Introduction
Recent years have witnessed a great success of supervised learning in various pattern recognition problems, such as image classification, object detection, and speech recognition. Standard supervised learning relies heavily on the
i.i.d.data assumption; however, datasetbias is unavoidable in many situations due to selection bias or mechanism changes. For example, this problem has been well recognized in the computer vision community
[Torralba and Efros2011, Khosla et al.2012]: the widely adopted vision datasets have their special properties and are not representative of the visual world. In medical diagnosis, the distribution of cell types varies from patient to patient, and we need to train a classifier on the data collected from previous patients that generalizes well to unseen patients
[Blanchard, Lee, and Scott2011, Muandet, Balduzzi, and Schölkopf2013]. These problems are known as domain generalization, in which the training set consists of data from heterogeneous source domains, say patients, and the test data distribution is different from that of the training data.To handle the distribution changes, many existing domain generalization methods aim to learn domaininvariant representations that have stable distributions across all source domains [Muandet, Balduzzi, and Schölkopf2013, Erfani et al.2016, Ghifary et al.2017]. The learned invariant representations are expected to generalize well to any unseen test set under the assumption that the changes of distribution across source and test domains are caused by some common factors whose effects are removed in the invariant representations. In computer vision, such factors could be illumination, camera viewpoints, and backgrounds. These methods have achieved good performance in computer vision [Ghifary et al.2015, Ghifary et al.2017] and medical diagnosis [Muandet, Balduzzi, and Schölkopf2013].
However, existing methods that learn domaininvariant representations assume that only changes across domains while the conditional distribution is rather stable. Thus, the conditional distribution is also invariant, and the learning problem reduces to ensuring that the marginal distribution is invariant across domains. This assumption greatly simplifies the problem, but it is unclear whether this assumption holds in practical situations. According to some recent results in causal learning [Schölkopf et al.2012, Janzing and Scholkopf2010], can be stable when changes in the situation where is the cause for , i.e., the causal structure is . This is because the mechanism that generates the cause, i.e., , is not coupled with the mechanism that generates the effect from the cause, i.e., , and not vice versa. That is to say, if is the cause and is the effect, often changes together with . In this situation, if changes, it is very likely that also changes across domains, which violates the stability of assumption. In practice, we have plenty of problems where the causal structure is
. For example, , in face recognition, Y is person id, X is the feature, and
is the viewpoint. Let us consider each viewpoint as a domain, then in each domain we have conditional distribution. According to Bayes theorem,
thus changes across domains. This conflicts with previous assumptions that keeps unchanged. There are also other examples, e.g. speaker recognition and person reidentification [Yang et al.2017].In this paper, we assume both and change across domains. We aim to find a feature transformation that has invariant classconditional distribution
. To achieve so, we propose to minimize two regularization terms that enforce distribution invariance across source domains. The first term measures the variance of each classconditional distribution across all source domains and then sums up the variances for all classes. The second term is the variance of class priornormalized marginal distribution
, which measures the global distribution discrepancy. The normalization of class priors is introduced to remove the effects brought by possible changes in across source domains. If the prior distribution does not change across source domains, the second term reduces to the common technique used in existing domaininvariant representation learning methods [Muandet, Balduzzi, and Schölkopf2013, Ghifary et al.2017]. To preserve the discriminative power of the learned representation, we also incorporate the intraclass and interclass distances used in kernel Fisher discriminant analysis (FDA)[Mika et al.1999].Compared to existing domaininvariant representation learning methods, our method does not require the assumption of stable by exploiting the labels on the source domains which were overlooked in the previous methods. Especially, if the prior distribution on the test sets is the same as that on the training set containing all source domains, our method is able to learn representations that have invariant joint distribution across all domains. We conduct a series of experiments on both synthetic and real data, and the results demonstrate the effectiveness of our method.
Related Work
Domain generalization has been widely applied in classification tasks [Xu et al.2014, Duan et al.2009, Muandet, Balduzzi, and Schölkopf2013, Ghifary et al.2017, Ghifary et al.2015, Erfani et al.2016]. Compared with standard supervised learning, domain generalization methods aim to reduce data bias across different domains and improve the generalization of the learned model to unseen but related domains. For example, [Xu et al.2014]assumed that positive samples within the same shared latent domain should have similar likelihood and proposed to exploit the lowrank structure from latent domains for domain generalization. [Muandet, Balduzzi, and Schölkopf2013] proposed domaininvariant component analysis (DICA) through learning an invariant feature representation , in which the difference between marginal distributions is minimized. [Ghifary et al.2017] proposed a unified framework called scatter component analysis for domain adaptation and domain generalization. The scatter component analysis combines domain scatter [Muandet, Balduzzi, and Schölkopf2013], kernel PCA [Schölkopf, Smola, and Müller1998], and kernel FDA [Mika et al.1999] in a single objective function. However, all these methods assume that the distribution between domains differs only in the marginal distribution while the conditional distribution keeps stable or unchanged across domains. This assumption can simplify the problem of domain generalization, but it is easily violated in realworld applications.
