Visual Domain Adaptation with Manifold Embedded
Distribution Alignment∗
Jindong Wang, Wenjie Feng, Yiqiang Chen†

Han Yu‡Institute of Computing Technology, CAS, Beijing, China{wangjindong,yqchen}@ict.ac.cn
fengwenjie@software.ict.ac.cn

School of Computer Science and Engineering, NTU
Singapore
han.yu@ntu.edu.sg

Meiyu Huang

Philip S. Yu§

Qian Xuesen Lab. of Space Technology, China Academy ofSpace Technology, Beijing, China
huangmeiyu@qxslab.cn

Department of Computer Science, UIC
Chicago, USA
psyu@uic.edu

ABSTRACTKEYWORDS

Visual domain adaptation aims to learn robust classifiers for thetarget domain by leveraging knowledge from a source domain. Existing methods either attempt to align the cross-domain distributions,or perform manifold subspace learning. However, there are two
significant challenges: (1) degenerated feature transformation, whichmeans that distribution alignment is often performed in the original
feature space, where feature distortions are hard to overcome. On
the other hand, subspace learning is not sufficient to reduce the distribution divergence. (2) unevaluated distribution alignment, whichmeans that existing distribution alignment methods only alignthe marginal and conditional distributions with equal importance,
while they fail to evaluate the different importance of these two
distributions in real applications. In this paper, we propose a Manifold Embedded Distribution Alignment (MEDA) approach toaddress these challenges. MEDA learns a domain-invariant classifier in Grassmann manifold with structural risk minimization,
while performing dynamic distribution alignment to quantitatively
account for the relative importance of marginal and conditional distributions. To the best of our knowledge, MEDA is the first attempt
to perform dynamic distribution alignment for manifold domain
adaptation. Extensive experiments demonstrate that MEDA shows
significant improvements in classification accuracy compared tostate-of-the-art traditional and deep methods.

Domain Adaptation, Transfer Learning, Distribution Alignment,
Subspace Learning∗ The first two authors contributed equally.
† Corr. to Y.Chen. J. Wang and Y. Chen are also affiliated with Beijing Key Lab. of Mobile

Computing and Pervasive Devices. W. Feng is also with CAS Key Lab. of Network
Data Science & Technology. J. Wang and W. Feng are also with University of Chinese
Academy of Sciences, Beijing, China.
‡ H. Yu is also with the Joint NTU-UBC Research Centre of Excellence in Active Living
for the Elderly (LILY), and the Alibaba-NTU Singapore Joint Research Institute.
§ P. Yu is also affiliated with Institute for Data Science, Tsinghua University, Beijing,
China.
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MM ’18, Oct. 22–26, 2018, Seoul, Republic of Korea© 2018 Association for Computing Machinery.
ACM ISBN 978-1-4503-5665-7/18/10. . . $15.00
https://doi.org/10.1145/3240508.3240512

ACM Reference format:
Jindong Wang, Wenjie Feng, Yiqiang Chen, Han Yu, Meiyu Huang, and Philip
S. Yu. 2018. Visual Domain Adaptation with Manifold Embedded Distribution Alignment. In Proceedings of 2018 ACM Multimedia Conference, Seoul,
Republic of Korea, Oct. 22–26, 2018 (MM ’18), 9 pages.
https://doi.org/10.1145/3240508.3240512

