EigenTransfer: A Unified Framework for Transfer Learning
Wenyuan Dai dwyak@apex.sjtu.edu.cn Ou Jin kingohm@apex.sjtu.edu.cn Gui-Rong Xue grxue@apex.sjtu.edu.cn Qiang Yang qyang@cse.ust.hk Yong Yu yyu@apex.sjtu.edu.cn () Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China () Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

Abstract
This paper proposes a general framework, called EigenTransfer, to tackle a variety of transfer learning problems, e.g. cross-domain learning, self-taught learning, etc. Our basic idea is to construct a graph to represent the target transfer learning task. By learning the spectra of a graph which represents a learning task, we obtain a set of eigenvectors that reflect the intrinsic structure of the task graph. These eigenvectors can be used as the new features which transfer the knowledge from auxiliary data to help classify target data. Given an arbitrary non-transfer learner (e.g. SVM) and a particular transfer learning task, EigenTransfer can produce a transfer learner accordingly for the target transfer learning task. We apply EigenTransfer on three different transfer learning tasks, cross-domain learning, cross-category learning and self-taught learning, to demonstrate its unifying ability, and show through experiments that EigenTransfer can greatly outperform several representative non-transfer learners.

texts, even though they may be more or less correlated and can help each other. In this case, transfer learning (Thrun, 1996; Caruana, 1997; Wu & Dietterich, 2004; Raina et al., 2006) can leverage the learned knowledge in one context to enhance the learning in different contexts and thus establish a channel between previous isolated learned tasks. In the past, a variety of transfer learning tasks have been investigated, including lifelong learning (Thrun, 1996), multi-task learning (Caruana, 1997), crossdomain learning (Wu & Dietterich, 2004; Daum´ III e & Marcu, 2006), cross-category learning (Raina et al., 2006) and self-taught learning (Raina et al., 2007), to name a few. Most the proposed transfer learning algorithms have achieved great improvement against traditional learning algorithms. However, each of the above specialized works only addresses a particular transfer learning problem, but given the many different approaches and algorithms for transfer learning, there is a lack of general frameworks that can unify the different transfer learning problems and provide corresponding solutions. The focus of this paper is to design such a unifying framework. In general, transfer learning makes use of the available auxiliary data to help the learning on the target data that include target training data and target test data. We observe that, to enable transfer learning, the target data and the auxiliary data usually share some common parts and relations between them. For example, in multi-task learning (Caruana, 1997), multiple learning tasks share the same feature space. Thus, we can construct a graph to represent the transfer learning task and model the relations between the target data and the auxiliary data. In this task graph, we use instances, features, and labels to serve as the nodes, while the edges are set based on the relations between the end nodes, connecting the target and auxiliary data in a unified graph structure. By learning the

1. Introduction
Traditional statistical learning makes a basic assumption that the training data and the test data should be governed by the same underlying distribution. Under this identical-distribution assumption, statistical learning has achieved quite a lot of success. However, the limitation of the identical-distribution assumption is that it isolates the learning tasks in different conAppearing in Proceedings of the 26 th International Conference on Machine Learning, Montreal, Canada, 2009. Copyright 2009 by the author(s)/owner(s).

EigenTransfer: A Unified Framework for Transfer Learning Table 1. The definition of the three representative transfer learning problems: cross-domain learning, cross-category learning, and self-taught learning. The notations and illustrative example have been included in this table. In the example, the pictures in the blue frames are labeled data, while the others are unlabeled. Cross-Domain Learning Instances Features Labels Xt = {xt }n i=1 F = {f (i) }s i=1 C = {c(i) }l i=1
(i)

Cross-Category Learning Xt = {xt }n i=1 F = {f (i) }s i=1 (i) Ct = {ct }l i=1
(i)

Self-taught Learning Xt = {xt }n i=1 F = {f (i) }s i=1 (i) Ct = {ct }l i=1
(i)

Training Data

Auxiliary Data

Instances Features Labels

Xa = {xt }m i=1 F = {f (i) }s i=1 C = {c(i) }l i=1

(i)

