変数の拡張に対する最適輸送を用いたドメイン適応
有竹

俊光1，日野英逸1,2

1統計数理研究所，2理化学研究所

情報論的学習理論と機械学習研究会（IBISML）
2022/01/18

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教師付き学習•

教師付き学習<latexit sha1_base64="dMLPiDD2AHXoN+9c3Fh6ZfGNK64=">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</latexit>s s Ns• 教師データ： DS = {(xi , yi )}i=1<latexit sha1_base64="RqoYdBBClZGiHAiFD1vR0rjZ0Lc=">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</latexit>

t Nt
D
=
{x
j }j=1• テストデータ： T•

教師，テストデータの分布が異なると予測精度が低下
ソースドメイン
教師データが得られる領域＋分布• ターゲットドメイン
テストデータが得られる領域＋分布•

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ドメイン適応•教師，テストデータの同時分布を近づける<latexit sha1_base64="FAox7p8TBvOngTU/vJ5X/0F1Ne0=">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</latexit>

0

0

0

0

ps (x , y ) ⇡ pt (x , y )•ドメイン不変な特徴の抽出• 密度比による標本の重み付け• 最適輸送による教師データの輸送•

テストデータのラベルは未知のもとで適応を行う（教師なしドメイン適応）

背景：最適輸送を用いたドメイン適応<latexit sha1_base64="8EiX/O24s7FqqYgvDhdGxgecYzY=">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</latexit>s s Ns• 教師データ {(xi , yi )}i=1 を

4

sourceターゲットドメインへ輸送•

輸送先で判別器を学習•

テストデータのラベルは未知•

PS (y|x) = PT (y|T (x)) を仮定•

最適輸送前後での各点でのラベル分布が一致<latexit sha1_base64="fuS/bLEDpmxOqzfIOlPoVuACb0o=">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</latexit>

target5

背景：最適輸送問題<latexit sha1_base64="LCspikYggBI8Ts6/Ah56CNOTFEg=">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</latexit>

⇤

⇡
ˆ =
<latexit sha1_base64="XbfaCzEHhdDre5HNT/5fxauzkYs=">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</latexit>

arg minNs X
NtXˆ S ,DT ) i=1 j=1⇡
ˆ 2⇧(Dn

Ns ⇥Nt PNs

ˆ S , DT ) ⌘ ⇡ 2 R
⇧(D
+

•

c
c
⇡
ˆij E(xsi ; xtj )
1
⇡
=
ij
i=1
Nt ,

PNt

1
⇡
=
ij
j=1
Ns

輸送計画により定まる輸送量と輸送コストの積の総和を最小化する問題ˆ 2 ⇧(DS , DT )輸送計画 ⇡
輸送元，輸送先の対応関係を決める
2
E(x
,
x
)
=
kx
x
k
s
t
s
t 2 などを利用• 輸送コストは<latexit sha1_base64="Y/McNeF/An3OBdZAD8YrA9xVeTs=">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</latexit>•<latexit sha1_base64="9H6DdbFI+kkW8L7YfiuCxetnSYc=">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</latexit>•

線形計画法によって解くことが可能

o6

最適輸送の例<latexit sha1_base64="h6QnYzn95oIcBna6CB+efwlRvp4=">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</latexit>t{x0i }Ni=1•

輸送コストとして<latexit sha1_base64="QYl29cayZeipVqoEpflXoFm4WtU=">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</latexit>

ci,j = c(xi , x0j ) = kxi

を利用

x0j k22<latexit sha1_base64="kESSfJJfbP5VjrLu/R4z67vgMx4=">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</latexit>s{xi }Ni=1•

最適輸送問題を解いてソース，ターゲットで対応する標本を推定<latexit sha1_base64="qnWcmFGgtx5VbE5i5RQ2TZLblsU=">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</latexit>• Ns 6= Nt のときは一対多の対応関係が生じる

背景：Joint Distribution Optimal

7

Transport (JDOT)<latexit sha1_base64="q/So/GTo2aWOk3NWyhQmrBl0XXM=">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</latexit>