1

INTRODUCTION

The rapid growth of online media and content sharing applications has stimulated a great demand for automatic recognition
and analysis for images and other multimedia data [8, 20]. Unfortunately, it is often expensive and time-consuming to acquiresufficient labeled data to train machine learning models. Thus, it isoften necessary to leverage the abundant labeled samples in someexisting domains to facilitate learning in a new target domain. Domain adaptation [27, 36] has been a promising approach to solve
such cross-domain learning problems.
Since the distributions of the source and target domains aredifferent, the key to successful adaptation is to reduce the distribution divergence. To this end, existing work can be summarized
into two main categories: (a) instance reweighting [9, 39], whichreuses samples from the source domain according to some weighting technique; and (b) feature matching, which either performs
subspace learning by exploiting the subspace geometrical structure [13, 15, 30], or distribution alignment to reduce the marginal orconditional distribution divergence between domains [23, 40]. Our
focus is on feature matching methods. There are two significant
challenges in existing methods, i.e. degenerated feature transformation and unevaluated distribution alignment.
Degenerated feature transformation means that both subspacelearning and distribution alignment can only reduce, but not remove
the distribution divergence [1]. Specifically, subspace learning [13,15, 30] conducts subspace transformation to obtain better feature
representations. However, feature divergence is not eliminated after
subspace transformation [22] since subspace learning only utilizes
the subspace or manifold structure, but fails to perform featurealignment. On the other hand, distribution alignment [23, 26, 37]濼瀁濴濿澳瀆瀂瀈瀅濶濸澳濴瀁濷澳濸瀇澳濷瀂瀀濴濼瀁the local property of subspaces and ignored conditional distribution alignment. CORAL [30] aligned subspaces in second-order
statistics, but it did not consider the distribution alignment. Scattercomponent analysis (SCA) [14] converted the samples into a set ofsubspaces (i.e. scatters) and then minimized the divergence between瀆瀂瀈瀅濶濸
濢瀅濼濺濼瀁濴濿澳瀆瀂瀈瀅濶濸澳濴瀁濷澳瀀濴瀅濺濼瀁濴濿
濣澻瀋濲瀆澼澳濴瀁濷澳濣澻瀋濲瀇澼澳濼瀆澳濶濿瀂瀆濸澿澳濣澻瀋濲瀆澼澳濴瀁濷澳濣澻瀋濲瀇澼澳濼瀆澳濶濿瀂瀆濸澿澳濶瀂瀁濷濼瀇濼瀂瀁濴濿濶瀂瀁濷濼瀇濼瀂瀁濴濿(a) Source(b)Target: Type I 濶瀂瀁濷濼瀇濼瀂瀁濴濿(c) Target:
Type II瀇濴瀅濺濸瀇澳濷瀂瀀濴濼瀁濣澻瀌瀏瀋濲瀆澼澳濴瀁濷澳濣澻瀌瀏瀋濲瀇澼澳濼瀆澳濹濴瀅濣澻瀌瀏瀋濲瀆澼澳濴瀁濷澳濣澻瀌瀏瀋濲瀇澼澳濼瀆澳濹濴瀅
them. GFK [15] extended the idea of sampled points in manifold [16]and proposed to learn the geodesic flow kernel between domains.
Figure 1: Examples of two different target domains w.r.t. theThework of [4] used a Hellinger distance to approximate the geosame source domain during distribution adaption.
desic distance in Riemann space. [3] proposed to use Grassmann fordomain adaptation, but they ignored the conditional distribution
usually reduces the distribution distance in the original featurealignment. Different from these approaches, MEDA can learn aspace, where features are often distorted [3] which makes it hard todomain-invariant classifier in the manifold and align both marginal
reduce the divergence between domains. Therefore, it is critical toand conditional distributions.
exploit both the advantages of subspace learning and distribution
Distribution alignment. MEDA substantially differs from existingalignment to further facilitate domain adaptation.
work that only align marginal or conditional distribution [26]. Joint
Unevaluated distribution alignment means that existing work [19,distribution adaptation (JDA) [23] proposed to match both distribu23, 33, 40] only attempted to align the marginal and conditionaltions with equal weights. Others extended JDA by adding regulardistributions with equal weights. But they failed to evaluate the relization [22], sparse representation [38], structural consistency [19],ative importance of these two distributions. For example, when two
domain invariant clustering [33], and label propagation [40]. The
domains are very dissimilar (Figure 1(a) → 1(b)), the marginal distrimain differences between MEDA and these methods are: 1) Thesebution is more important to align. When the marginal distributions
work treats the two distributions equally. However, when there
are close (Figure 1(a) → 1(c)), the conditional distribution should
is a greater discrepancy between both distributions, they cannot
be given more weight. However, there is no alignment method
evaluate their relative importance and thus lead to underminedwhich can quantitatively account for the importance of these two
performance. Our work is capable of evaluating the quantitative imdistributions in conjunction.
portance of each distribution via considering their different effects.
As far as we know, there has been no previous work that tackle
2) These methods are designed only in the original space, wherethese two challenges together. In this paper, we propose a novel
feature distortion will hinder the performance. MEDA can align the
Manifold Embedded Distribution Alignment (MEDA) method
distributions in the manifold to overcome the feature distortions.
to address the challenges of both degenerated feature transforDomain-invariant classifier learning. The recent work of ARTL [22],mation and unevaluated distribution alignment. MEDA learns a
DIP [2, 3], and DMM [7] also aimed to build a domain-invariant
domain-invariant classifier in Grassmann manifold with structuralclassifier. However, ARTL and DMM can be undermined by fearisk minimization, while performing dynamic distribution alignture distortion in original space, and they failed to leverage thement by considering the different importance of marginal anddifferent importance of distributions. DIP mainly focused on feature
conditional distributions. We also provide a feasible solution totransformation and only aligned marginal distributions. MEDA isquantitatively evaluate the importance of distributions. To our best
able to avoid the feature distortion and quantitatively evaluate theknowledge, MEDA is the first attempt to reveal the relative imporimportance of marginal and conditional distribution alignment.
tance of marginal and conditional distributions in transfer learning.
This work makes the following contributions:
1) We propose the MEDA approach for domain adaptation. MEDA
3 MANIFOLD EMBEDDED DISTRIBUTION
is capable of addressing both the challenges of degenerated feature
ALIGNMENT
transformation and unevaluated distribution alignment.
2) We propose the first quantitative evaluation of the relative
3.1 Problem Definition
importance of marginal and conditional distributions in domainn and an unlabeled
Given a labeled source domain Ds = {xsi , ysi }i=1
adaptation. This is significantly useful in future research on transfer
n+m , assume the feature space X = X ,targetdomainD={x}tt j j=n+1
stlearning.
label space Ys = Yt , but marginal probability Ps (xs ) , Pt (xt ) with
3) Extensive experiments on 7 real-world image datasets demonconditional probability Q s (ys |xs ) , Q t (yt |xt ). The goal of domain
strate that compared to several state-of-the-art traditional and deep
adaptation is to learn a classifier f : xt 7→ yt to predict the labels
methods, MEDA achieves a significant improvement of 3.5% in
yt ∈ Yt for the target domain Dt using labeled source domain Ds .
average classification accuracy.
According to the structural risk minimization (SRM) [35], f =argminf ∈HK ℓ(f (x), y) + R(f ), where the first term indicates the
2 RELATED WORK
loss on data samples, the second term denotes the regularization
MEDA substantially distinguishes from existing feature matching
term, and HK is the Hilbert space induced by kernel function K(·, ·).
domain adaptation methods in several aspects:
Since there is no labels on Dt , we can only perform SRM on Ds .
Subspace learning. Subspace Alignment (SA) [13] aligned the baseMoreover, due to the different distributions between Ds and Dt , itvectors of both domains, but failed to adapt feature distributions.
is necessary to add other constraints to maximize the distribution
Subspace distribution alignment (SDA) [31] extended SA by adding
consistency while learning f .
the subspace variance adaptation. However, SDA did not consider

Source12Target1 Features in the original
Figure 2: The main idea of MEDA. ○space are transformed into manifold space by learning the
2 Dynamic distribution alignment (bymanifold kernel G. ○learning µ) with SRM is performed in manifold to learn the
final domain-invariant classifier f .