Domain adaptation is a related problem which has been extensively studied in the literature [Baktashmotlagh et al.2013, Huang et al.2007, Pan et al.2011, Long et al.2017, Shao, Kit, and Fu2014, Shao et al.2016, Luo et al.2017, Liu, Yang, and Tao2017]. Assuming that only changes, the distribution changes can be corrected by importance reweighting [Huang et al.2007] or domaininvariant feature learning [Pan et al.2011, Baktashmotlagh et al.2013], using unlabeled data from source and target domains. Recently, several works attempted to work in the situation where both and change across domains [Zhang et al.2013, Gong et al.2016, Long et al.2017]. [Zhang et al.2013] and [Gong et al.2016] proposed to consider the domain adaptation problem in the generalized target shift (GeTarS) scenario where the causal direction is . In this scenario, both the change of distribution and conditional distribution are considered to reduce the data bias across domains. [Zhang et al.2013] made an assumption that features from source domains can be transferred to the target domain by a locationscale transformation, which is restricted in realworld applications because of the presence of noises in features. [Gong et al.2016] proposed to learn components whose conditional distribution
is invariant across domains and estimate the target label distribution
through labeled source domain data and unlabeled target domain data. Since there are no labels in the target domain to match classconditionals, the invariance of is achieved by minimizing the discrepancy of the marginal distribution under some untestable assumptions. [Long et al.2017] proposed an iterative way to match the conditionals by using the predicted labels from previous iterations as pseudo labels. Different from the domain adaptation methods, domain generalization does not require unlabeled data from the target domains.Conditional Invariant Domain Generalization
In this section, we first establish the basic notations of domains and formally introduce the definition of domain generalization. Then we give a detailed description of the proposed conditional invariant domain generalization (CIDG) method.
Problem Definition
Denote and as the input feature and label spaces, respectively. A domain defined on
can be represented by a joint probability distribution
. For simplicity, we denote the joint probability distribution of the th source domain as . The domain is associated with a sample , where and denotes the sample size of the domain . Then we can define domain generalization as follows.Definition 1 (Domain Generalization).
Given multiple related source domains and each domain is associated with a sample , where . The goal of domain generalization is to learn a classification function from source domain datasets and apply it to an unseen but related target domain .
Kernel Mean Embedding
Before introducing the proposed method, we briefly review the kernel mean embedding of distributions, which is an important mathematical tool to represent and compare distributions [Song, Fukumizu, and Gretton2013, Sriperumbudur et al.2010]. Let denote a characteristic reproducing kernel Hilbert space (RKHS) on associated with a kernel , and be an associated mapping such that . Suppose we have two observations and from domain , then we have . The kernel embedding of a distribution can be formulated as the following:
(1) 
where denotes for simplicity. If a kernel is characteristic, then the mean embedding is injective. All the information about the distribution can be preserved [Sriperumbudur et al.2010]. The kernel embedding cannot be computed directly and is usually estimated from observations. Given a sample , where is the sample size of the domain, and the kernel embedding can be empirically estimated as the following:
(2) 
Proposed Approach
The proposed conditional invariant domain generalization (CIDG) method aims to find a conditional invariant representation
(a linear transformation of the original features) to reduce the variance of the conditional distribution
across source domains. Suppose we can learn a perfect conditional invariant representation , which satisfies , and denotes the target domain. We can gather all the source domains to construct a new single domain with a joint distribution . Therefore, under the condition , the learned has the invariant joint distribution across training and test domains. Contrarily, the previous method can only guarantee that is invariant, and whether is invariant remains unknown. If is different from , our method cannot guarantee the invariance of the joint distribution either. Nevertheless, our method can at least guarantee invariant classconditional distributions, which is still better than previous methods. This is because is usually not very sensitive to the changes in the prior if is highly correlated with .The learning of conditional invariant representations is achieved mainly through two regularization terms: total scatter of classconditional distributions and scatter of class priornormalized marginal distributions. The first term measures the variance of locally, while the second term measures the variance of globally. In addition to these two terms, we also incorporate several terms that measure the discriminative power of the representation as done in the previous works. By minimizing the distribution variance across domains and maximizing the discriminative power in one objective function, we can obtain the conditional invariant representation which is predictable for the labels on unseen target domains.
