 gives an example of the enhanced visual tree for CIFAR100. In this tree, the categories are organized in a hierarchical structure according to their similarities.
According to the construction procedure of the visual tree, categories on the same branch are more similar to each other than the ones on other branches. Thus, it is reasonable to perform multitask metric learning over the sibling child nodes under the same parent node to utilize the internode visual correlation among them. The authors exploit the same strategy as mtLMNN [31] which decomposes the metric into a common part and an individual part as
where M_{0} defines the common metric shared among all sibling child nodes and M_{ t } defines the nodespecific metric.
For root node, the joint objective function is then defined as
where the parameters γ_{0} and γ_{ t }’s control the regularization on the common part and individual part respectively.
For nonroot nodes at the midlevel of the visual tree, besides the internode correlations, the interlevel visual correlations between the parent node and its sibling child nodes at the next level should be also exploited. Since all nodes on the same branch are similar, any node p characterizes the common visual properties of its sibling child nodes. On the other hand, the taskspecific metric M_{ p } for node p contains the taskspecific composition. Thus, it is reasonable to utilize the taskspecific metric of node p to help the learning of its sibling child nodes. Based on this idea, the regularization β∥M_{0}−M_{ p }∥^{2} is added into the objective of (6) for nonroot nodes, where M_{0} is the common metric shared among the sibling child nodes under parent node p and M_{ p } is the taskspecific metric for node p at the upper level.
This method introduces the hierarchical visual tree into multitask metric learning, which is used to guide the multitask learning and thus provides a more powerful capability of describing the relationship among tasks.
Task relationship learning and regularization
Transfer metric learning by learning task relationship (TML) Zhang et al. [40, 41] proposes a multitask metric learning by learning task relationship. This model is also a direct adaptation of a traditional multitask learning approach to the metric learning task. The authors proposes a multitask relationship learning (MTRL) [13] in their previous work which assumes all the parameter vectors to follow a matrixvariant normal distribution [42] and automatically learns the relationships between tasks by a regularization. Since the parameter to be learned in metric learning is a matrix rather than a vector, the authors concatenate all columns of the Mahalanobis matrix to form a vector for each task \(\tilde {\mathbf {M}}_{t}=\text {vec}(\mathbf {M}_{t})\) and then apply the regularization of MTRL to it: \(\tilde {\mathbf {M}}\mathbf {\Omega }^{1}\tilde {\mathbf {M}}^{\top }\) where \(\tilde {\mathbf {M}}=\left [\text {vec}(\mathbf {M}_{1}),\ldots,\text {vec}(\mathbf {M}_{T})\right ]\). It is equivalent to apply the following matrixvariant normal prior distribution to \(\tilde {\mathbf {M}}_{t}\)’s.
In this definition, the row covariance matrix I_{ d }^{2} models the relationships between features and the column covariance matrix Ω models the relationships between different vectorized Mahalanobis matrices \(\tilde {\mathbf {M}}\)’s. Thus, Ω indeed determines the relationships between tasks. Since it cannot be given a priori in most cases, the authors propose to estimate it from data automatically.
The obtained model is shown in (7) and can be solved by alternating optimization.
In that paper, the authors further propose a transfer metric learning based on this model by training the concatenated Mahalanobis matrix of only target task while leaving other matrices fixed as source tasks. The idea of learning the relationship between tasks is interesting, but the covariance between the vectorized Mahalanobis matrices does not explain well from the perspective of distance metric.
Multitask maximally collapsing metric learning (MtMCML) Ma et al. [43] proposes a multitask metric learning approach using the graphbased regularization. To be specific, a graph is constructed to describe the relations between the tasks, where each node corresponds to a Mahalanobis matrix of one task, and an edge connecting two nodes represents the similarity between the two tasks. Thus an adjacency matrix W(0≤W(i,j)≤ 1) is obtained where a higher W(i,j) indicates that metrics i and j are more related. The regularization is
where \(\tilde {\mathbf {M}}_{i}=\text {vec}(\mathbf {M}_{i})\) converts the Mahalanobis matrix of the itask into a vector in a columnwise manner, DIA is a diagonal matrix where \(\mathbf {DIA}(i,i)=\sum _{j=1}^{T}{\mathbf {W}(i,j)}\), and thus the matrix L=DIA−W indeed defines the graph Laplacian matrix. The model can be optimized by alternating method.