3.2

Main Idea

MEDA consists of two fundamental steps. Firstly, MEDA performs
manifold feature learning to address the challenge of degenerated
feature transformation. Secondly, MEDA performs dynamic distribution alignment to quantitatively account for the relative importanceof marginal and conditional distributions to address the challenge of
unevaluated distribution alignment. Eventually, a domain-invariant
classifier f can be learned by summarizing these two steps with theprinciple of SRM. Figure 2 presents the main idea of the proposed
MEDA approach.
Formally, if we denote д(·) the manifold feature learning functional, then f can be represented as2
f = arg
min ℓ(f (д(xi )), yi ) + η|| f ||KÍf∈ni =1 HK(1)+ λD f (Ds , Dt ) + ρR f (Ds , Dt )2 is the squared norm of f . The term D (·, ·) represents
where || f ||Kfthe proposed dynamic distribution alignment. Additionally, we
introduce R f (·, ·) as a Laplacian regularization to further exploit
the similar geometrical property of nearest points in manifold G [5].
η, λ, and ρ are regularization parameters accordingly.
The overall learning process of MEDA is in Algorithm 1. In
next sections, we first introduce manifold feature learning (learn
д(·)). Then, we present the dynamic distribution alignment (learn
D f (·, ·)). Eventually, we articulate the learning of f .

3.3

Manifold Feature Learning

Manifold feature learning serves as the preprocessing step to eliminate the threat of degenerated feature transformation. MEDA learnsд(·) in the Grassmann manifold G(d) [18] since features in the
manifold have some geometrical structures [5, 18] that can avoid
distortion in the original space. And G can facilitate classifier learning by treating the original d-dimensional subspace (i.e. feature
vector) as its basic element [4]. Additionally, feature transformationand distribution alignment often have efficient numerical forms
and can thus facilitate domain adaptation on G(d) [18]. There areseveral approaches to transform the features into G [4, 16], among
which we embed Geodesic Flow Kernel (GFK) [15] to learn д(·) for
its computational efficiency. We only introduce the main idea of
GFK and the details can be found in its original paper.
When learning manifold features, MEDA tries to model the domains with d-dimensional subspaces and then embed them into

G. Let Ss and St denote the PCA subspaces for the source andtarget domain, respectively. G can thus be regarded as a collectionof all d-dimensional subspaces. Each original subspace can be seen
as a point in G. Therefore, the geodesic flow {Φ(t) : 0 ≤ t ≤ 1}
between two points can draw a path for the two subspaces. If welet Ss = Φ(0) and St = Φ(1), then finding a geodesic flow fromΦ(0) to Φ(1) equals to transforming the original features into an
infinite-dimensional feature space, which eventually eliminates thedomain shift. This kind of approach can be seen as an incremental
way of ‘walking’ from Φ(0) to Φ(1). Specifically, the new features
can be represented as z = д(x) = Φ(t)T x. From [15], the innerproduct of transformed features zi and zj gives rise to a positive
semidefinite geodesic flow kernel:∫ 1⟨zi , zj ⟩ =(Φ(t)T xi )T (Φ(t)T xj ) dt = xTi Gxj(2)0Thus, the feature in original space√ can be transformed into Grassmann manifold with z = д(x) = Gx. G can be computed
efficiently√
by singular value decomposition [15]. Note that G is only an expression form and cannot be computed directly, while its squareroot is calculated by Denman-Beavers algorithm [10].

3.4

Dynamic Distribution Alignment

The purpose of dynamic distribution alignment is to quantitatively
evaluate the importance of aligning marginal (P) and conditional (Q)distributions in domain adaptation. Existing methods [23, 40] failed
in this evaluation by only assuming that both distributions areequally important. However, this assumption may not be realistic
for real applications. For instance, when transferring from Figure1(a) to 1(b), there is a large difference between datasets. Therefore,
the divergence between Ps and Pt is more dominant. In contrast,from Figure 1(a) to 1(c), the datasets are similar. Therefore, thedistribution divergence in each class (Q s and Q t ) is more dominant.
The adaptive factor:
In view of this phenomenon, we introduce an adaptive factor
to dynamically leverage the importance of these two distributions.
Formally, the dynamic distribution alignment D f is defined as
D f (Ds , Dt ) = (1 − µ)D f (Ps , Pt ) + µ

C
Õ
c=1

(c)

D f (Q s , Q t )(3)where µ ∈ [0, 1] is the adaptive factor and c ∈ {1, · · · , C} is the classindicator. D f (Ps , Pt ) denotes the marginal distribution alignment,(c)and D f (Q s , Q t ) denotes the conditional distribution alignment forclass c.
When µ → 0, it means that the distribution distance between thesource and the target domains is large. Thus, marginal distributionalignment is more important (Figure 1(a) → 1(b)). When µ → 1,
it means that feature distribution between domains is relatively
small, so the distribution of each class is dominant. Thus, the conditional distribution alignment is more important (Figure 1(a) → 1(c)).
When µ = 0.5, both distributions are treated equally as in existing
methods [23, 40]. Hence, the existing methods can be regarded as
the special cases of MEDA. By learning the optimal adaptive factorµopt (which we will discuss later), MEDA can be applied to different
domain adaptation problems.

