Order Learning and Its Application to Age Estimation

1 INTRODUCTION . 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 = { θ1 , θ2 , · · · , θn } , is ordered , where each class θi represents one or more object instances . Between two classes θi and θj , there are three possibilities : θi > θj or θi < θj 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 . 2 RELATED WORK . Pairwise comparison : It is a fundamental problem to estimate the priorities ( or ranks ) of objects through pairwise comparison . In the classic paper , Saaty ( 1977 ) noted that , even when direct estimates of certain quantities are unavailable , rough ratios between them are easily obtained in many cases . Thus , he proposed the scaling method to reconstruct absolute priorities using only relative priorities . The scaling method was applied to monocular depth estimation ( Lee & Kim , 2019a ) and aesthetic assessment ( Lee & Kim , 2019b ) . Ranking from a pairwise comparison matrix has been studied to handle cases , in which the matrix is huge or some elements are noisy ( Braverman & Mossel , 2008 ; Jamieson & Nowak , 2011 ; Negahban et al. , 2012 ; Wauthier et al. , 2013 ) . On the other hand , the pairwise approach to LTR learns , between two documents , which one is more relevant to a query ( Liu , 2009 ; Herbrich et al. , 1999 ; Burges et al. , 2005 ; Tsai et al. , 2007 ) . The proposed order learning is related to LTR , since it also predicts the order between objects . But , while LTR sorts multiple objects with unknown ranks and focuses on the sorting quality , order learning compares a single object x with optimally selected references with known ranks to estimate the rank of x. Ordinal regression : Ordinal regression predicts an ordinal variable ( or rank ) of an instance . Suppose that a 20-year-old is misclassified as a 50-year old and a 25-year old , respectively . The former error should be more penalized than the latter . Ordinal regression exploits this characteristic in the design of a classifier or a regressor . In Frank & Hall ( 2001 ) and Li & Lin ( 2007 ) , a conversion scheme was proposed to transform an ordinal regression problem into multiple binary classification problems . Ordinal regression based on this conversion scheme has been used in various applications , including age estimation ( Chang et al. , 2010 ; 2011 ; Niu et al. , 2016 ; Chen et al. , 2017 ) and monocular depth estimation ( Fu et al. , 2018 ) . Note that order learning is different from ordinal regression . Order learning performs pairwise comparison between objects , instead of directly estimating the rank of each object . In age estimation , ordinal regression based on the conversion scheme is concerned with the problem , “ Is a person ’ s age bigger than a threshold θ ? ” for each θ . In contrast , order learning concerns “ Between two people , who is older ? ” Conceptually , order learning is easier . Technically , if there are N ranks , the conversion scheme requires N − 1 binary classifiers , but order learning needs only a single ternary classifier . Moreover , whereas ordinal regression assumes that ranks form a total order , order learning can be used even in the case of a partial order ( Hrbacek & Jech , 1984 ) . Metric learning : A distance metric can be learned from examples of similar pairs of points and those of dissimilar pairs ( Xing et al. , 2003 ) . The similarity depends on an application and is implicitly defined by user-provided examples . If a learned metric generalizes well to unseen data , it can be used to enforce the desired similarity criterion in clustering ( Xing et al. , 2003 ) , classification ( Weinberger et al. , 2006 ) , or information retrieval ( McFee & Lanckriet , 2010 ) . Both metric learning and order learning learn important binary relations in mathematics : metric and order ( Hrbacek & Jech , 1984 ) . However , a metric decides whether an object x is similar to or dissimilar from another object y , whereas an order tells whether x is greater than or smaller than y . Thus , a learned metric is useful for grouping similar data , whereas a learned order is suitable for processing ordered data . Age estimation : Human ages can be estimated from facial appearance ( Kwon & da Vitoria Lobo , 1994 ) . Geng et al . ( 2007 ) proposed the aging pattern subspace , and Guo et al . ( 2009 ) introduced biologically inspired features to age estimation . Recently , deep learning has been adopted for age estimation . Niu et al . ( 2016 ) proposed OR-CNN for age estimation , which is an ordinal regressor using the conversion scheme . Chen et al . ( 2017 ) proposed Ranking-CNN , which is another ordinal regressor . While OR-CNN uses a common feature for multiple binary classifiers , Ranking-CNN employs a separate CNN to extract a feature for each binary classifier . Tan et al . ( 2018 ) grouped adjacent ages via the group-n encoding , determined whether a face belongs to each group , and combined the results to predict the age . Pan et al . ( 2018 ) proposed the mean-variance loss to train a CNN classifier for age estimation . Shen et al . ( 2018 ) proposed the deep regression forests for age estimation . Zhang et al . ( 2019 ) developed a compact age estimator using the two-points representation . Also , Li et al . ( 2019 ) proposed a continuity-aware probabilistic network for age estimation . 3 ORDER LEARNING . 3.1 WHAT IS ORDER ? . Let us first review mathematical definitions and concepts related to order . An order ( Hrbacek & Jech , 1984 ; Bartle , 1976 ) , often denoted by ≤ , is a binary relation on a set Θ = { θ1 , θ2 , · · · , θn } that satisfies the three properties of • Reflexivity : θi ≤ θi for every θi ∈ Θ ; • Antisymmetry : If θi ≤ θj and θj ≤ θi , then θi = θj ; • Transitivity : If θi ≤ θj and θj ≤ θk , then θi ≤ θk . In real-world problems , an order describes ranks or priorities of objects . For example , in age estimation , θi ≤ θj means that people in age class θi look younger than those in θj . We may use the symbol→ , instead of≤ , to denote an order on a finite set Θ . Then , the order can be represented by a directed graph ( Gross & Yellen , 2006 ) using elements in Θ as nodes . If θi → θj , there is a directed edge from node θi to node θj . The order graph is acyclic because of antisymmetry and transitivity . For example , for n , m ∈ N , let n → m denote that m is a multiple of n. Note that it is an order on any subset of N. Figure 1 ( a ) is the graph representing this order on { 1 , . . . , 9 } . Elements θi and θj are comparable if θi → θj or θj → θi , or incomparable otherwise . In Figure 1 ( a ) , 6 and 8 are incomparable . In age estimation , it is difficult to compare apparent ages of people in different ethnic groups or of different genders . An order on a set Θ is total ( or linear ) if all elements in Θ are comparable to one another . In such a case , Θ is called a linearly ordered set . In some real-world problems , orders are not linear . In this work , a subset Θc of Θ is referred to as a chain , if Θc is linearly ordered and also maximal , i.e . there is no proper superset of Θc that is linearly ordered . In Figure 1 ( a ) , nodes 1 , 2 , 4 , and 8 form a chain . In Figure 1 ( b ) , the entire set is composed of three disjoint chains .

Summary: The paper proposes a method for learning partial orders based on learning fuzzy pairwise comparisons (smaller/greater/approximatively equal to), and the retaining of a set of representants in each chain in order to allow consistent result by consistency maximization. The method is applied to the problem of estimating the age based on faces. Extensions proposed are the learning of multiple disjoint chains on this dataset, manually partitioned or learned through an iterative assignment algorithm that has similarities with a soft expectation-maximization principle. Extensive experiments are done on the age estimation problem with comparison to exisiting approaches, and an application to aesthetic assessment is proposed in appendix.