Total scatter of classconditional distributions
Suppose we have related domains on . The marginal distribution on of the th domain is denoted as . Suppose the class labels of each domain vary from to . For simplicity, the th class conditional distribution of the th domain is denoted as . The total scatter of classconditional distributions across domains can be formulated as:
(3) 
where and is called the domain scatter [Ghifary et al.2017] or distributional variance [Muandet, Balduzzi, and Schölkopf2013]. Instead of measuring the domain scatter w.r.t. the marginal distributions as done in previous works like [Ghifary et al.2017], we measure the domain scatter w.r.t. each classconditional distribution and then sum them together.
Before introducing the computation of the above scatter, we first give the formulation of the learned feature transformation. Denote the feature matrix as the data matrix of samples from source domains, where is the dimension of the feature space and . Define a set of functions related to the feature map . We aim to find a linear feature transformation transforming into a finite subspace , that is
. According to the kernel principal component analysis (KPCA)
[Schölkopf, Smola, and Müller1998], the linear transformation can be formulated as the linear combination of , where is the coefficient matrix. By using this representation, we can avoid explicitly computing the feature map and use the kernel trick instead.For simplicity, denote as ,
(4)  
where is trace operator. To measure the distribution scatter of the distributions of , we apply the linear feature transformation to the above scatter and obtain
(5) 
where is:
(6) 
in which and can be computed according to the empirical estimation shown in equation (2). Denote as the th sample belonging to the th class in the th domain, where and . Let denote the sample size of the th class from the th domain, we have:
(7) 
where denotes the indicies of examples in the th class.
Scatter of class priornormalized marginal distributions
The scatter of each classconditional distribution is estimated locally using the samples from that class. When the number of examples in each class is small, optimizing (Total scatter of classconditional distributions) can easily overfit the data. To further improve the estimation accuracy, we propose another regularization term which measures the scatter of classprior normalized marginal distributions. The new regularization term is able to measure the global distance between all classconditionals. In the th domain, the marginal distribution is defined as
(8) 
If the class prior distribution does not change across domains, and we can also find a feature representation that has an invariant classconditional across source domains, we can say that is also domaininvariant, but not vice versa. Nevertheless, searching for a representation that reduces the discrepancy between the marginal distributions can to some extent reduce the discrepancy of class conditional distributions, though the original purpose was to match marginal distributions only [Muandet, Balduzzi, and Schölkopf2013, Ghifary et al.2017]. However, if the class prior changes across source domains, the above statements are no longer true. That is to say, even if the class conditionals are domaininvariant, the marginal distribution are not invariant because of the changes in . To mitigate this issue, we propose to match the classprior normalized marginal distribution, which is defined as follows:
(9) 
It can be seen that the classprior normalized marginal distribution enforces the same prior probability for each class. Therefore, the changes in the prior distribution across source domains are adjusted, which guarantees that the priornormalized marginal distribution is domaininvariant when the class conditionals are invariant. By embedding the class priornormalzied marginal distribution into a Hilbert space, the scatter of the normalized marginal distribution across domains can be formulated as:
(10) 
where , and is the priornormalized marginal distribution of the th domain. is the kernel mean of the class priornormalized marginal distribution of all domains. To learn the domaininvariant representation, we apply the linear feature transformation to the above scatter, resulting in:
(11) 
where can be formulated as follows:
(12) 
in (12) can be empirically estimated from the observations as:
(13) 
Note that if are identical for all , that is the classes are balanced, the class priornormalized marginal distribution reduces to the empirical estimate of the original marginal distribution adopted in [Muandet, Balduzzi, and Schölkopf2013, Ghifary et al.2017].
Preserving Discriminative Power
In addition to the above proposed two domaininvariance regularization terms, we also consider extra terms to preserve the discriminativeness of the learned representation. There have been plenty of works in supervised dimension reduction in the case, and kernel Fisher discriminant analysis [Mika et al.1999] is a representative method which has been used in domain generalization [Ghifary et al.2017]. Becasue the focus of our method is to better learn the domaininvariant representations, we incorporate kernel Fisher discriminant analysis for fair comparison to existing methods. Specifically, the examples with the same label should be similar and the examples with different labels should be well separated. These two constraints can be formulated as two regularization terms: withinclass scatter and betweenclass scatter, which are briefly described as follows.