Xa = {xt }m i=1 F = {f (i) }s i=1 (i) Ca = {ca }r i=1

(i)

Xa = {xt }m i=1 F = {f (i) }s i=1 N/A

(i)

Test Data

Instances Features Labels

Xu = {xu }k Xu = {xu }k Xu = {xu }k i=1 i=1 i=1 (i) s (i) s F = {f }i=1 F = {f }i=1 F = {f (i) }s i=1 N/A, but are expected to be drawn from the same label set as the training data

(i)

(i)

(i)

spectra of this task graph, we can obtain an eigen feature representation for all the nodes in the task graph. This feature representation reflects the intrinsic structure of the task graph that includes the information about target data, auxiliary data, and the relations among them. Under the new feature representation, the knowledge from the auxiliary data can be transferred to help the learning on the target data. We refer to our framework as EigenTransfer, and conducted extensive experiments to evaluate the effectiveness of our framework on three representative transfer learning problems: cross-domain learning (Daum´ III e & Marcu, 2006), cross-category learning (Raina et al., 2006), and self-taught learning (Raina et al., 2007). We show that the general framework can indeed model the different transfer learning tasks well. We also show through experimental results that, when compared to non-transfer learners, EigenTransfer can achieve great improvements in general.

in cross-domain learning (Daum´ III & Marcu, 2006), e the auxiliary and the target data (training and test data) share the same categories, i.e. horse and sheep, but are in different domains, i.e. real animals and cartoon animals. In cross-category learning (Raina et al., 2006), the auxiliary data are labeled but drawn from different categories from the target data. For example, in Table 1, the target data are about horse and sheep, while the auxiliary data are about building and sunrise. In self-taught learning (Raina et al., 2007), the difference from cross-category learning is that the auxiliary data are unlabeled. A formal definition of these problems including the notations and the illustrative example, is shown in Table 1. In this table, all the instances x  X = Xt  Xa  Xu are represented by the features in the feature space F = {f (i) }s . i=1 The objective of transfer learning is to minimize the prediction error on the test set Xu , with the help of the auxiliary data Xa .

2. Problem Formulation
In transfer learning, we have two target data sets: a target training data set Xt = {xi }n with labels, and t i=1 a target test data set Xu = {xi }k to be predicted. u i=1 Different from traditional machine learning, we also have an auxiliary data set Xa = {xi }m to help the t i=1 target learning. Generally, different transfer learning problems differ in the form of auxiliary data. As shown in Table 1,

3. Constructing the Task Graphs
In this work, we use spectral learning technique (Chung, 1997) to tackle the transfer learning problems. In spectral learning (Chung, 1997), the input is a weighted graph G = (V, E) to represent the target task, where V = {vi }n and E = {eij }n i=1 i,j=1 represent the node set and the edge set in the graph, respectively. By calculating the eigenvectors of the graph

EigenTransfer: A Unified Framework for Transfer Learning
xt xt f
(1) (1)

xt xt

(1)

ct ct

(1)

xt xt f (1)

(1)

ct ct

(1)

(2)

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(2)

(2)

(2)

. . . xt
(n)

c(1)

f (1)

. . . xt
(n)

. . . ct
(l)

. . . xt
(n)

. . . ct
(l)

c(2) f (2)
(1) xa

. . . c
(l)

f (2) xa f (3) . . . f
(s-1) (1)

f (2) ca
(1)

xa f (3) . . . f
(s-1)

(1)

f (3) . . . f
(s-1)

xa . . . xa

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xa . . . xa

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ca . . . ca

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xa . . . xa

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(m)

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(m)

xu f (s) xu . . . xu

(1)

xu f (s) xu . . . xu

(1)

xu f (s) xu . . . xu

(1)

(2)

(2)

(2)

(k)

(k)

(k)

(a) Cross-Domain Learning

(b) Cross-Category Learning

(c) Self-taught Learning

Figure 1. The graph construction for three different transfer learning problems: (a) cross-domain learning; (b) crosscategory learning; (c) self-taught learning.