Ns X
Nt
X

arg min<latexit sha1_base64="P29jYfMiIRgK18VvmHBGVR37ESc=">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</latexit>

ˆ S ,D f ),f 2F i=1 j=1⇡
ˆ 2⇧(DTcc⇡
ˆij E↵ (xsi , ysi ; xtj , ŷtj )E↵ (xcsi , ysi ; xctj , ŷtj ) = ↵kxcsi

xctj k22 + L(ysi , ŷtj )source• 教師なしドメイン適応では
一般には PS (y|x) 6= PT (y|T (x))<latexit sha1_base64="mEJalF9cOpuCeuTVRsRc7Nl/Sko=">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</latexit>

テストデータのラベルは不明• 輸送コストにラベルの
ŷ
=
f
(x)
• 疑似ラベル不一致度を加える をテスト
(x, y) の同時分布がより近くなる•データのラベルとして利用<latexit sha1_base64="0JvgVXqvBa32Ff7q/G1uhQ7rZ5k=">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</latexit><latexit sha1_base64="CHCmGOfa6vhEcCMj1yHEn4oQtPw=">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</latexit><latexit sha1_base64="vVWMf3ZQpeNypbzHZqirDwyf8CM=">AAAECXichVO/T9tAFH7BLaSUn+1SqQvCAiGG6IIQoE6oXTomoSFINEK2uYRT/Eu2E5RY/gfK2qpDp1bqUHXqVnVl6T/QgT8BMQapCwPfXdxUgIjvZN+79973+d33fKZvizBi7Cw3pj14OD6RfzT5eGp6ZnZu/slu6LUDi1ctz/aCPdMIuS1cXo1EZPM9P+CGY9q8ZrZeyXitw4NQeO6bqOvzumM0XdEQlhHBVW4czOmswNRYuGsUU0OndJS8+VyX3tIheWRRmxzi5FIE2yaDQsx9KhIjH746xfAFsISKc0poEtg2sjgyDHhbeDex20+9LvaSM1RoC1+x8QRALtAS+8O+sT77zb6zc3Z1L1esOGQtXazmAMv9g9l3z3b+ZqIcrBEd/UeNQJiYzohTxUO2ANEGntEKRMjZUicXUMJXHqmJNaim0/vY33lRWYqX2Rd2ATU+szN2Cj3czqX1tcwrn8Au+V2gjpV6jqrARb9i+E2lhuxQPd1LfQ+HSsldTDriSVrpfUyOqlCotac6K70eauY4qzyz7Nq/bq4qJW4jkoxq5b/TTHHZ/Dezs7glS5DBGg+zEtyS4u07cdfYXSsUNwrr5XV9+2V6X/L0nBZpBYpu0ja9phJVUQ2nE3pPH7QT7Yf2U/s1SB3LpZindGNop9ctJ+Be</latexit>•

ような輸送を期待モデル f も同時に学習

target8変数の拡張に対するドメイン適応
ラベル ys<latexit sha1_base64="57/DonMKCFiBmWmIbstfjgC8JXo=">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</latexit>•

ターゲットドメインでのみ新しい変数を観測•

センサー，調査項目の追加

共通変数

xcs

<latexit sha1_base64="ZpQJdEcZVdNjPogec7lsMuW3b/Y=">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</latexit>e

新規変数 xs

source

分布の変化

新規変数の観測

target
ラベル yt<latexit sha1_base64="PKIUqiHOgo8lYlhE9ztFnv237KE=">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</latexit>•<latexit sha1_base64="xfJoVjJ+KXLkAbVX1pQTBc71Pe4=">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</latexit>c
共通変数 xt<latexit sha1_base64="AAExQ7wCAQ1RQoee1SglBVSbILA=">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</latexit><latexit sha1_base64="VaPo7sguFwXKVXw8Q2uD2n03bDU=">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</latexit>e

