ICML2018论文GAIN补充材料
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Supplementary Materials
GAIN: Missing Data Imputation using Generative Adversarial Nets
Jinsung Yoon
1 *
James Jordon
2 *
Mihaela van der Schaar
1 2 3
1. Proofs
1.1. Proof of Lemma 1
For t ∈ {0, 1} define the set M
i
t
= {m ∈ {0, 1}
d
: m
i
= t}.
V (D, G) = E
˜
X,z,M,H
h
M
T
log D
G(
˜
X, M), H
+ (1 − M)
T
log
1 − D
G(
˜
X, M), H
i
= E
ˆ
X,M,H
h
M
T
log D(
ˆ
X, H) + (1 − M)
T
log
1 − D(
ˆ
X, H)
i
=
Z
X
X
m∈{0,1}
d
Z
H
m
T
log D(x, h) + (1 − m)
T
log(1 − D(x, h))
p(x, m, h)dhdx
=
Z
X
Z
H
X
m∈{0,1}
d
X
i:m
i
=1
log D(x, h)
i
+
X
i:m
i
=0
log(1 − D(x, h)
i
)
!
p(x, m, h)dhdx
(1)
=
Z
X
Z
H
d
X
i=1
X
m∈M
i
1
log D(x, h)
i
+
X
m∈M
i
0
log(1 − D(x, h)
i
)
p(x, m, h)dhdx
=
Z
X
Z
H
d
X
i=1
log D(x, h)
i
X
m∈M
i
1
p(x, m, h)
+
log(1 − D(x, h)
i
)
X
m∈M
i
0
p(x, m, h)
dhdx
=
Z
X
Z
H
d
X
i=1
log D(x, h)
i
p(x, h, m
i
= 1) + log(1 − D(x, h)
i
)p(x, h, m
i
= 0)dhdx
where (1) follows from switching the order of summation. We then note that
y 7→ a log y + b log(1 − y)
achieves its
maximum in [0, 1] at
a
a+b
and so V (D, G) is maximized (for fixed G) when
D(x, h)
i
=
p(x, h, m
i
= 1)
p(x, h, m
i
= 0) + p(x, h, m
i
= 1)
(1)
for each i ∈ {1, ..., d}. .
*
Equal contribution
1
University of California, Los Angeles, CA, USA
2
University of Oxford, UK
3
Alan Turing Institute, UK.
Correspondence to: Jinsung Yoon <jsyoon0823@gmail.com>.
Proceedings of the
35
th
International Conference on Machine Learning, Stockholm, Sweden, PMLR 80, 2018. Copyright 2018 by the
author(s).
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