IJCNN Poster 07/19/2022

Domain Adaptation with Optimal Transport
for Extended Variable Space
Toshimitsu Aritake1, Hideitsu Hino1, 2
1The Institute of Statistical Mathematics, 2RIKEN AIP

Abstract
Domain Adaptation

Methods & Results

In supervised learning, the accuracy of predictions degrades
when the distributions of the training and test data are different.
Domain adaptation aims at learning a model from training datathat well performs on the test data.

We consider bi-directional optimal transport (OT)
to transfer label information from the training
data to the test data. Our proposed method
enables the use of extra features of test data forbetter prediction.

We consider the case where new features are observed for test
data; hence, the distribution of training and test data havedifferent dimensionalities. We also assume the labels are given
only for the training data (unsupervised domain adaptation).

Experimental results show that the proposed
method successfully estimate the good decisionboundary using extra features.

Problem Setup

MethodExperiments

Problem

Artificial Data

Common variables areobserved in both thesource and target domains•

Distribution of common
variables may differ••

Joint Distribution Optimal Transport(JDOT)
P⇤<latexit sha1_base64="E/GEqdbpHkavNyS8xJkKHCeyajA=">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</latexit>⇡
ˆ = arg minf 2F ,ˆ⇡2⇧(Dˆ S ,DT )n

Ns ⇥Nt PNs

ˆij E↵,ij
i,j ⇡

1
ˆ
⇧(DS , DT ) ⌘ ⇡ 2 R+
i=1 ⇡ij = Nt ,• Minimize the sum of transportation cost•<latexit sha1_base64="5/a8BCj0Ny/pbtc3iueosP/+KkA=">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</latexit>

PNt

1
j=1 ⇡ij = Ns

Use distance between variable andlabel discrepancy as transportation cost
(Joint Distribution Optimal Transport; [1])E↵,ij = ↵kxcsi(1)oCommon feature

Experiment with Realistic Dataset•

Gas Sensor Array Drift Dataset[2] is used•••

xctj k22 + L(ysi , ŷtj = f (xctj ))Use the output of the model f as a
pseudo-label of the target label
Train the model using the transported source labels•

Incorrectly transported labels are modified iteratively

Extra feature

Extra variables are observedin the target domain•<latexit sha1_base64="KQuDLFoA6IU0IhZaq+1d0jTSI70=">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</latexit>

Our proposed method successfully learned a good prediction
model in the target domain that uses extra features•<latexit sha1_base64="bwnwfMzOKy7MgaHnrmIZV3HlxSY=">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</latexit>•

Binary classification of the types of gasses
Original data is timeseries of sensor values and transientfeatures are extracted from original data
Each domain has different degree of deterioration,
which results in domain shift• Comparison with baseline (no domain adaptation) andprior works [1, 3, 4]

Proposed Method
1. Transfer the target extra features to the source domain<latexit sha1_base64="cnrqHAVFF6mPymctvWOiXs4wNrM=">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</latexit>(2)

domains

Baseline

2. Transfer the source labels to the target domainecc 2E↵,ij = ↵(kxsi xtj k2 + kx̂si xetj k22 ) + L(ysi , f (xctj , xetj ))

1! 2
1! 3
1! 4
2! 1
2! 3
2! 4
3! 1
3! 2
3! 4
4! 1
4! 2
4! 3

83.33
52.28
64.49
52.13
56.84
63.55
51.06
68.67
94.39
50.00
42.97
92.98

c
E↵,ij = ↵kxsi

c 2
ce
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e

•

e

c

The distribution of extra features given common features is
identical for each class before and after OT
There always exists the destination of OT whose transportation costof extra variables is zero

•
•

•

c

Assumption PS (x |x , y) = PT (x |T (x , y))<latexit sha1_base64="2yywkD6RE08eN78S4m+z2a8HgyM=">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</latexit>

The above bi-directional OT is reduced to theunidirectional OT from the source to the target domain

Algorithm
Repeat the following two steps until the stopping criterion is met
1. Solve the joint distribution OT problem Eq. (1), and transport
the source label information to the target domain
At the initial iteration, the distance between features is used ascc 2transportation cost E↵,ij = ↵kxsi xtj k2
Label discrepancy is added to the cost function after the initial
c
c 2
(n)
c
e
E
=
↵kx
x
k
+
L(y
,
f
(x
,
x
iteration, ↵,ij
si
si
tj 2
tj
tj ))

•

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2. Train the model f using the transported source label• Barycentric mapping is used to determine starget labelst>1 >⇡ 1)) ⇡
ˆ Ỹ )
to train the model (X , (diag(ˆ• Ŷ s : a matrix whose rows are one-hot encoded source labels<latexit sha1_base64="pBr9CvMC2rJKf3rlR+iWIAc1bQo=">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</latexit><latexit sha1_base64="gLlEWfJ7SIDCfx9GeqiSVgIr1qI=">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</latexit>

JDOT
no extra
77.71
93.45
60.75
79.26
89.36
69.16
92.02
81.92
77.57
52.66
71.08
82.81

CCA

DSFT

Proposed

66.87
43.27
58.88
56.91
25.03
51.40
92.55
67.67
96.26
48.93
32.93
57.31

42.97
57.78
94.39
62.77
84.56
41.12
66.49
86.94
40.19
51.60
32.93
42.69

78.31
96.02
87.85
84.04
89.47
71.96
94.15
88.76
81.31
53.72
74.30
82.57

JDOT
ideal
83.73
94.15
85.98
85.64
90.99
74.77
95.74
88.55
80.37
80.85
73.89
82.57

Conclusion••

Domain adaptation method to use the extra variables
given for the target domain is proposed
Bi-directional OT for domain adaptation is reduced tothe unidirectional OT under the assumption<latexit sha1_base64="2yywkD6RE08eN78S4m+z2a8HgyM=">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</latexit>

•

e

c

e

c

PS (x |x , y) = PT (x |T (x , y))Upper-bound on the target error is provided for the
proposed algorithm

References[1] Courty, et al., “Joint distribution optimal transportation for domain adaptation”, Adv. Neural Inf. Process. Syst. (2017)[2] Vergara, et al., “Chemical gas sensor drift compensation using classifier ensembles”, Sensors and Actuators B:
Chemical (2012)[3] Yeh, et al., “Heterogeneous domain adaptation and classification by exploiting the correlation subspace”,
IEEE Trans. Image Process. ( 2014)[4] Wei, et al., “A General Domain Specific Feature Transfer Framework for Hybrid Domain Adaptation”, IEEE Trans Knowl
Data Eng., (2019)