新規変数 xt

既存の変数はソース，
ターゲットドメインで共通•

同一のセンサ等により同じ情報を観測している• 計測値の分布は変化する（計測対象の状態の違いなど）

9

本研究の貢献•

共通変数の分布の変化，新規変数の観測に対する最適輸送を用いたドメイン適応法の提案•

ソース，ターゲット間双方向の最適輸送を利用• 実際には１方向の輸送により実現可能•

ターゲットドメインでの予測モデルのテスト誤差の上界の導出

10

変数の拡張に対するドメイン適応
ラベル ys<latexit sha1_base64="57/DonMKCFiBmWmIbstfjgC8JXo=">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</latexit>•

次元の異なる変数間の輸送コストの定義が必要••<latexit sha1_base64="xfJoVjJ+KXLkAbVX1pQTBc71Pe4=">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</latexit>

共通変数

xcs

<latexit sha1_base64="ZpQJdEcZVdNjPogec7lsMuW3b/Y=">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</latexit>e

新規変数 xs

sourcetargetベクトル埋め込み等では変数の共通性を考慮できない

ドメイン適応<latexit sha1_base64="an8SIRIcRqq0S1kj8WGa5bVZFWM=">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</latexit>ce

疑似ラベル ŷ = f (xt , xt )ソースドメインの新規変数を推定してドメイン適応を行う
新規変数の予測はターゲットドメインから
ソースドメインへの最適輸送により行う• JDOTと同様に擬似ラベルを利用した最適輸送を考える•

11

双方向の輸送によるドメイン適応
ラベル ys<latexit sha1_base64="57/DonMKCFiBmWmIbstfjgC8JXo=">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</latexit>•

ソース，ターゲット間
双方向の輸送によるドメイン適応1. ターゲットからソースへ
新規変数を輸送<latexit sha1_base64="57/DonMKCFiBmWmIbstfjgC8JXo=">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</latexit><latexit sha1_base64="xfJoVjJ+KXLkAbVX1pQTBc71Pe4=">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</latexit>

共通変数<latexit sha1_base64="xfJoVjJ+KXLkAbVX1pQTBc71Pe4=">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</latexit>

xcs<latexit sha1_base64="ZpQJdEcZVdNjPogec7lsMuW3b/Y=">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</latexit><latexit sha1_base64="ZpQJdEcZVdNjPogec7lsMuW3b/Y=">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</latexit>e

新規変数 xs

source2

1target<latexit sha1_base64="Uz28kw9qLEeJhR+41FmFSWK00N0=">AAADfnichVLLThRBFL1D+0AEGWRDYmImDhAgOtYYAsTEhMjGJQ8HSJhh0tXUQIV+pbtmdOjMzpU/4MIVJC6ApfwBG3/ABZ9gXGLixoWnahoIEIaqVNd9ndv3nls8dGWsGDvJdFl37t673/2g52Fv36P+7MDj5TioR44oOYEbRKvcjoUrfVFSUrliNYyE7XFXrPDtOe1faYgoloH/XjVDUfHsTV/WpGMrmKrZp+UtWyXN1pvaWJnz3MdqvO48P5PEeDWbZwVmVu66UEyFPKVrPhjIDFOZNiggh+rkkSCfFGSXbIqx16hIjELYKpTAFkGSxi+oRT3A1hElEGHDuo3vJrS11OpD1zljg3bwFxcnAjJHI+wn22en7Ac7ZL/YvxtzJSaHrqWJm7exIqz2fx5a+nsrysOtaOsC1QHBsb0OXSXn2SJ4azidGVCImTGdSzARGovmxGlX09j5crr0enEkGWV77DfY2GUn7Bh8+I0/zrcFsfgV2XV+H6gPhj3PVOBjXgns3LChJ1RJdc3vxjlTWksoD38rrfSmTJ6pUJp7x0xWWwPULNCr7llP7WyaE4aJq4gW3l7x6ku7Liy/KhSnCpMLk/nZt+kr7KYn9IzGUOc0zdI7mqcSavxEB/SdjiyyRq0X1st2aFcmxQzSpWXN/Afzg8Yi</latexit><latexit sha1_base64="Uz28kw9qLEeJhR+41FmFSWK00N0=">AAADfnichVLLThRBFL1D+0AEGWRDYmImDhAgOtYYAsTEhMjGJQ8HSJhh0tXUQIV+pbtmdOjMzpU/4MIVJC6ApfwBG3/ABZ9gXGLixoWnahoIEIaqVNd9ndv3nls8dGWsGDvJdFl37t673/2g52Fv36P+7MDj5TioR44oOYEbRKvcjoUrfVFSUrliNYyE7XFXrPDtOe1faYgoloH/XjVDUfHsTV/WpGMrmKrZp+UtWyXN1pvaWJnz3MdqvO48P5PEeDWbZwVmVu66UEyFPKVrPhjIDFOZNiggh+rkkSCfFGSXbIqx16hIjELYKpTAFkGSxi+oRT3A1hElEGHDuo3vJrS11OpD1zljg3bwFxcnAjJHI+wn22en7Ac7ZL/YvxtzJSaHrqWJm7exIqz2fx5a+nsrysOtaOsC1QHBsb0OXSXn2SJ4azidGVCImTGdSzARGovmxGlX09j5crr0enEkGWV77DfY2GUn7Bh8+I0/zrcFsfgV2XV+H6gPhj3PVOBjXgns3LChJ1RJdc3vxjlTWksoD38rrfSmTJ6pUJp7x0xWWwPULNCr7llP7WyaE4aJq4gW3l7x6ku7Liy/KhSnCpMLk/nZt+kr7KYn9IzGUOc0zdI7mqcSavxEB/SdjiyyRq0X1st2aFcmxQzSpWXN/Afzg8Yi</latexit>ce