G, we obtain the graph spectra for the target task for further learning. Traditional spectral learning algorithms, such as spectral clustering (Shi & Malik, 2000), usually construct the task graph G(V, E) based on nearest neighbor graph or similarity graph. However, for transfer learning, these two kinds of graphs are no longer capable. Thus, we have to design new graph construction strategies which take into account target data (including training and test data), auxiliary data and the relations among these data. In this work, we construct a task graph G(V, E) in the following way. As shown in Figure 1, the nodes V in the graph G represent instances, features or class labels, while the edges E represent the relations between these nodes, where the weights of the edges are based on the number of co-occurrences between the end nodes in the target and auxiliary data. Consequently, the task graph G(V, E) contains almost all the information for the transfer learning task, including the target data and the auxiliary data. Usually, the task graph G is sparse, symmetric, real and positive semi-definite. Thus, the spectra of the graph G can be calculated very efficiently. In the next several subsections, we model three representative transfer learning problems, cross-domain learning (Daum´ III & Marcu, 2006), cross-category e learning (Raina et al., 2006), and self-taught learning

(Raina et al., 2007), with the task graph representation. For many other transfer learning problems, the task graphs can be constructed in similar ways, although they will not be modeled in this paper due to a lack of space. 3.1. Cross-Domain Learning As shown in Figure 1(a), in the task graph G(V, E) (i) for cross-domain learning, the target instances xt , (i) (i) the auxiliary instances xa , the test instances xa , the features f (i) and the class labels c(i) are used as the nodes. We have V = Xt  Xa  Xu  F  C. (1) The edges E are based on the co-occurrences between any two end nodes in the target training data, the auxiliary data and the target test data. Let eij denote the weight of the edge between the two nodes vi and vj . We have   vi ,vj vi  Xt  Xa  Xu  vj  F     vj ,vi vi  F  vj  Xt  Xa  Xu    vi  Xt  vj  C  C(vi ) = vj  1 1 vi  C  vj  Xt  C(vj ) = vi eij = (2)   1 vi  Xa  vj  C  C(vi ) = vj     1 vi  C  vj  Xa  C(vj ) = vi    0 otherwise where x,f represents the importance of the feature f  F appearing in the instance x  Xt  Xa  Xu , and

EigenTransfer: A Unified Framework for Transfer Learning

C(x) means the true label of the instance x. 3.2. Cross-Category Learning Since the problem definition of cross-category learning is different from that of cross-domain learning, its corresponding graph differs consequently. However, the construction of the graph for cross-category learning is still similar as for cross-domain learning. Specifically, we construct a graph G(V, E) to represent the problem of cross-category learning as shown in Figure 1(b), where V = Xt  Xa  Xu  F  Ct  Ca , and   vi ,vj     vj ,vi     1 1 eij =   1     1    0 vi  X t  X a  X u  v j  F vi  F  v j  X t  X a  X u vi  Xt  vj  Ct  C(vi ) = vj vi  Ct  vj  Xt  C(vj ) = vi vi  Xa  vj  Ca  C(vi ) = vj vi  Ca  vj  Xa  C(vj ) = vi otherwise (3)

to form an eigen feature representation. In this work, we use the normalized cut (Shi & Malik, 2000) technique to learn the new feature representation to enable transfer learning. 4.1. Normalized Cut Given a transfer learning task, based on the graph construction in Section 3, we can obtain a task graph G(V, E) which reflects the information about the transfer learning task, consisting of the target data, the auxiliary data, and the relations between the two kinds of data. Then, we have an adjacency matrix W  Rn×n with respect to the graph G(V, E)   w11 . . . w1n  . .  , where w = e . (7) .. .  W = . ij ij . . . wn1 ... wnn Let D = (dij )1i,jn be the diagonal matrix with respect to the adjacency matrix W . We have dij =
n t=1