In this work, the authors empirically set the adjacency matrix as W(i,j)=1, which indeed defines every pair of tasks are related. It is not difficult to prove that such a regularization is just a variant of the regularization of mtLMNN. Therefore, these two methods are closely related in this special case.
This work naturally introduces the graph regularization into multitask learning by applying a Laplacian to the vectorized Mahalanobis matrices. However, the relationship between two metrics is still vague, and the Laplacian matrix L is not easy to be reasonably determined.
Regularization with a common metric
A framework for approaches based on common metric Yang et al. [44] proposes a general framework for multitask metric learning to solve the scenario that all metrics are similar to a common one. The optimization problem is shown in (9) where M_{ t } is the Mahalanobis matrix of the tth task where M_{0} is the common one.
In this framework, the loss L and constraints \(\mathcal {C}_{t}\) are used to incorporate the sideinformation from training samples into the learning process, while the regularization D(M_{ t },M_{ c }) encourages the metric of each task to be similar to a common one M_{ c }, and D(M_{0},M_{ c }) further regularizes the common metric to be close to a predefined metric. Without more prior information available, M_{0} is set to the identity I to define a Euclidean metric.
The mtLMNN can be easily included as a special case of this framework by \(D(\mathbf {X},\mathbf {Y})=\\mathbf {X}\mathbf {Y}\_{\mathrm {F}}^{2}\). The only difference exists on the constraints: the Mahalanobis matrix of the tth task in mtLMNN is M_{0}+M_{ t }, where both the two parts are positive semidefinite; the Mahalanobis matrix of the tth task in (9) with Frobenius norm is M_{ t } and the positive semidefiniteness of only this matrix is required. The authors indicate that the latter actually provides a more reasonable model because the constraints in mtLMNN is unnecessary to be so strict.
Geometry preserving multitask metric learning (GPmtML) Yang et al. [44] proposes the geometry preserving multitask metric learning approach based on the general framework (9). Different from most previous approaches, the GPmtML considers the multitask metric learning problem from the perspective of propagating the relative distance. The authors indicate that learning of a metric is a process of embedding the supervised information from training samples into the learnt metric, and thus it is reasonable to couple the multiple tasks by sharing the embedded supervised information among metrics. As we have illustrated, it is an important class of metric learning approaches which learn the metric from relative distances given by triplets, and thus it is reasonable to propagate such relationships about the relative distance between metrics. Motivated by this, the authors propose the concept of geometry preserving probabilistic [44, 45] to measure such kind of characteristic between two metrics defined by Mahalanobis matrices A and B.
where (x_{1},y_{1},x_{2},y_{2})∼f and ∧ denotes the logical “and” operator.
Then the geometry preserving multitask metric learning is proposed which aims to increase the geometry preserving probabilistic. The method is obtained by using the von Neumann divergence [46, 47] (10) as regularization in (9).
By a series of theoretical analysis, this method is proved to be able to encourage a higher geometry preserving geometry, and thus more likely to keep the relative distances of samples across different metrics. The introduced regularization is jointly convex and thus the problem can be effectively solved by alternating optimization.
This is the first paper that attempts to construct the multitask metric learning by sharing the supervised sideinformation among tasks. It provides a reasonable explanation from the perspective of metric learning. However, the macrostructure of the model is too simple and thus cannot describe more complex relationship among tasks.
Sharing transformation
According to (1). Learning a Mahalanobis distance is equivalent to learning a corresponding linear transformation. There are indeed some metric learning algorithms that aim to learn such a transformation directly, and it naturally provides a way to construct a multitask metric learning by sharing some parts of transformation.
Multitask metric learning based on common subspace (mtMLCS) Yang et.al [48] proposes a multitask learning method based on the assumption of common subspace. The idea is motivated by multitask feature learning [11] which learns a common sparse representations across multiple tasks. Based on the same assumption that all the tasks share a common lowdimensional subspace, the authors propose a multitask framework for metric learning by transformation.