疑似ラベル ŷ = f (xs , xs )<latexit sha1_base64="P29jYfMiIRgK18VvmHBGVR37ESc=">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</latexit>E↵ (xcsi , ysi ; xctj , ŷtj ) = ↵kxcsi

xctj k22 + L(ysi , ŷtj )2. ソースからターゲットへラベルを輸送<latexit sha1_base64="uRPfbProtd3Hi1q2Dm3xvxs+Rho=">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</latexit>E↵ (xcsi , x̂esi , ysi ; xctj , xetj , ŷtj ) = ↵(kxcsi

xctj k22 + kx̂esi

3. 輸送したラベルによりモデル f を学習<latexit sha1_base64="fnUw72CUboT/aZsqPG0nWG9Hzj8=">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</latexit>

xetj k22 ) + L(ysi , ŷtj )12

双方向の輸送によるドメイン適応•

仮定<latexit sha1_base64="APpnJ4fF4m9LVTw8M7uY0tsthSU=">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</latexit>e

ラベル ys<latexit sha1_base64="57/DonMKCFiBmWmIbstfjgC8JXo=">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</latexit>

c

e

•

c

<latexit sha1_base64="ZpQJdEcZVdNjPogec7lsMuW3b/Y=">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</latexit>(x , y) の輸送先の xe の<latexit sha1_base64="uYIu5lWDf5SZVnYvRHGFCFN8tyc=">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</latexit>ctarget(x , y) の輸送先に輸送元の xe<latexit sha1_base64="taRv1mJDeNpDmsvIhx5lt0Zmbrk=">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</latexit><latexit sha1_base64="taRv1mJDeNpDmsvIhx5lt0Zmbrk=">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</latexit><latexit sha1_base64="Uz28kw9qLEeJhR+41FmFSWK00N0=">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</latexit>e<latexit sha1_base64="uYIu5lWDf5SZVnYvRHGFCFN8tyc=">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</latexit>cの輸送コストが0となり (x , y) を用いた
c
e
(x
,
x
, y) を用いた輸送が等価
輸送と

• x

e

<latexit sha1_base64="3ckLAZdTsqL6bV6CKenH+PmIW5M=">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</latexit>ce

疑似ラベル ŷ = f (xs , xs )と同じ x をもつ点が存在<latexit sha1_base64="taRv1mJDeNpDmsvIhx5lt0Zmbrk=">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</latexit>e