We use the maximum mean discrepancy (MMD) [6] to empirically calculate the distribution divergence between domains. As
a nonparametric measurement, MMD has been widely applied inmany existing methods [14, 26, 40], and its theoretical effectivenesshas been verified in [17]. The MMD distance between distributions p and q is defined as d 2 (p, q) = (Ep [ϕ(zs )] − Eq [ϕ(zt )])2H
K
where HK is the reproducing kernel Hilbert space (RKHS) inducedby feature map ϕ(·). Here, E[·] denotes the mean of the embedded samples. In order to compute an MMD associated with f ,we adopt projected MMD [28] and compute the marginal distri2 . Simbution alignment as D f (Ps , Pt ) = ∥E[f (zs )] − E[f (zt )]∥ H
(c)

(c)

2 . Then, dynamic distribution alignment∥E[f (zs )] − E[f (zt )]∥ H
K
can be expressed as2
D f (Ds , Dt ) =(1 − µ)∥E[f (zs )) − E[f (zt )]∥ H

+µ

C
Õ
c=1

(c)

(c)

Learning Classifier f

After manifold feature learning and dynamic distribution alignment,
f can be learned by summarizing SRM over Ds and distributionalignment. Adopting the square loss l 2 , f can be represented as
f = arg minnÕ2(yi − f (zi ))2 + η|| f ||Kf ∈HK i=1

(7)

+ λD f (Ds , Dt ) + ρR f (Ds , Dt )In order to perform efficient learning, we now reformulate eachterm in detail.
SRM on the Source Domain: Using the representer theorem [5],f admits the expansion

K

2∥E[f (zs )] − E[f (zt )]∥ H
K

f (z) =

(4)

n+m
Õ

βi K(zi , z)

(8)

i=1

Note that since Dt has no labels, it is not feasible to evaluate
the conditional distribution Q t = Q t (yt |zt ). Instead, we follow theidea in [37] and use the class conditional distribution Q t (zt |yt ) toapproximate Q t . In order to evaluate Q t (zt |yt ), we apply prediction
to Dt using a base classifier trained on Ds to obtain soft labels
for Dt . The soft labels may be less reliable, so we iteratively refine
the prediction. Note that we only use the base classifier in the first
iteration. After that, MEDA can automatically refine the labels for
Dt using results from previous iteration.
The quantitative evaluation of the adaptive factor µ:
We can treat µ as a parameter and tune its value by crossvalidation techniques. However, there is no labels for the target domain in unsupervised domain adaptation problems. It is extremely
hard to calculate the value of µ. In this work, we made the firstattempt towards calculating µ (i.e. µ̂) by exploiting the global andlocal structure of domains. We adopted the A-distance [6] as the basic measurement. The A-distance is defined as the error of building
a linear classifier to discriminate two domains (i.e. a binary classification). Formally, we denote ϵ(h) the error of a linear classifier hdiscriminating the two domains Ds and Dt . Then, the A-distancecan be defined asd A (Ds , Dt ) = 2(1 − 2ϵ(h))

3.5Kilarly, the conditional distribution alignment is D f (Q s , Q t ) =(c)be of significant help in future research on transfer learning anddomain adaptation.

(5)

We can directly compute the marginal A-distance using above
equation, which is denoted as d M . For the A-distance between
conditional distributions, we denote dc as the A-distance for the
(c)
(c)
(c)
cth class. It can be calculated as dc = d A (Ds , Dt ), where Ds

where β = (β 1 , β 2 , · · · )T ∈ R(n+m)×1 is the coefficients vector and
K is a kernel. Then, SRM on Ds can benÕ2(yi − f (zi ))2 + η|| f ||K
i=1
n+m
Õ

=

i=1

(9)

= ||(Y − β T K)A||F2 + ηtr(β T Kβ)
where ||·||F is the Frobenious norm. K ∈ R(n+m)×(n+m) is the kernel
matrix with Ki j = K(zi , zj ), and A ∈ R(n+m)×(n+m) is a diagonal
domain indicator matrix with Aii = 1 if i ∈ Ds , otherwise Aii = 0.
Y = [y1 , · · · , yn+m ] is the label matrix from source and the target
domains. tr(·) denotes the trace operation. Although the labels for
Dt are unavailable, they can be filtered out by the indicator matrix
A.
Dynamic distribution alignment: Using the representer theorem and kernel tricks, dynamic distribution alignment in equation (4) becomes
D f (Ds , Dt ) = tr β T KMKβ(10)where M = (1 − µ)M0 + µ
element calculated byÍC

c=1 Mc is the MMD matrix with its

1 ,zi , zj ∈ Ds
 n2(M0 )i j = m12 ,zi , zj ∈ Dt
 − 1 , otherwise
 mn

(c)
and Dt denote samples from class c in Ds and Dt , respectively.

Eventually, µ can be estimated asdMµ̂ ≈ 1 −(6)Í
dM + C
c=1 dcThis estimation has to be conducted at every iteration of the
dynamic distribution adaptation, since the feature distribution mayvary after evaluating the conditional distribution each time. To
be noticed, this is the first solution to quantitatively estimate therelative importance of each distribution. In fact, this estimation can

2
Aii (yi − f (zi ))2 + η|| f ||K(Mc )i j =

(c)


1 ,



nc2

 1


,

 mc2


(11)

(c)

zi , zj ∈ Ds(c)zi , zj ∈ Dt
(
(c)
(c)
zi ∈ Ds , zj ∈ Dt
− mc1nc ,

(c)
(c)


zi ∈ Dt , zj ∈ Ds



 0,
otherwise(c)where nc = |Ds | and mc = |Dt |.

(12)

Table 1: Statistics of the seven benchmark datasets.
Algorithm 1 Manifold Embedded Distribution Alignment
Input: Data matrix X = [Xs , Xt ], source domain labels ys , manifold subspace dimension d, regularization parameters λ, η, ρ,
and #neighbor p.
Output: Classifier f .
1: Learn manifold feature transformation kernel G via equa√tion (2), and get manifold feature Z = GX.
2: Train a base classifier using Ds , then apply prediction on Dt
to get its soft labels ŷt .
3: Construct kernel K using transformed features Zs = Z1:n,: andZt = Zn+1:n+m,: .
4: repeat5:
Calculate the adaptive factor µ̂ using equation (6). and compute M0 and Mc by equations (11) and (12).
6:Compute β ⋆ by solving equation (16) and obtain f via therepresenter theorem in equation (8).
7:Update the soft labels of Dt : ŷt = f (Zt ).
8: until Convergence
9: return Classifier f .