(4)

wit

0

i=j . i=j

(8)

where x,f represents the importance of the feature f  F appearing in the instance x  Xt  Xa  Xu , and C(x) means the true label of the instance x. 3.3. Self-taught Learning The only difference between the graph construction of self-taught learning and that of cross-category learning is, in self-taught learning, the nodes with respect to the (i) auxiliary labels ca are removed, as shown in Figure 1(c), since the auxiliary data are all unlabeled in selftaught learning. Let G(V, E) be the task graph for self-taught learning. Then, V = Xt  Xa  Xu  F  Ct . and   vi ,vj    vj ,vi  1 eij =   1    0 vi  X t  X a  X u  v j  F vi  F  v j  X t  X a  X u vi  Xt  vj  Ct  C(vi ) = vj vi  Ct  vj  Xt  C(vj ) = vi otherwise (5)

Then, the graph Laplacian of G can be calculated as follows. L = D - W. (9)

The normalized cut (Shi & Malik, 2000) algorithm calculates the first N eigenvectors v1 , . . . , vN of the generalized eigenproblem Lv = Dv. (10)

(6)

The first N eigenvectors v1 , . . . , vN form a new feature representation. Under the new feature representation, traditional learners such as support vector machines (SVM) (Boser et al., 1992) can be used to train classifiers based on the target training data Xt to predict the labels of target test data Xu . Since the eigen feature representation takes into account the entire transfer learning task, traditional learners under the eigen feature representation are wrapped into transfer learners. The detailed description for our algorithm EigenTransfer is presented in Algorithm 1. In our algorithm, the non-zero occurrences in L is linear to the input, and thus L is usually very sparse. Moreover, L is real, symmetric and positive semi-definite. To solve the eigenproblem for this kind of matrices, we used the Lanczos algorithm (Saad, 1992) in this work. Generally, the computational cost of Lanczos algorithm is linear to the input, and thus our algorithm EigenTransfer is efficient and scalable.

where x,f represents the importance of the feature f  F appearing in the instance x  Xt  Xa  Xu , and C(x) means the true label of the instance x.

4. Learning Graph Spectra
After the graph construction for transfer learning tasks, we need to learn the spectra of the task graph

EigenTransfer: A Unified Framework for Transfer Learning

Algorithm 1 EigenCluster: a unified framework for transfer learning Input: a target clustering task, including the target (i) data set Xt = {xt }n , the auxiliary data set Xa = i=1 (i) (i) {xa }m , and the test data set Xu = {xu }k . i=1 i=1 Output: classification result on Xu . 1: Construct the task graph G(V, E) based on the target clustering task (c.f. Section 3). 2: Calculate the adjacent matrix W based on the task graph G(V, E). 3: D  diag(W e); L  D - W . 4: Calculate the first N eigenvectors v1 , . . . , vN which satisfy the generalized eigenproblem: Lv = Dv. 5: Let U be the matrix with v1 , . . . , vN as columns. (i) 6: for each xt  Xt do (i) 7: Let yt be the corresponding row in U with re(i) spect to xt . 8: end for (i) 9: Train a classifier based on Yt = {yt }n instead i=1 (i) of Xt = {xt }n using a traditional classification i=1 algorithm, e.g. SVM (Boser et al., 1992), and then (i) classify the test data Xu = {xu }k . i=1

Table 2. The description of the data sets used for evaluating cross-domain learning. Due to the space limitation, we omit some details of sub-directories. Data Set Target Data Auxiliary Data real-aviation real-auto cdl-sraa1 simulated-aviation simulated-auto simulated-auto real-auto cdl-sraa2 simulated-aviation real-aviation cdl-20ng1 rec.*, talk.* rec.*, talk.* cdl-20ng2 rec.*, sci.* rec.*, sci.* cdl-20ng3 comp.*, talk.* comp.*, talk.* cdl-20ng4 comp.*, sci.* comp.*, sci.* cdl-20ng5 comp.*, rec.* comp.*, rec.* cdl-20ng6 sci.*, talk.* sci.*, talk.* cdl-reuters1 orgs.*, places.* orgs.*, places.* cdl-reuters2 people.*, places.* people.*, places.* cdl-reuters3 orgs.*, people.* orgs.*, people.*