新規変数 xs<latexit sha1_base64="taRv1mJDeNpDmsvIhx5lt0Zmbrk=">AAADZXichVK7ThtBFL32hkcA80iiKFIKLIxRlMIaRyhEqVDSUAKOMRI21s4ywIh9aXftACv/QLpUkUIFEkWUz0iTH0jhL0AoJZHSUHBmvBgBspnR7NzXuXvvucN9W4YRY+1U2ng0MDg0/HhkdCwzPjE59WQt9BqBJcqWZ3vBOjdDYUtXlCMZ2WLdD4TpcFtU+N5H5a80RRBKz/0UHfii5pg7rtyWlhnBVKlynt3fFPXJHCswvbL3hWIi5ChZy95UapaqtEUeWdQghwS5FEG2yaQQe4OKxMiHrUYxbAEkqf2CWjQCbANRAhEmrHv47kDbSKwudJUz1GgLf7FxAiCzlGd/2A92wX6zn+ycXfbMFescqpYD3LyDFX594suL0v8HUQ7uiHZvUH0QHNvp01XczRbAu43Tn4EIMe905xJM+NqiOLE61TQPv12U3q/m4zl2wv6CjWPWZr/Ah9v8Z52uiNUjZFf5XaA+a/YcXYGLecWwc82GmlAt0RW/W12mlBZTDv5WUmmvTI6uUOr7UE9WWT3ULNCr6llN7XqarzUTdxEtvL3i3Zd2X1h7Uyi+LcyvzOcWPySvcJhe0gy9Qp0LtEhLtExlzeRX+k5H6TMjYzwznndC06kE85RuLWP6CrZUvcM=</latexit>

分布が同一
•

共通変数

xcs

sourcec

PT (x |x , y) = PS (x |T (x , y))<latexit sha1_base64="uYIu5lWDf5SZVnYvRHGFCFN8tyc=">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</latexit><latexit sha1_base64="xfJoVjJ+KXLkAbVX1pQTBc71Pe4=">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</latexit>

13

双方向の輸送によるドメイン適応• 双方向の輸送はソースからターゲットへの1方向の輸送により実現可能
1. ソースからターゲットへラベルを輸送
E↵ (xcsi , ysi ; xctj , ŷtj ) = ↵kxcsi xctj k22 + L(ysi , ŷtj )<latexit sha1_base64="P29jYfMiIRgK18VvmHBGVR37ESc=">AAAEmXichVPNbtNAEB63Bkr4aVouSLlYhKAAJdpEFSAQUgEhVYhD25C2Up1aa3ebmPpP9iaQun4BLhw5cAKJA+IxuPACHPoIiGNBXDgwuzamTdVkV/bOzM73efYbrxk4dsQJ2VcmJtVTp89MnS2cO3/h4nRxZnY18nuhxVqW7/jhukkj5tgea3GbO2w9CBl1TYetmTuPxf5an4WR7XvP+SBgbZd2PHvbtijHkFF8o7uUdy3qxE8SQ6dO0KVV3TS1V5uWEUd2MjeQy/08xl8kc5repTweJNK7/qCQ4vS9w8BbhxH6ntHYbGg3tfxrz5JqyjxEZhTLpEbk0I4b9cwoQzaW/BllADpsgQ8W9MAFBh5wtB2gEOHcgDoQCDDWhhhjIVq23GeQQAGxPcximEExuoPvDnobWdRDX3BGEm3hVxx8QkRqUCHfyCdyQL6Sz+Q7+XMiVyw5RC0DXM0UywJj+vXl5u+xKBdXDt3/qBEIE6c74lRxzhbi7jY+oxXgmHNXntxGJQIZEZpYaTX93bcHzXsrlfga+UB+oBrvyT75gnp4/Z/Wx2W28g7ZBb+HqJdSPVdW4GG/YoybUg3RoXbmC323cqWEF0MZ95Os0pOYXFmhLddd2VkR9bFmhmcVZxZd+9fNG1KJYUQyplrx73Qy3Hj+o9njuAVLOIY1zrMSvCX14Ttx3Fht1Oq3a/PL8+WFR9l9mYISXIEqKnoHFmARlqCF1fxSSspVpaKW1Ifqovo0TZ1QMswlODLU5l/6TBO3</latexit>