To couple multiple tasks with a common lowdimensional subspace, the authors notice that for any lowrank Mahalanobis matrix M, the corresponding linear transformation matrix L is of full row rank and has the size of r×d, where r=rank(M) is the dimension of the subspace. Applying a compact SVD to L, there is L=UΛV^{⊤} where V is a d×r matrix defining a projection to the lowdimensional subspace, and UΛ defines a transformation in the subspace. This fact motivates a straightforward multitask strategy with common subspace: to share a common projection matrix V and learn an individual transformation \(\mathbf {R}_{t}\doteq \mathbf {U}_{t}\mathbf {\Lambda }_{t}\) for each task.
However, it is computationally complex to apply an orthogonal constraint to V. On the other hand, it’s notable that the orthogonality is not necessary for V to define a subspace. As well as V is of the size r×d and r<d, it indeed defines a subspace of dimensionality no more than r with some extra fullrank transformation in the subspace. Therefore, a common matrix L of size r×d is used to realize the common projection instead of V^{⊤}, and the extra transformation can be absorbed by R_{ t }. The obtained model for multitask metric learning is to a transformation for each task L_{ t }=R_{ t }L_{0} where L_{0} defines the common subspace and R_{ t } defines the taskspecific metric. This strategy is then incorporated into the LMCA [49] which is a variant of LMNN [2] by learning the transformation.
This approach is simple to implement. Compared with the approaches that learn metrics by learning Mahalanobis matrices, mtMLCS does not require the symmetric positivedefinite constraints, and thus is much easier to optimize. However, this model is not convex and thus the global optimum cannot be obtained.
Coupled projection multitask metric learning (CPmtML) Bhattarai et al. [50] proposes a multitask metric learning approach which also focuses on the methods that learns a linear transformation. In this paper, the authors refer the transformation in (1) as “projection”, and the idea to couple different tasks is to decompose it into a common projection and a taskspecific projection. Different from mtMLCS in which the common projection and taskspecific projection are concatenated, CPmtML decomposes the projection in the manner of distance:
It is easy to show that the relation among different tasks is the same as mtLMNN where both of them obtain the distance by summing the squared distances of common and taskspecific parts:
The authors pointed out that there are important differences between the two approaches. First, the sideinformation of mtLMNN is based on triplets while CPmtML is based on similar/dissimilar pairs. Second, using the formulation of projection, it is easy to obtain a lowrank metric. Third, the authors propose a scalable SGD based learning algorithm. Finally, it can work in online setting.
Since this method learns the metric by optimizing on the transformation L, it has the similar merits and faults as mtMLCS. It is also designed for the simple case where the tasks are correlated by a common Mahalanobis matrix.
Deep multitask metric learning (DMML) Soleimani et al. [51] proposes a multitask learning version of deep metric learning. The method is constructed based on the discriminative deep metric learning (DDML) [29]. For any pair of points, the DDML transforms the two points with a neural network, and then the distance is defined to be the Euclidean distance of their transformations. Thus the process of metric learning is done by learning the parameters of the network.
The DMML uses a straightforward way to construct a multitask version of DDML by sharing the same first layer. Assuming there are T tasks, the outputs for two points x_{i,t},x_{j,t} in the tth task are \(\mathbf {h}_{1,t}^{(1)}=s\left (\mathbf {W}^{(1)}\mathbf {x}_{i,t}+\mathbf {b}^{(1)}\right)\) and \(\mathbf {h}_{2,t}^{(1)}=s\left (\mathbf {W}^{(1)}\mathbf {x}_{j,t}+\mathbf {b}^{(1)}\right)\), where all tasks share a common weights matrix W^{(1)} and a common bias vector b^{(1)}, and s is a nonlinear operator such as tanh. Then the outputs the second layer is calculated separately for each task as \(h_{1,t}^{(2)}=s\left (\mathbf {W}_{t}^{(2)}\mathbf {h}_{1,t}^{(1)}+\mathbf {b}_{t}^{(2)}\right)\) and \(h_{2,t}^{(2)}=s\left (\mathbf {W}_{t}^{(2)}\mathbf {h}_{2,t}^{(1)}+\mathbf {b}_{t}^{(2)}\right)\), where each task use the taskspecific weights matrix \(\mathbf {W}_{t}^{(2)}\) and bias vector \(\mathbf {b}_{t}^{(2)}\), and s is the nonlinear operator again. The obtained distance now can be calculated by
Then the model is learned by the following optimization problem:
where \(g(z)=\frac {1}{\beta }\log (1+\exp (\beta z))\) is the smoothed approximation for [ z]_{+}= max(z,0), and β controls its sharpness.