5. Experiments
In this section, we empirically evaluate our algorithm on three different transfer learning problems: crossdomain learning (Daum´ III & Marcu, 2006), crosse category learning (Raina et al., 2006), and self-taught learning (Raina et al., 2007). We show that, our framework EigenTransfer can greatly outperform nontransfer learners in general. 5.1. Cross-Domain Learning The first experiment was conducted for cross-domain learning. We used three data sets, SRAA1 , 20 Newsgroups (Lang, 1995) and Reuters-215782 , to generate the cross-domain learning tasks. All three data sets have hierarchical structures. We used the top directories as categories, while splitting the data into target and auxiliary data sets based on sub-directories which we refer to as domains. As a result, the target and auxiliary data share the same categories, i.e. top directories, but they belong to different domains, i.e. sub-directories. The description of the data sets for cross-domain learning is presented in Table 2.
1 2

We evaluated three baseline classifiers, naive Bayes classifier (NBC) (Lewis, 1992), support vector machine (SVM) (Boser et al., 1992), and transductive support vector machine (TSVM) (Joachims, 1999). For each baseline method, we use EigenTransfer to form an eigen feature representation, and then apply the baseline learner to the target data under the eigen feature representation. Therefore, as shown in Table 3, for each baseline method, we have two versions, "NonTransfer" and "EigenTransfer". Here, "Non-Transfer" means to simply apply the baseline method to the original data; "EigenTransfer" means to apply the baseline method to the data in the new feature representation learned by EigenTransfer. The performance in Table 3 is measured in error rate by averaging 10 random repeats on each data set by each evaluation method. For each repeat, we randomly selected 30 instances per category as the target training data, in order to simulate the setting for transfer learning where the target labeled data are few. We also include the standard deviations of the repeats in this table. The negative transfer cases, for when transfer learning lowers the learning performance, have been set to Italic font. From this table, we can see that in most cases, our framework EigenTransfer can perform better than non-transfer learners by a large margin. However, EigenTransfer cannot prevent negative transfer completely, a reasonable fact that is also pointed out in several previous transfer learning literatures such as (Caruana, 1997; Rosenstein et al., 2005). 5.2. Cross-Category Learning In the experiments for cross-category learning, we used two data sets, the 20 Newsgroups (Lang, 1995) data set and the ohscal data set from OHSUMED (Hersh et al., 1994), to generate the tasks.

http://www.cs.umass.edu/~mccallum/code-data.html http://www.daviddlewis.com/resources/testcollections/