2. 輸送したラベルによりモデル f を学習<latexit sha1_base64="fnUw72CUboT/aZsqPG0nWG9Hzj8=">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</latexit><latexit sha1_base64="qntmFRcL6TKA+B87mGizxF8JQt8=">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</latexit>

argminNs X
NtXˆ S ,D f ),f 2F i=1 j=1⇡
ˆ 2⇧(DT<latexit sha1_base64="Wel6QdKp06KtpNF+NeKzvL0+l1Q=">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</latexit>E↵ (xcsi , ysi ; xctj , ŷtj ) = ↵kxcsicc⇡
ˆij E↵ (xsi , ysi ; xtj , ŷtj )xctj k22 + L(ysi , ŷtj = f (xctj , xetj ))14

数値実験（人工データ）•

2次元の2つの円が異なる
クラスとなるデータ•

共通変数，新規変数がそれぞれ1次元•ex
ソースの新規変数とターゲットのラベル y は輸送，推定には利用しない<latexit sha1_base64="11Anxin+jwlEl7oMtPI+PoWEokE=">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</latexit><latexit sha1_base64="5NFtBfmfBTqLqkI+ePnh2j/9sNU=">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</latexit>•

sourceモデル集合 F としてRBFカーネルを用いたSVMを利用して推定<latexit sha1_base64="QLiCMxmRogqluxgdv/bv2vMNChs=">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</latexit>

target15

数値実験（人工データ）•輸送したラベルの一部が不正確
でも学習後のモデルは真のラベルを正しく予測可能•

初期値等によっては誤った
判別境界が得られる場合もある•

新規変数を用いないJDOTよりも精度のよい推定が可能<latexit sha1_base64="ROPizjGM8WhEpJoJteAaNG16V9E=">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</latexit>i`Mb72`
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16

数値実験（実データ）•

2種類のガスの判別問題•

元データは時系列データ• 時系列データから1つのセンサにつき6個の特徴量を抽出しガスの種類を判別

計測時期の違いによりVergara et al., Sensor and Actuators B:
センサの劣化度合いが異なる Chemical (2012)• 16個のセンサのうち8個を共通のセンサ
残りの8個をテスト時のみ追加されるとして実験••

RBFカーネルを用いたSVMをモデルとして利用

17

数値実験（実データ）•

結果•

Baseline
ドメイン適応を行わない

JDOT
新規変数を用いずに適応を行った場合• Proposed
提案法により新規変数を考慮した場合••<latexit sha1_base64="Lr7tb5eiQHumfvdhjtT5GYF8HW4=">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</latexit>/QKBMb
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一部の適応が不要な場合を除いて新規変数を考慮することで予測精度が向上することを確認

18

まとめ
•

ターゲットドメインでの新規変数の観測に対する最適輸送を用いたドメイン適応法を提案•

ソース，ターゲットで共通となる変数の均質性を考慮• 双方向の輸送問題によりソースの新規変数とターゲットのラベルを推定してドメイン適応を実行
e c
e
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, y)) という仮定のもとではT
S•
双方向の輸送はソースからターゲットへの輸送で実現可能• 実験的に新規変数を考慮しないJDOTと比較して精度のよい推定が可能なことを確認<latexit sha1_base64="APpnJ4fF4m9LVTw8M7uY0tsthSU=">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</latexit>•

ターゲットドメインでの予測モデルの汎化誤差の上界の導出

Appendix

20

最適輸送問題•ある質量の山の土を別の山に移す問題•

土の移動コスト•

移す元と移す先の距離• 移す量•

最小コストを達成する
山の移し方は？

21

Mongeの問題<latexit sha1_base64="mzp07/wDEyD4coh800ikMWRWQbc=">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</latexit>