Betweenclass scatter:
(14) 
where matrix can be computed as:
(15) 
and denotes the number of examples in the th class from all domains. Note that and can be empirically estimated as and .
Withinclass scatter:
(16) 
where the matrix can be computed as:
(17) 
Objective Function and Optimization
In this subsection, we first formulate our objective function with the above regularization terms and then find the solutions by maximizing the objective function.
The proposed CIDG aims to learn an invariant feature transformation by solving the following optimization problem:
(18) 
The numerator enforces the distance between features in different classes to be large. The denominator aims to learn a conditional invariant feature representation and reduce the distance between features in the same class simultaneously.
Replace the scatters with equation, (Total scatter of classconditional distributions), (Scatter of class priornormalized marginal distributions), (14), (16) and introduce several tradeoff parameters , the above objective function can be reformulated as follows:
(19) 
where , are tradeoff parameters, which need to be selected according to the validation set.
Note that the above objective function is invariant when rescaling , where is a constant. Consequently, (19) can be reformulated as the following constrained optimization problem:
(20)  
which yields Lagrangian:
(21)  
where
is an identity matrix of dimension
and is a diagonal matrix with the Lagrange multipliers aligned in the diagonal. Solving (21) by setting the derivative w.r.t.to be zero, we arrive at a standard eigenvalue decomposition problem:
(22) 
In practice, the term is added by a small constant to get a more stable solution, becoming . We summarize the algorithm of our CIDG in Algorithm 1.
domain index  domain 1  domain 2  domain 3  

class index  
x  (1,0.3)  (2, 0.3)  (3, 0.3)  (3.5, 0.3)  (4.5, 0.3)  (5.5, 0.3)  (8, 0.3)  (9.5, 0.3)  (10, 0.3) 
y  (2,0.3)  (1, 0.3)  (2, 0.3)  (2.5, 0.3)  (1.5, 0.3)  (2.5, 0.3)  (2.5, 0.3)  (1.5, 0.3)  (2.5, 0.3) 
# samples  30  20  30  20  60  40  40  40  40 
Experiments
In this section, we conduct experiments on one synthetic data and two realworld image classification datasets to demonstrate the effectiveness of our conditional invariant domain generalization (CIDG) method. The synthetic data are two dimensional, which facilitate the comparison of the performance of different methods through the visualization of the data distribution. The two realworld image classification datasets are the VLCS and Office+Caltech datasets, which are widely used datasets to evaluate the performance of domain generalization and domain adaptation [Ghifary et al.2017, Gong et al.2016, Khosla et al.2012]. We compare our CIDG with several stateoftheart domain generalization methods, which are summarized below.

Knearest neighbors (KNN) using the original features, which servers as the baseline method.

Kernel principal component analysis (KPCA) [Schölkopf, Smola, and Müller1998] which finds the dominant components of the original features. KNN is applied for classification on the KPCA features.

UndoBias [Khosla et al.2012], which is a multitask learning method aims to reduce the data bias. Because undobias is a binary classification algorithm, we use the onevsrest strategy for multiclass classification.

Domain invariant component analysis (DICA) [Muandet, Balduzzi, and Schölkopf2013], which is a domain generalization method learns an domaininvariant feature representation in terms of marginal distributions. We use KNN to do classification on the learned feature representation.

Scatter component analysis (SCA) [Ghifary et al.2017], which is a another method that learns domaininvariant features in terms of marginal distributions. The method incorporates discriminative terms and domain scatter terms into a unified framework.
Note that we have also conducted experiments using kernel finsher discriminant analysis (FDA), however, it performs worse than KPCA. Consequently, we do not report the results of KLDA in this paper.