EigenTransfer: A Unified Framework for Transfer Learning Table 3. The experimental results in error rate for cross-domain learning. The results are the averages of 10 random repeats, as well as their standard deviations. The results with Italic font means negative transfer. All the methods were well-tuned using 10-fold cross-validation. Data Set cdl-sraa1 cdl-sraa2 cdl-20ng1 cdl-20ng2 cdl-20ng3 cdl-20ng4 cdl-20ng5 cdl-20ng6 cdl-reuters1 cdl-reuters2 cdl-reuters3 average NBC Non-Transfer EigenTransfer 0.271 ± 0.032 0.146 ± 0.021 0.251 ± 0.045 0.170 ± 0.028 0.273 ± 0.068 0.060 ± 0.008 0.150 ± 0.013 0.061 ± 0.020 0.199 ± 0.019 0.079 ± 0.034 0.266 ± 0.025 0.082 ± 0.032 0.180 ± 0.020 0.065 ± 0.027 0.160 ± 0.033 0.050 ± 0.035 0.357 ± 0.049 0.232 ± 0.054 0.342 ± 0.065 0.277 ± 0.058 0.299 ± 0.024 0.249 ± 0.024 0.250 ± 0.036 0.134 ± 0.031 SVM Non-Transfer EigenTransfer 0.205 ± 0.034 0.097 ± 0.023 0.213 ± 0.064 0.059 ± 0.009 0.209 ± 0.034 0.028 ± 0.005 0.145 ± 0.050 0.026 ± 0.016 0.194 ± 0.059 0.030 ± 0.008 0.204 ± 0.038 0.074 ± 0.048 0.119 ± 0.038 0.029 ± 0.005 0.086 ± 0.020 0.013 ± 0.002 0.240 ± 0.020 0.252±0.037 0.218 ± 0.027 0.217 ± 0.028 0.259 ± 0.044 0.219 ± 0.017 0.190 ± 0.039 0.095 ± 0.018 TSVM Non-Transfer EigenTransfer 0.164 ± 0.011 0.063 ± 0.016 0.129 ± 0.014 0.047 ± 0.001 0.201 ± 0.060 0.097 ± 0.004 0.071 ± 0.007 0.061 ± 0.002 0.129 ± 0.129 0.025 ± 0.014 0.113 ± 0.052 0.092 ± 0.075 0.041 ± 0.002 0.022 ± 0.001 0.043 ± 0.002 0.033 ± 0.004 0.201 ± 0.033 0.248±0.040 0.219 ± 0.045 0.202 ± 0.027 0.231 ± 0.067 0.215 ± 0.024 0.140 ± 0.038 0.101 ± 0.019

Table 4. The description of the data sets used for evaluating cross-category learning. Data Set ccl-20ng1 ccl-20ng2 ccl-20ng3 ccl-20ng4 ccl-20ng5 ccl-ohs1 ccl-ohs2 ccl-ohs3 ccl-ohs4 ccl-ohs5 Target Data comp.graphics, rec.sport.baseball comp.sys.ibm.pc.hardware, rec.sport.baseball sci.crypt, talk.politics.guns alt.atheism, sci.space sci.med, talk.politics.mideast Carcinoma, Pregnancy Prognosis, Receptors In-Vitro, Molecular-Sequence-Data Antibodies, DNA DNA, In-Vitro

0.4

cross-domain learning (cdl-20ng5) cross-category learning (ccl-20ng5) self-taught learning (stl-ohs3)

0.3 Error Rate

0.2

0.1

0

20

40

60 80 Number of Eigenvectors

100

120

The 20 Newsgroups data set has 20 categories, and the oshcal data set contain 10 categories. We used the following strategy to generate cross-category learning tasks. For each task, we selected some categories as the target data, including target training data and test data. The other categories in the data set were used as auxiliary labeled data. Table 4 shows the description of the cross-category learning tasks in our experiments. In this table, we only listed the target categories. For each task, the categories that have not been listed in this table were used as the auxiliary ones. Table 5 shows the experimental results for the crosscategory learning when there are 30 labeled data per category in the target training set. The format of this table is the same as that of cross-domain learning (c.f. Table 3 and Section 5.1). From the table, we can see that, in general, EigenTransfer can improve the nontransfer learners when solving cross-category learning problems, resulting in few negative transfer cases. 5.3. Self-taught Learning The experiments for self-taught learning were also conducted on the 20 Newsgroups (Lang, 1995) data set,

Figure 2. The influence of the number of eigenvectors used to form the eigen feature representation.

and the ohscal data set from OHSUMED (Hersh et al., 1994). The only difference from the data sets of crosscategory learning in Section 5.2 is that we removed all the labels from the auxiliary data. Thus, for the details of the data sets used for self-taught learning, we can refer to Table 4. Table 6 shows the experimental results with respect to self-taught learning when there are 30 labeled data per category in the target training set. From this table, we can see that although there are two negative transfer cases, generally, EigenTransfer can greatly outperform non-transfer learners in the situation of self-taught learning as well. 5.4. Eigenvectors In EigenTransfer, there is only one parameter to set which is the number of eigenvectors used to form the eigen-feature representation. In the experiments in Sections 5.1, 5.2 and 5.3, we tuned this number using a 10-fold cross validation. However, we also tested