Laplacian Regularization: Additionally, we add a Laplacian
regularization term to further exploit the similar geometrical property of nearest points in manifold G [5]. We denote the pair-wiseaffinity matrix as(sim(zi , zj ), zi ∈ Np (zj ) or zj ∈ Np (zi )Wi j =(13)0,otherwise
where sim(·, ·) is a similarity function (such as cosine distance) tomeasure the distance between two points. Np (zi ) denotes the setof p-nearest neighbors to point zi . p is a free parameter and must
be set in the method. By introducing Laplacian matrix L = D − WÍwith diagonal matrix Dii = n+m
j=1 Wi j , the final regularization canbe expressed by
R f (Ds , Dt ) ==n+m
Õi, j=1n+m
Õ

Wi j (f (zi ) − f (zj ))2f (zi )Li j f (zj )(14)i, j=1= tr β T KLKβOverall Reformulation: Substituting with equations (9), (10)and (14), f in equation (7) can be reformulated as
f = arg min ||(Y − β T K)A||F2 + η tr(β T Kβ)
f ∈HK



+ tr β T K(λM + ρL)Kβ#Sample#Feature#Class

Domain

1,410
1,123
4,652
1,800
2,000
7,341
3,376

800 (4,096)
800 (4,096)
4,096
256
256
4,096
4,096

10
10
31
10
10
5
5

A, W, DC
A, W, DUSPS (U)MNIST (M)ImageNet (I)VOC (V)

4

(15)

(16)

MEDA has a nice property: it can learn the cross-domain function
directly without the need of explicit classifier training. This makes
it significantly different from most existing work such as JGSA [40]and CORAL [30] that further needs to learn a certain classifier.

EXPERIMENTS AND EVALUATIONSIn this section, we evaluate the performance of MEDA throughextensive experiments on large-scale public datasets. The sourcecode for MEDA is available at http://transferlearning.xyz/.

4.1

Data Preparation

We adopted seven publicly image datasets: Office+Caltech10, USPS+ MNIST, ImageNet + VOC2007, and Office-31. These datasets are
popular for benchmarking domain adaptation algorithms and havebeen widely adopted in most existing work such as [15, 22, 40, 41].
Table 1 lists the statistics of the seven datasets.
Office-31 [29] consists of three real-world object domains: Amazon (A), Webcam (W) and DSLR (D). It has 4,652 images with 31
categories. Caltech-256 (C) contains 30,607 images and 256 categories. Since the objects in Office and Caltech follow different distributions, domain adaptation can help to perform cross-domain recognition. There are 10 common classes in the two datasets. For our
experiments, we adopted the Office+Caltech10 datasets from [15]which contains 12 tasks: A → D, A → C,..., C → W. In the rest of
the paper, we use A → B to denote the knowledge transfer fromsource domain A to the target domain B.
USPS (U) and MNIST (M) are standard digit recognition datasets
containing handwritten digits from 0-9. Since the same digits across
two datasets follow different distributions, it is necessary to performdomain adaptation. USPS consists of 7,291 training images and 2,007
test images of size 16 × 16. MNIST consists of 60,000 training images
and 10,000 test images of size 28 × 28. We construct two tasks: U→ M and M → U.
ImageNet (I) and VOC2007 (V) are large standard image recognition datasets. Each dataset can be treated as one domain. The
images from the same classes of two domains follow different distributions. In our experiments, we adopt the sub-datasets presented
in [12] to construct cross-domain tasks. Five common classes are
extracted from both datasets: bird, cat, chair, dog, and person. Eventually, we have two tasks: I → V and V → I.

4.2

Setting derivative ∂ f /∂β = 0, we obtain the solutionβ ⋆ = ((A + λM + ρL)K + ηI)−1 AYTDatasetOffice-10
Caltech-10
Office-31
USPS
MNIST
ImageNet
VOC2007

State-of-the-art Comparison Methods

We compared the performance of MEDA with several state-of-theart traditional and deep domain adaptation approaches.
Traditional learning methods:• 1NN, SVM, and PCA• Transfer Component Analysis (TCA) [26], which performs
marginal distribution alignment• Geodesic Flow Kernel (GFK) [15], which performs manifold
feature learning• Joint distribution alignment (JDA) [23], which adapts bothmarginal and conditional distribution