下限

inf

⇢Z

c(x, T (x))dµ(x) T# µ = ⌫Xを達成する輸送写像 T : X ! Y を求めよ
<latexit sha1_base64="E5KPIyLeERmSMBkoBmz/P3J8FFA=">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</latexit>• µ ：距離空間 X 上の確率測度<latexit sha1_base64="lmnL0PZYfcnRQVyF1L6TWliUlUU=">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</latexit><latexit sha1_base64="4m8Yo9ezdDzCo1EiSKC4xhHXX9Y=">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</latexit>• ⌫ ：距離空間 Y 上の確率測度<latexit sha1_base64="KkfAMMhIEApAwIhmwlTy1V0fiqs=">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</latexit><latexit sha1_base64="Kp1H0N/KPSGrI0wGLAq44FmcycU=">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</latexit>• c : X ⇥ Y ! [0, 1]：輸送コストを表す可測関数<latexit sha1_base64="VUfi8K8OjYoBXEwE07HkdL8dQLo=">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</latexit>• T# µ ：T による µ の像測度（押し出し）<latexit sha1_base64="PknMIsm3NDi0D1fnvmU3e4oWjjQ=">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</latexit><latexit sha1_base64="rUVyB4LkJv0dZEv9oKiTk0PJpAo=">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</latexit><latexit sha1_base64="lmnL0PZYfcnRQVyF1L6TWliUlUU=">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</latexit>

22

Mongeの問題<latexit sha1_base64="mzp07/wDEyD4coh800ikMWRWQbc=">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</latexit>

下限

inf

⇢Z

c(x, T (x))dµ(x) T# µ = ⌫Xを達成する輸送写像 T : X ! Y を求めよ
<latexit sha1_base64="E5KPIyLeERmSMBkoBmz/P3J8FFA=">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</latexit>•

輸送の過程は考慮しない•解が存在しない場合がある→ µ がDirac測度，⌫ が非Dirac測度<latexit sha1_base64="lmnL0PZYfcnRQVyF1L6TWliUlUU=">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</latexit><latexit sha1_base64="Kp1H0N/KPSGrI0wGLAq44FmcycU=">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</latexit>•

輸送射像を求める代わりに輸送計画を求める問題を解く（Kantrovichの定式化）

23

Kantorovichの問題<latexit sha1_base64="tOgdUvtg0XNlgRB99wZNiDUuMY0=">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</latexit>

下限 inf⇢Zc(x, y)d (x, y)X⇥Yを達成するカップリング<latexit sha1_base64="VfhGOP1+Sysy2NcwCMww4RYMfno=">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</latexit>

2 (µ, ⌫)を求めよ
<latexit sha1_base64="zHZb9ut1LJdzWuux1ex5ragkrFU=">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</latexit><latexit sha1_base64="QkwwICI+Q0x+ECYrlxPBjTFKfRY=">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</latexit>•(µ, ⌫)：X ⇥ Y 上の確率測度かつR(x, y)dy = µ をみたす
Y<latexit sha1_base64="ZSjBlqbPnp92l1Nw6DpFvCyyJAQ=">AAADZ3ichVK7ThtBFL32hvAIxJBICCmNg0MUUVhjhAhKhaBJyctgBBbaXQYYsbO72h0bwco/QEtBkVAEiQLxGWnyAxR8AqIEiYaCM+MFFBBmRrNzX+fuveeOE3oiVoydZ7LWm7a37R2dXe+6e97nevs+LMRBLXJ52Q28IKo4dsw94fOyEsrjlTDitnQ8vuhsTWn/Yp1HsQj8ebUT8qq0N3yxLlxbwbRUWVFC8ji/tNpbYEVmVv65UEqFAqVrOujLfKEVWqOAXKqRJE4+Kcge2RRjL1OJGIWwVSmBLYIkjJ9Tg7qArSGKI8KGdQvfDWjLqdWHrnPGBu3iLx5OBGSehtgZO2FX7B87ZRfs9sVcicmha9nB7TSxPFzN7Q3M3byKkrgVbT6iWiAcbNmiq+QhWwTvOk5rBhRixk3nAkyExqI5cZvV1HcPruZ+zA4lX9kRuwQbf9g5+ws+/Pq1ezzDZ38hu87vA7Vt2JOmAh/zSmB3DBt6QtVU1/yuPTCltYQK8DfSSl/KJE2Fwty7ZrLaGqBmjl51z3pq99McNkw8RTTw9kpPX9pzYWGkWBorjsyMFiYm01fYQZ9okL6hzu80QT9pmsqoUdI+/abD7IWVs/qtgWZoNpNiPtJ/y/p8B4Ezvqk=</latexit><latexit sha1_base64="H9mr7hPb3qG8xJvGF5XqHQFbvTg=">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</latexit><latexit sha1_base64="X77Bwj7KPV7k8XtJlLUi5jPxJwA=">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</latexit><latexit sha1_base64="VfhGOP1+Sysy2NcwCMww4RYMfno=">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</latexit>