This method is based on a simple yet effective idea which a part of the network weights are shared across multiple tasks. It is not difficult to implement by slightly modify the original network architecture. However, only the first layer is shared across different tasks in this model, which may be not the optimal choice and it is not easy to determine how many layers should be shared.
Deep convernets metric learning with multitask learning (mtDCML) McLaughlin et al. [52] proposes to introduce auxiliary tasks in the model to help the metric learning task. The central idea is also to learn the distance metric by learning a feature representation. Denoting the subnetwork to transform the sample to the feature representation as G, and the network parameters as w, the learned distance can be calculated by the Euclidean distance of their representations as
The network is trained using sample pairs in the training dataset. The cost for similar/dissimilar pairs are shown below:
Then the cost function is written as
To improve the metric learning task, the authors include other related auxiliary tasks into the objective and obtain the multitask version:
where \(\mathcal {T}_{t}\) is an auxiliary task which helps to learn a better representation.
The selection of the auxiliary task depends on the problem of interest and there are a variety of choices. For the example in [52] where the main task is a metric learning for face verification, all the auxiliary tasks involve assigning one of several mutually exclusive labels to each training image. Thus the following softmax regression cost function is used
where z is the feature representation of the input image \(G\left (\mathbf {x}_{l}^{i}\right)\) or \(G\left (\mathbf {x}_{2}^{i}\right)\), L_{ t } is the label set for the tth task, and 1{l^{t}=j} is an indicator function that takes value one when j is equal to the ground truth l^{t} and zero otherwise. Using this framework, several auxiliary tasks can be included by using different label set L_{ t }, such as identification, attributes, pose, etc. Please refer to [52] for more details.
The strategy to construct the multitask metric learning used in this paper is common in the community of multitask learning. It is a flexible model by using different auxiliary tasks. However, for some task, it is difficult to choose a proper auxiliary task, and a bad auxiliary task may induce deterioration of the performance.
Applications
Multitask metric learning has been widely used in a variety of practical applications, and we would like to introduce some representative works in this section.
Semantic categorization and social tagging with knowledge transfer among tasks Wang et al. [32] uses their proposed multitask multifeature similarity learning to solve the large scale visual applications. The metrics for visual categorization and automatic tagging are learned jointly based on the framework, which benefits from several perspectives. First, M^{2}SL learns a metric for each feature instead of concatenating the multiple features into one feature. This effectively reduces the computation complexity growth from O(M^{2}d^{2}) to O(Md^{2}) and also the risk of overfitting. Second, the multitask framework is more flexibility to explore the intrinsic model sharing and feature weighting relations on image data with large amount of classes. Third, the knowledge is transferred among semantic labels and social tagging information by the model. This combines the information fusion from both sides for effective image understanding.
The authors compare the performances of two versions of M^{2}SL (linear and kernelized) with some other methods and the experimental results are shown in Fig. 3. From the results, the kernelized M^{2}SL always achieves the best performance, especially when the number of tasks are greater. For the linear M^{2}SL, it also outperforms the singletask MSL. Thus, the knowledge transfer by multitask learning effectively improves the performance of metric learning.
Person reidentification over camera networks Ma et al. [43] uses their proposed multitask maximally collapsing metric learning to solve the person reidentification over camera networks. Person reidentification in a camera network is a challenging problem because the data are collected from different cameras. The method to use a common metric overlooks the differences between cameras, and thus the authors propose to use a multitask learning approach for this problem. With the MtMCML, an particular metric is learned for each pair of cameras, while the common information can be shared among them. The experimental results show that the multitask approach works substantially better than other stateoftheart methods as shown in Fig. 4.