Source  Target  1NN  KPCA  DICA  Undobias  SCA  CIDG 

L,C,S  V  
V,C,S  L  
V,L,S  C  
V,C,L  S  
C,S  V,L  
C,L  V,S  
C,V  L,S  
L,S  V,C  
L,V  S,C  
V,S  L,C 
Source  Target  1NN  KPCA  DICA  Undobias  SCA  CIDG 

W,D,C  A  
A,W,D  C  
A,W,C  D  
A,C,D  W  
A,C  D,W  
D,W  A,C  
A,W  C,D  
A,D  C,W  
C,W  A,D  
C,D  A,W 
Synthetic Dataset
In this section, we randomly generate two dimensional examples for source domains and target domain from different Gaussian distributions
, where is the mean andis the standard deviation. The values of mean
and standard deviation pairs of different classes in three domains are shown in Table 1. We consider the first two domains as source domains and the third one as a target domain. The first row of Figure 1 visualizes the samples from three different domains corresponding to three different colors (yellow, magenta, cyan), and the domains are domain , domain and domain from left to right. The second row of Figure 1 shows that each domain has three clusters (green, red, blue) corresponding to three different classes and the domains are represented by different shapes (star, circle, cross). The first column illustrates the raw feature distributions.We compare our CIDG with KNN, KPCA, DICA, and SCA to evaluate the distributions of the learned feature representation across domains. Since UndoBias is a SVMbased method that does not need to explicitly learn a feature representation, we do not compare the results with UndoBias on synthetic data. We use the RBF kernel for all the methods involving computation of kernel matrices. In all experiments, domain 1 and domain 2 are used as source domains and domain 3 is used as the unseen target domain. From the results in Figure 1, we can see that the proposed CIDG achieves the best accuracy of . KPCA almost has no improvement over the baseline KNN method on the synthetic dataset. DICA can cluster one class (blue) well but performs badly for the other two classes. SCA can learn better feature distribution but the blue class and the green class are mixed in the learned representation. Additionally, the samples in the same class lie in a line rather than reside in a clear cluster. Our CIDG can learn more robust feature representations and the learned features in the same class are distributed in a wellshaped cluster.
VLCS Dataset
VLCS is an image classification dataset widely used for evaluating the performance of domain generalization. This dataset contains images from four different subdatasets corresponding to four domains: PASCAL VOC2007 (V) [Everingham et al.2010], LabelMe (L) [Russell et al.2008], Caltech101 (C) [Griffin, Holub, and Perona2007], and SUN09 (S) [Choi et al.2010]. Five shared classes (bird, car, chair, dog and person) are selected from these four datasets. The images are preprocessed by subtracting the mean values and cropped on the central region out of the resized images. Then the preprocessed images are fed into the DeCAF network and extracted the 4096 dimensional DeCAF_{6} features [Donahue et al.2014]. We randomly select of the data as training set from each domain and repeat the random selection five times. The mean classification accuracy and standard deviation of the five random selection are given for each method. All parameters are selected through validation, in which of the training data is selected as validation set. All kernel methods use a RBF kernel and the learned features are classified using KNN except for UndoBias. The results are shown in Table 2.
From the results in Table 2, we can see that our conditional invariant domain generalization (CIDG) performs the best on 9 of the 10 domain generalization tasks. KPCA performs the best when L,S are source domains and V,C are target domains. Note that almost all the domain generalization methods outperform the 1NN on raw features. However, some methods on several domain tasks perform even worse than 1NN on raw features. This is mainly because that features of real world images are complicated and noisy. The learned features are not discriminative when generalized to target domains.
Office+Caltech Dataset
The Office+Caltech image dataset consists of ten overlapping categories between the Office dataset and the Caltech256 dataset (C). Because the Office dataset contains three subdatasets: AMAZON (A), DSLR (D), and WEBCAM (W), we have four different domains in total. Similarly, We randomly select
of the data as training set from each domain and repeat the random selection five times. The mean classification accuracy and standard deviation of the five random selection are reported for each method. The feature extraction is the same as that used for the VLCS dataset except we use the CAFFE network
[Jia et al.2014] instead of the DeCAF network. The other settings are the same as those in experiments on the VLCS dataset.From the results in Table 3, we can find that the proposed CIDG achieves the best performance on 9 of the 10 domain generalization tasks. This further validates that enforcing conditional invariance is more reasonable than enforcing only marginal invariance. Note that Undobias is a SVMbased method. It is possibly the main reason why it outperforms CIDG when using D,W as source domains and A,C as target domains.
Conclusion
In this paper, we have proposed a conditional invariant domain generalization approach considering the situation that both and change across domains. Different from previous works which assume that only changes, our proposed method can learn representations that have invariant joint distribution across domains if the prior distribution does not change between the source domains and the target domains. Two regularization terms that enforce classconditional distribution invariance across domains are proposed and validated on both synthetic and real datasets.
Acknowledgments
This work was supported by National Key Research and Development Program of China 2017YFB1002203, NSFC No.61572451, No.61390514, and No. 61632019, Youth Innovation Promotion Association CAS CX2100060016, Fok Ying Tung Education Foundation WF2100060004, and Australian Research Council Projects FL170100117, DP180103424, DP140102164, LP150100671.
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