We propose order learning to determine the order graph of classes, representing ranks or priorities, and classify an object instance into one of the classes. To this end, we design a pairwise comparator to categorize the relationship between two instances into one of three cases: one instance is `greater than,' `similar to,' or `smaller than' the other. Then, by comparing an input instance with reference instances and maximizing the consistency among the comparison results, the class of the input can be estimated reliably. We apply order learning to develop a facial age estimator, which provides the state-of-the-art performance. Moreover, the performance is further improved when the order graph is divided into disjoint chains using gender and ethnic group information or even in an unsupervised manner. To measure the quality of something, we often compare it with other things of a similar kind. Before assigning 4 stars to a film, a critic would have thought, "It is better than 3-star films but worse than 5-stars." This ranking through pairwise comparisons is done in various decision processes (Saaty, 1977) . It is easier to tell the nearer one between two objects in a picture than to estimate the distance of each object directly (Chen et al., 2016; Lee & Kim, 2019a) . Also, it is easy to tell a higher pitch between two notes, but absolute pitch is a rare ability (Bachem, 1955) . Ranking through comparisons has been investigated for machine learning. In learning to rank (LTR), the pairwise approach learns, between two documents, which one is more relevant to a query (Liu, 2009) . Also, in ordinal regression (Frank & Hall, 2001; Li & Lin, 2007) , to predict the rank of an object, binary classifications are performed to tell whether the rank is higher than a series of thresholds or not. In this paper, we propose order learning to learn ordering relationship between objects. Thus, order learning is related to LTR and ordinal regression. However, whereas LTR and ordinal regression assume that ranks form a total order (Hrbacek & Jech, 1984) , order learning can be used for a partial order as well. Order learning is also related to metric learning (Xing et al., 2003) . While metric learning is about whether an object is 'similar to or dissimilar from' another object, order learning is about 'greater than or smaller than.' Section 2 reviews this related work. In order learning, a set of classes, Θ = {θ 1 , θ 2 , · · · , θ n }, is ordered, where each class θ i represents one or more object instances. Between two classes θ i and θ j , there are three possibilities: θ i > θ j or θ i < θ j or neither (i.e. incomparable). These relationships are represented by the order graph. The goal of order learning is to determine the order graph and then classify an instance into one of the classes in Θ. To achieve this, we develop a pairwise comparator that determines ordering relationship between two instances x and y into one of three categories: x is 'greater than,' 'similar to,' or 'smaller than' y. Then, we use the comparator to measure an input instance against multiple reference instances in known classes. Finally, we estimate the class of the input to maximize the consistency among the comparison results. It is noted that the parameter optimization of the pairwise comparator, the selection of the references, and the discovery of the order graph are jointly performed to minimize a common loss function. Section 3 proposes this order learning. We apply order learning to facial age estimation. Order learning matches age estimation well, since it is easier to tell a younger one between two people than to estimate each person's age directly (Chang et al., 2010; Zhang et al., 2017a) . Even when we assume that age classes are linearly ordered, the proposed age estimator performs well. The performance is further improved, when classes are divided into disjoint chains in a supervised manner using gender and ethnic group information or even in an unsupervised manner. Section 4 describes this age estimator and discusses its results. Finally, Section 5 concludes this work. Order learning was proposed in this work. In order learning, classes form an ordered set, and each class represents object instances of the same rank. Its goal is to determine the order graph of classes and classify a test instance into one of the classes. To this end, we designed the pairwise comparator to learn ordering relationships between instances. We then decided the class of an instance by comparing it with reference instances in the same chain and maximizing the consistency among the comparison results. For age estimation, it was shown that the proposed algorithm yields the stateof-the-art performance even in the case of the single-chain hypothesis. The performance is further improved when the order graph is divided into multiple disjoint chains. In this paper, we assumed that the order graph is composed of disjoint chains. However, there are more complicated graphs, e.g. Figure 1 (a), than disjoint chains. For example, it is hard to recognize an infant's sex from its facial image (Porter et al., 1984) . But, after puberty, male and female take divergent paths. This can be reflected by an order graph, which consists of two chains sharing common nodes up to a certain age. It is an open problem to generalize order learning to find an optimal order graph, which is not restricted to disjoint chains.
The notion of order learning is proposed and it is applied to regression problems in computer vision
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