Largescale face retrieval Bhattarai et al. [50] uses their proposed coupled projection multitask metric learning to solve the largescale face retrieval. They use the multitask framework to learn different tasks on heterogeneous datasets simultaneously, where a common projection is used to share information among these tasks. The tasks include face identity, age recognition, and expression recognition. By jointly learning these tasks, the authors get an improved performance as shown in Fig. 5.
Offline signature verification Soleimani et al. [51] aims to deal with the offline signature verification problem using the deep multitask metric learning. For offline signature verification, there are writerdependent (WD) approaches and writerindependent (WI) approaches. These two approaches benefits from their particular advantages respectively. These two approaches are well integrated in this model where the shared layer acts as a WI approach while the separated layers learn WD factors. In the experiments, the DMML achieves better performance than other methods. For example, on the UTSig dataset and using the HOG feature, the DMML achieves equal error rate (ERR) of 17.45% while the SVM achieves ERR of 20.63%; using the DRT feature, the DMML achieves ERR of 20.28% while the SVM achieves ERR of 27.64%.
Hierarchical largescale image classification Zheng et al. [39] uses their proposed hierarchical multitask metric learning to solve the largescale image classification problem. To deal with the largescale problem, the authors first learn a visual tree to organize large number of image categories hierarchically in a coarsetofine fashion. Then a series metrics are learnt hierarchically. Using the HMML, both the internode visual correlations and the interlevel visual correlations are utilized. The internode correlation is obtained directly from the multitask framework, while the interlevel correlation is obtained by passing the taskspecific part into the next level. The experimental results shown in Fig. 6 demonstrate that the multitask model obtain better performance on largescale classification.
Person reidentification with auxiliary tasks McLaughlin et al. [52] uses the multitask learning to improve the performance of person reidentification. Using their proposed deep convernets metric learning with multitask learning, the authors train the network to jointly perform verification and identification and to recognize attributes related to the clothing and pose of the person in each image. The main job of the network is to learn a metric using similar and dissimilar pairs. With the help of auxiliary tasks (attribute recognition), the network learn a metric to give a satisfactory performance. Figure 7 shows the experimental results. It is obvious that the accuracy is effectively improved by introducing auxiliary tasks.
Conclusion
In this paper, we have systematically reviewed multitask metric learning. Following a brief overview of metric learning, various multitask learning approaches are categorized into four families and introduced respectively. We then review the motivations, models, and algorithms of them, and also discuss and compare some closely related approaches. Finally some representative applications of multitask metric learning are illustrated.
For future work, we suggest potential issues for exploration. First, the theoretical analysis of multitask metric learning should be addressed. There has long been an important issue yielding multiple results [53–56], with most studies focusing on how multitask learning improves the generalization [57] of a conventional algorithm. However, as mentioned earlier, the metric learning improves the performances of the algorithms who use the metric indirectly. This makes these results difficult for application to metric learning algorithms. There has also been some research [58–61] on the theoretical analysis of metric learning, however it has been to difficult to explain these in the context of multitask learning, Whilst Yang et al. [44] has attempted to provide an intuitive explanation, the issue pertaining to multitask learning remains unresolved. Second, how to avoid the negative transfer among tasks. Existing approaches are designed to couple multiple metrics without considering the problem of negative transfer, and thus it is likely to deteriorate the performances when the tasks are not related. Third, most existing multitask metric learning approaches are designed for global linear metrics. Thus it should be extended to more types of metric learning approaches, including local metric learning and nonlinear metric learning. Finally, increased applications of multitask metric learning are expected to be discovered.
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Funding
The paper was partially supported by National Natural Science Foundation of China (NSFC) under grant no.61403388, no.61473236, Natural science fund for colleges and universities in Jiangsu Province under grant no. 17KJD520010, Suzhou Science and Technology Program under grant no. SYG201712, SZS201613, Key Program Special Fund in XJTLU (KSFA01), and UK Engineering and Physical Sciences Research Council (EPSRC) grant numbers EP/I009310/1, EP/M026981/1.
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PY carried out the whole structure of the idea and the mainly drafted the manuscript. KH provided the guidance of the whole manuscript and revised the draft. AH participated the discussion and gave valuable suggestion on the idea. All authors read and approved the final manuscript.
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Correspondence to Peipei Yang.
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Keywords
 Multitask learning
 Metric learning
 Review