Table 2: Accuracy (%) on Office+Caltech10 datasets using SURF features.
Task

1NN

SVM

PCA

TCA

GFK

JDA

TJM

CORAL

SCA

ARTL

JGSA

MEDA

C→A
C→W
C→D
A→C
A→W
A→D
W→C
W→A
W→D
D→C
D→A
D→W

23.7
25.8
25.5
26.0
29.8
25.5
19.9
23.0
59.2
26.3
28.5
63.4

53.1
41.7
47.8
41.7
31.9
44.6
28.8
27.6
78.3
26.4
26.2
52.5

39.5
34.6
44.6
39.0
35.9
33.8
28.2
29.1
89.2
29.7
33.2
86.1

45.6
39.3
45.9
42.0
40.0
35.7
31.5
30.5
91.1
33.0
32.8
87.5

46.0
37.0
40.8
40.7
37.0
40.1
24.8
27.6
85.4
29.3
28.7
80.3

43.1
39.3
49.0
40.9
38.0
42.0
33.0
29.8
92.4
31.2
33.4
89.2

46.8
39.0
44.6
39.5
42.0
45.2
30.2
30.0
89.2
31.4
32.8
85.4

52.1
46.4
45.9
45.1
44.4
39.5
33.7
36.0
86.6
33.8
37.7
84.7

45.6
40.0
47.1
39.7
34.9
39.5
31.1
30.0
87.3
30.7
31.6
84.4

44.1
31.5
39.5
36.1
33.6
36.9
29.7
38.3
87.9
30.5
34.9
88.5

51.5
45.4
45.9
41.5
45.8
47.1
33.2
39.9
90.5
29.9
38.0
91.9

56.5
53.9
50.3
43.9
53.2
45.9
34.0
42.7
88.5
34.9
41.2
87.5

Average

31.4

41.1

43.6

46.2

43.1

46.8

46.3

48.8

45.2

44.3

50.0

52.7

Table 3: Accuracy (%) on USPS+MNIST and ImageNet+VOC2007 datasets.
Task

1NN

SVM

PCA

TCA

GFK

JDA

TJM

CORAL

SCA

ARTL

JGSA

MEDA

U→M
M→U
I→V
V→I

44.7
65.9
50.8
38.2

62.2
68.2
52.4
42.7

45.0
66.2
58.4
65.1

51.2
56.3
63.7
64.9

46.5
61.2
59.5
73.8

59.7
67.3
63.4
70.2

52.3
63.3
63.7
73.0

30.5
49.2
59.6
70.3

48.0
65.1
-

67.7
88.8
62.4
72.2

68.2
80.4
52.3
70.6

72.1
89.5
67.3
74.7

Average

49.9

56.3

58.7

59.0

60.2

65.1

63.1

52.4

-

72.8

67.9

75.9

Table 4: Accuracy (%) on Office+Caltech10 datasets using DeCaf6 features.
1NN

SVM

PCA

TCA

GFK

Traditional Methods
JDA
TJM SCA

ARTL

JGSA

CORAL

DMM

AlexNet

DDC

C→A
C→W
C→D
A→C
A→W
A→D
W→C
W→A
W→D
D→C
D→A
D→W

87.3
72.5
79.6
71.7
68.1
74.5
55.3
62.6
98.1
42.1
50.0
91.5

91.6
80.7
86.0
82.2
71.9
80.9
67.9
73.4
100.0
72.8
78.7
98.3

88.1
83.4
84.1
79.3
70.9
82.2
70.3
73.5
99.4
71.7
79.2
98.0

89.8
78.3
85.4
82.6
74.2
81.5
80.4
84.1
100.0
82.3
89.1
99.7

88.2
77.6
86.6
79.2
70.9
82.2
69.8
76.8
100.0
71.4
76.3
99.3

89.6
85.1
89.8
83.6
78.3
80.3
84.8
90.3
100.0
85.5
91.7
99.7

88.8
81.4
84.7
84.3
71.9
76.4
83.0
87.6
100.0
83.8
90.3
99.3

89.5
85.4
87.9
78.8
75.9
85.4
74.8
86.1
100.0
78.1
90.0
98.6

92.4
87.8
86.6
87.4
88.5
85.4
88.2
92.3
100.0
87.3
92.7
100.0

91.4
86.8
93.6
84.9
81.0
88.5
85.0
90.7
100.0
86.2
92.0
99.7

92.0
80.0
84.7
83.2
74.6
84.1
75.5
81.2
100.0
76.8
85.5
99.3

92.4
87.5
90.4
84.8
84.7
92.4
81.7
86.5
98.7
83.3
90.7
99.3

91.9
83.7
87.1
83.0
79.5
87.4
73.0
83.8
100.0
79.0
87.1
97.7

91.9
85.4
88.8
85.0
86.1
89.0
78.0
84.9
100.0
81.1
89.5
98.2

92.0
90.6
89.3
84.1
91.8
91.7
81.2
92.1
100.0
80.3
90.0
98.5

92.4
91.1
91.4
84.7
79.3
82.8
-

92.8
91.6
91.7
84.8
80.2
82.5
-

93.4
95.6
91.1
87.4
88.1
88.1
93.2
99.4
99.4
87.5
93.2
97.6

Average

71.1

82.0

81.7

85.6

81.5

88.2

86.0

85.9

90.7

90.0

84.7

89.4

86.1

88.2

90.1

-

-

92.8

Task

• Transfer Joint Matching (TJM) [24], which adapts marginal
distribution with source sample selection• Adaptation Regularization (ARTL) [22], which learns domain classifier in original space• CORrelation Alignment (CORAL) [30], which performs secondorder subspace alignment• Scatter Component Analysis (SCA) [14], which adapts scatters in subspace• Joint Geometrical and Statistical Alignment (JGSA) [40],which aligns marginal & conditional distributions with label
propagation• Distribution Matching Machine (DMM) [7], which learns atransfer SVM to align distributions
And deep domain adaptation methods:• AlexNet [21], which is a standard convnet• Deep Domain Confusion (DDC) [34], which is a single-layer
deep adaptation method with MMD loss
• Deep Adaptation Network (DAN) [25], which is a multi-layer
adaptation method with multiple kernel MMD

Deep Methods
DAN DCORAL

DUCDA

MEDA

• Deep CORAL (DCORAL) [32], which is a deep neural network with CORAL loss
• Deep Unsupervised Convolutional Domain Adaptation (DUCDA)[41], which is based on attention and CORAL loss

4.3

Experimental Setup

For fair comparison, we follow the same protocols as [37, 40, 41] toadopt the extracted features for MEDA and other traditional methods. To be specific, 256 SURF features are used for USPS+MNIST
datasets; for Office+Caltech10 datasets, both 800 SURF and 4,096
DeCaf6 [11] features are used; for Office-31 dataset, 4,096 DeCaf6
features are used; for ImageNet+VOC datasets, 4,096 DeCaf6 features are used. Deep methods can be used to the original images.
Parameter setting: The optimal parameters of all comparison
methods are set according to their original papers. As for MEDA, we
set the manifold feature dimension d = 20, 30, 40 for Office+Caltech10,
USPS+MNIST, and ImageNet+VOC datasets, respectively. The iteration number are set to T = 10. We use the RBF kernel withthe bandwidth set to be the variance of inputs. The regularization
parameters are set as p = 10, λ = 10, η = 0.1, and ρ = 1. The

MNIST (M(DA)

C

D (M(DA)

ImageNetVOC (M(DA)

MNIST (JGSA)C

D (JGSA)

ImageNetVOC (JGSA)

90

90

80

80

Accuracy (%)

Accuracy (%)

USPS
USPS

70
60
50
40

Table 5: Mean and standard deviation of accuracy in featurelearning in both original and manifold space.