EigenTransfer: A Unified Framework for Transfer Learning Table 5. The experimental results in error rate for cross-category learning. The results are the averages of 10 random repeats, as well as their standard deviations. The results with Italic font means negative transfer. All the methods were well-tuned using 10-fold cross-validation. Data Set ccl-20ng1 ccl-20ng2 ccl-20ng3 ccl-20ng4 ccl-20ng5 ccl-ohs1 ccl-ohs2 ccl-ohs3 ccl-ohs4 ccl-ohs5 average NBC Non-Transfer EigenTransfer 0.153 ± 0.041 0.037 ± 0.018 0.130 ± 0.054 0.041 ± 0.019 0.181 ± 0.052 0.064 ± 0.022 0.188 ± 0.039 0.056 ± 0.012 0.256 ± 0.052 0.132 ± 0.039 0.204 ± 0.053 0.057 ± 0.012 0.088 ± 0.013 0.052 ± 0.015 0.156 ± 0.020 0.125 ± 0.032 0.286 ± 0.025 0.240 ± 0.033 0.217 ± 0.026 0.184 ± 0.046 0.186 ± 0.038 0.099 ± 0.025 SVM Non-Transfer EigenTransfer 0.120 ± 0.030 0.021 ± 0.012 0.125 ± 0.043 0.020 ± 0.006 0.077 ± 0.024 0.017 ± 0.011 0.090 ± 0.019 0.031 ± 0.011 0.174 ± 0.031 0.084 ± 0.022 0.083 ± 0.019 0.042 ± 0.010 0.094 ± 0.040 0.040 ± 0.008 0.130 ± 0.036 0.080 ± 0.025 0.235 ± 0.047 0.176 ± 0.029 0.186 ± 0.030 0.144 ± 0.026 0.131 ± 0.032 0.065 ± 0.016 TSVM Non-Transfer EigenTransfer 0.045 ± 0.006 0.016 ± 0.009 0.032 ± 0.008 0.012 ± 0.004 0.037 ± 0.005 0.029 ± 0.015 0.060 ± 0.003 0.048 ± 0.015 0.121 ± 0.023 0.080 ± 0.013 0.202 ± 0.001 0.199 ± 0.003 0.066 ± 0.002 0.058 ± 0.001 0.081 ± 0.005 0.082±0.013 0.217 ± 0.039 0.201 ± 0.036 0.179 ± 0.008 0.187 ± 0.017 0.104 ± 0.010 0.091 ± 0.013

Table 6. The experimental results in error rate for self-taught learning. The results are the averages of 10 random repeats, as well as their standard deviations. The results with Italic font means negative transfer cases. All the methods were well-tuned using 10-fold cross-validation. Data Set stl-20ng1 stl-20ng2 stl-20ng3 stl-20ng4 stl-20ng5 stl-ohs1 stl-ohs2 stl-ohs3 stl-ohs4 stl-ohs5 average NBC Non-Transfer EigenTransfer 0.138 ± 0.050 0.039 ± 0.012 0.105 ± 0.016 0.048 ± 0.021 0.193 ± 0.036 0.116 ± 0.048 0.223 ± 0.048 0.052 ± 0.017 0.292 ± 0.042 0.126 ± 0.027 0.180 ± 0.049 0.083 ± 0.046 0.084 ± 0.007 0.063 ± 0.033 0.153 ± 0.015 0.123 ± 0.029 0.298 ± 0.050 0.248 ± 0.055 0.220 ± 0.033 0.170 ± 0.032 0.189 ± 0.035 0.107 ± 0.032 SVM Non-Transfer EigenTransfer 0.112 ± 0.033 0.025 ± 0.018 0.104 ± 0.037 0.018 ± 0.008 0.079 ± 0.021 0.041 ± 0.010 0.111 ± 0.034 0.026 ± 0.014 0.178 ± 0.031 0.092 ± 0.017 0.066 ± 0.022 0.061 ± 0.026 0.082 ± 0.032 0.035 ± 0.004 0.132 ± 0.018 0.082 ± 0.019 0.209 ± 0.038 0.189 ± 0.041 0.189 ± 0.031 0.133 ± 0.018 0.126 ± 0.030 0.070 ± 0.017 TSVM Non-Transfer EigenTransfer 0.041 ± 0.005 0.014 ± 0.005 0.032 ± 0.006 0.015 ± 0.004 0.039 ± 0.004 0.070±0.050 0.061 ± 0.004 0.045 ± 0.006 0.133 ± 0.033 0.113 ± 0.105 0.203 ± 0.001 0.200 ± 0.005 0.065 ± 0.002 0.056 ± 0.005 0.080 ± 0.004 0.085±0.019 0.230 ± 0.027 0.208 ± 0.023 0.180 ± 0.021 0.179 ± 0.020 0.106 ± 0.011 0.098 ± 0.024