70
60
50
40

0

0.2

0.4
0.6
0.8
(Original space)

1

0

0.2

0.4
0.6
0.8
(Manifold space)

1

Figure 3: Accuracy in original (left) and manifold space(right) with different µ. Dashed lines are best baseline.
80
Only SRM
SRM + DA
SRM + DA + Lap

Accuracy (%)

70

60

50

40
A

D

C

A

I

V

U

M

Task

Figure 4: Evaluation of each component.
approach of setting these parameters are in the supplementary file.
Additionally, the experiments on parameter sensitivity and convergence analysis in later experiments (Section 4.6 and 4.7) indicatethat MEDA stays robust with a wide range of parameter choices.
We adopt classification Accuracy on Dt as the evaluation metric,
which is widely used in existing literatures [15, 26, 37]: Accuracy =|x:x∈ Dt ∧ŷ(x)=y(x) |, where y(x) and ŷ(x) are the truth and predicted|x:x∈ Dt |labels for target domain, respectively.

4.4

Experimental Results and Analysis

The classification accuracy results on the aforementioned datasets
are shown in Tables 2, 3, and 4, respectively1 . From those results,
we can make several observations as follows.
Firstly, MEDA outperformed all other traditional and deep comparison methods in most tasks (21/28 tasks). The average classification accuracy of MEDA on 28 tasks was 73.2%. Compared tothe best baseline method JGSA (69.7%), the average performance
improvement was 3.5%, which showed a significant average error
reduction of 11.6%. Note that the results on Office-31 dataset werein the supplementary file 2 due to space constraints, and the observations are the same. Since these results were obtained from awide range of image datasets, it demonstrates that MEDA is capable of significantly reducing the distribution divergence in domain
adaptation problems.
Secondly, the performances of distribution alignment methods(TCA, JDA, ARTL, TJM, JGSA, and DMM) and subspace learning
methods (GFK, CORAL, and SCA) were generally worse than MEDA.
Each kind of methods has its limitations and cannot handle domain
adaptation in specific tasks. This indicates the disadvantages of
those methods to cope with degenerated feature transformation
1 Symbol ‘-’ denotes the result is not available since there is no code or results.
2 Supplementary file is at https://www.jianguoyun.com/p/DRuWOFkQjKnsBRjkr2E.

Task
C→A
C→W
C → A (DeCaf)
C → W (DeCaf)
U→M
I→V

Original Space44.9 (2.1)
33.5 (4.5)
92.5 (0.2)
88.4 (1.7)
64.1 (9.2)
63.0 (2.5)

Manifold Space56.5 (0.5)
54.0 (0.4)
93.4 (0.1)
95.5 (0.3)
71.2 (4.2)
63.7 (2.2)

Improvement
25.8% (-76.9%)
61.4% (-90.9%)
1.0% (-58.3%)
8.1% (-82.6%)
11.1% (-54.5%)
1.1% (-13.2%)

Table 6: Performance comparison between µopt and µ̂.
Task
C→A
µopt
57.0
µ̂
56.5
Performance-0.9%
Variation

W→D
89.2
88.5

C → A (DeCaf)
93.4
93.4

W → C (DeCaf)
88.0
93.2

M→U
89.4
89.5

I→V
67.6
67.3

-0.8%

0

+5.9%

+0.1%

-0.4%

and unevaluated distribution alignment. After manifold or subsapce
learning, there still exists large domain shift [3]; while feature
distortion will undermine the distribution alignment methods.
Thirdly, MEDA also outperformed the deep methods (AlexNet,
DDC, DAN, DCORAL, and DUCDA) on Office+Caltech10 datasets.
Deep methods often have to tune a lot of hyperparameters before obtaining the optimal results. Compared to them, MEDA only
involves several parameters that can easily be set by human experience or cross-validation. This implies the accuracy and efficiency
of MEDA in domain adaptation problems over other deep methods.

4.5

Effectiveness Analysis

4.5.1 Manifold Feature Learning. We investigate the effectiveness of manifold feature learning in handling the degenerated feature transformation challenge. To this end, we ran MEDA withand without manifold feature learning on randomly selected tasks.
Table 5 showed the mean, standard deviation, and performance improvement of classification accuracy with µ ∈ {0, 0.1, · · · , 1}. For
instance, the improvement of mean accuracy on task C → A was:
(56.5 − 44.9)/44.9 × 100% = 25.8%. From these results, we can observe that: 1) The performance of all the tasks were improved withmanifold feature learning, indicating that transforming features
into the manifold alleviates domain shift to some extent and facilitates distribution alignment; 2) The standard deviation of methods
that adopted manifold learning with different µ could be dramatically reduced. 3) MEDA can also reach a comparable performance
without manifold learning, while adding manifold learning would
produce better results. This reveals the effectiveness of manifold
feature learning to alleviate degenerated feature transformation.
4.5.2 Dynamic Distribution Alignment. We verify the effectiveness of dynamic distribution alignment in handling the unevaluated distribution alignment challenge. We ran MEDA by searchingµ ∈ {0, 0.1, · · · , 0.9, 1.0} and compared the performances with thebest baseline method (JGSA). From the results in Figure 3, we canclearly observe that the classification accuracy varied with different
choice of µ. This indicates the necessity to consider the different
effects between marginal and conditional distributions. We canalso observe that the optimal µ value varied on different tasks(µ = 0.2, 0, 1 for three tasks, respectively). Thus, it is necessary todynamically adjust the distribution alignment between domains
according to different tasks. Moreover, the optimal value of µ is notunique on certain task. The classification results may be the sameeven for different µ.

90

80
70

C
A
MNIST
USPS
C
W (DeCaf)
ImageNet
VOC

60

MNIST
USPS
C
A
C
A (DeCaf)
ImageNet
VOC

80
70
60

50
10 20 30 40 50 60 70 80 90 100

2000

MNIST
USPS
C
A
C
A (DeCaf)
ImageNet VOC

80
70

1000
500

60

50
2

4

8

16

d

p

(a) Subspace dimension d(b) #neighbor p

32

64

C
A
C
A (DeCaf)
MNIST
USPS
ImageNet VOC

1500

Objective

100

90

Accuracy(%)

100

90

Accuracy(%)

Accuracy(%)

100

50
0.1

0.5

1

10

100

1000

0

5

10

15

20

#Iteration(c) λ(d) Convergence

Figure 5: (a)∼(c): classification accuracy w.r.t. d, p, and λ, respectively. (d) convergence analysis.
The estimation of µ: We evaluate our solution of estimatingµ (equation (6)). Since the optimal µ is not unique, we can notdirectly compare the value of µopt and µ̂ to evaluate our solution.
Instead, we compare the performances (accuracy values) achievedby µopt and µ̂. The results in Table 6 indicated that the performanceof estimated µ̂ was very close to µopt , and sometimes it is better
than grid search (M → U). For instance, the performance variationof C → A was (57.0 − 56.5)/57.0 × 100% = 0.9%. This demonstrates
that the effectiveness in estimating µ. This estimation solution canbe directly applied to future research.
4.5.3 Evaluation of Each Component. When learning the finalclassifier f , MEDA involves three components: the structural risk
minimization (SRM), the dynamic distribution alignment (DA), andLaplacian regularization (Lap). We empirically evaluated the importance of each component. We randomly selected several tasks
and reported the results in Figure 4. Note that we did not run thisexperiment on the Decaf features of Office+Caltech10 dataset since
its results are already satisfied.
Those results clearly indicated that each component is important in MEDA, and they are indispensable. Moreover, we observethat in all tasks, it is more important to align the distributions. The
reason is that there exists large distribution divergence between
two domains. The results also suggests that adding Laplacian regularization is more beneficial in capturing the manifold structure.
Additionally, combining the effectiveness of manifold feature learning (Section 4.5.1), it is clear that all components are important for
improving the accuracy in domain adaptation tasks.

4.6

Parameter Sensitivity

As with other state-of-the-art domain adaptation algorithms [14,22, 40], MEDA also involves several parameters. In this section, we
evaluate the parameter sensitivity.
4.6.1 Subspace Dimension and #neighbor. We investigated thesensitivity of manifold subspace dimension d and #neighbor p
through experiments with a wide range of d ∈ {10, 20, · · · , 100}and p ∈ {2, 4, · · · , 64} on randomly selected tasks. From the resultsin Figure 5(a) and 5(b), it can be observed that MEDA was robustwith regard to different values of d and p. Therefore, they can beselected without knowledge in real applications.
4.6.2 Regularization Parameters. We ran MEDA with a wide
range of values for regularization parameters λ, η, and ρ on several
random tasks and compare its performance with the best baseline
method. For the lack of space, we only report the results of λ inTable 7: Running time (s) of ARTL, JGSA, and MEDA.
Task#Sample × #Feature ARTL
JGSA
MEDA

C→A
M→U
I→V

2,081 × 800
3,800 × 256
10,717 × 4,096

29.2
29.1
2,648.8

95.2
14.6> 10,000

32.3
31.4
2,931.7

Figure 5(c), and the results of ρ and η can be found in the supplementary file. We observed that MEDA can achieve a robust performancewith regard to a wide range of parameter values. Specifically, thebest choices of these parameters are: λ ∈ [0.5, 1, 000], η ∈ [0.01, 1],and ρ ∈ [0.01, 5]. To sum up, the performance of MEDA stays robustwith a wide range of regularization parameter choice.

4.7

Convergence and Time Complexity

We validated the convergence of MEDA through empirical analysis.
From the results in Figure 5(d), it can be observed that MEDA canreach a steady performance in only a few (T < 10) iterations. It
indicates the training advantage of MEDA in cross-domain tasks.
We also empirically checked the time complexity of MEDA andcompared it with other top two baselines ARTL and JGSA on different tasks. The environment was an Intel Core i7-4790 CPU with
24 GB memory. Note that the time complexity of deep methods are
not comparable with MEDA since they require a lot of backpropagations. The results in Table 7 reveal that except its superiority
in classification accuracy, MEDA also achieved a running timecomplexity comparable to top two best baseline methods.

5

CONCLUSIONSIn this paper, we propose a novel Manifold Embedded Distribution
Alignment (MEDA) approach for visual domain adaptation. Compared to existing work, MEDA is the first attempt to handle thechallenges of both degenerated feature transformation and unevaluated distribution alignment. MEDA can learn the domain-invariant
classifier with the principle of structural risk minimization while
performing dynamic distribution alignment. We also provide a feasible solution to quantitatively calculate the adaptive factor. We
conducted extensive experiments on several large-scale publicly
available image classification datasets. The results demonstrate thesuperiority of MEDA against other state-of-the-art traditional anddeep domain adaptation methods.

ACKNOWLEDGMENTS
This work is supported by National Key R & D Program of China(2016YFB1001200), NSFC (61572471,61702520,61672313), NSF throughgrants IIS-1526499, IIS-1763325, CNS-1626432, and Nanyang Assistant Professorship (NAP) of Nanyang Technological University.

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