the sensitivity of this parameter. In Figure 2, we varied the number of eigenvectors and tested the performance of EigenTransfer on three data sets, cdl-20ng5, ccl-20ng5 and stl-ohs3. From the figure, we can see that when the number of eigenvectors is greater than 60, the performance of EigenTransfer is very stable. In our experiments, based on cross validation, the number of eigenvectors is mostly set to the values between 40 to 120. 5.5. Labeled Target Data In Sections 5.1, 5.2 and 5.3, we have shown the results when there are 30 labeled data per target category. However, we also want to see the influence of different sizes of target training data. In Figure 3, we varied the number of labeled target training data instances per category from 10 to 100, and show the performance of EigenTransfer on three data sets cdl-20ng5, ccl-20ng5 and stl-ohs3, respectively. From the figure, we can see EigenTransfer always outperforms the non-transfer learner SVM. The less the target training data are, the more improvement EigenTransfer can achieve. More-

over, EigenTransfer achieves the most significantly improvements on cross-domain learning, and obtains the least significantly improvements on self-taught learning. We believe this is because the quality of auxiliary data in cross-domain learning is better than that in cross-category learning, which is better than that in self-taught learning.

6. Conclusions and Discussions
In this paper, we proposed a general transfer learning framework, which can model a variety of existing transfer learning problems and solutions. In our framework, first, a task graph is constructed to represent the transfer learning task. Then, we learn an eigen feature representation from the task graph based on spectral learning theory (Chung, 1997). Under the new feature representation, the knowledge from the auxiliary data tends to be transferred to help the target learning. Our experimental results show that the EigenTransfer framework can unify a number of well known transfer learning problems, can be used to wrap any

EigenTransfer: A Unified Framework for Transfer Learning
0.4 SVM EigenTransfer(SVM) 0.3 0.3 Error Rate Error Rate 0.2 Error Rate 0.4 SVM EigenTransfer(SVM) 0.3 0.4 SVM EigenTransfer(SVM)

0.2

0.2

0.1

0.1 0.1

0 0

20 40 60 80 100 Number of Labeled Data per Category

0 0

20

40 60 80 Number of Eigenvectors

100

0

20

40 60 80 Number of Eigenvectors

100

(a)

(b)

(c)

Figure 3. The influence of different number of target training data: (a) cross-domain learning on the cdl-20ng5 data set; (b) cross-category learning on the ccl-20ng5 data set; (c) self-taught learning on the stl-ohs3 data set.

non-transfer learning algorithms for a transfer learning task that we consider, and can greatly outperform non-transfer learners in many experiments. In our work, the task graph is constructed to fully reflect the transfer learning task, including target data, auxiliary data, and their commonness. There is little information loss during the graph construction. The graph spectra can represent the intrinsic structure of the task graph and the transfer learning task. This is the rationale of EigenTransfer being an effective transfer learning approach. Acknowledgement Wenyuan Dai and Gui-Rong Xue are supported by the grants from National Natural Science Foundation of China (NO. 60873211). We also thank the anonymous reviewers for their greatly helpful comments.

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