X(x, y)dx = ⌫の集合(x, y)：x 2 X からy 2 Y への輸送量を表す<latexit sha1_base64="kMFeDw4yB+5AA7bjmDdKXfGdxjE=">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</latexit>•

R<latexit sha1_base64="kOHD6zhjhDag/MPWy5Tq5xwCMyY=">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</latexit>

24

Kantorovichの問題⇢Z<latexit sha1_base64="tOgdUvtg0XNlgRB99wZNiDUuMY0=">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</latexit>

下限 inf2 (µ, ⌫)c(x, y)d (x, y)X⇥Yを達成するカップリング<latexit sha1_base64="VfhGOP1+Sysy2NcwCMww4RYMfno=">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</latexit>

を求めよ<latexit sha1_base64="b0iaG73cMhtQu6kxW4lZsURZNe4=">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</latexit>•

実用上は X = Y ，かつ c(x, y) を適当な距離 d(x, y) とすることが多い→Wasserstein距離の計算と等価<latexit sha1_base64="yebxEUb6Gg19ljM5Du2DMzRGVbU=">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</latexit><latexit sha1_base64="iexYlKStHJBfWwcg0rVlvz7PEdo=">AAADZXichVJPTxNBFH/togKCtEKIiQcbCqQQ00wbA8QT0YtHaCltAg3Z3U5h0v2X3W2hbPoFvHkygZMmHogfw4tfwAOfgBCPmHDx4G+ma40Qykxm5/37vX3v98bwLBGEjJ0nktrIg4ePRsfGH09MPplKpZ9uB27bN3nFdC3Xrxl6wC3h8EooQovXPJ/rtmHxqtF6K/3VDvcD4TpbYdfjdVvfd0RTmHoIU7WRO3qZ6S7tpbIsz9TK3BYKsZCleG246cQ87VKDXDKpTTZxciiEbJFOAfYOFYiRB1udIth8SEL5OfVoHNg2ojgidFhb+O5D24mtDnSZM1BoE3+xcHwgM7TAfrAzdsW+s6/skv2+M1ekcshauriNPpZ7e1Pvn5Wv70XZuEM6+IcagjCw7SFdRYNsPrxNnOEMhIhZU50LMOEpi+TE7FfTOf54VX5dWogW2Wf2E2x8YufsG/hwOr/ML5u8dIrsMr8D1KFiz1YVOJhXBLuh2JATqse65LcxYEpqEWXh78WV3pXJVhUKdR+ryUqri5o5epU9y6n9neayYuImooe3V7j50m4L28V8YSVf3HyVXX8Tv8JRek5zlEOdq7RO72iDKorJD3RCp8kLbVKb0Wb7oclEjJmm/5b24g8cQ704</latexit><latexit sha1_base64="WANtN6uYJ3RaWFR1H9EpIumE4l4=">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</latexit>

Wp (µ, ⌫) =✓inf

2 (µ,⌫)Zpd(x, y) d (x, y)
M ⇥M

◆1/p

25

離散化した場合<latexit sha1_base64="Cgj49Pnma0RjhsvVVXFlB7VsRag=">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</latexit>

minimize
n⇥m
2R

•

n X
m
X

ci,j i,j

i=1 j=1
PnPmsubject toi=1 i,j = ⌫j ,j=1

i,j = µi

離散分布から離散分布へ輸送

Pnn
m W n = {w|w 2 Rn ,µ 2 W ,⌫ 2 W ,+i=1 wi = 1}
<latexit sha1_base64="Nc+62I6g0FKn0ILWFq/qMkJ6r+4=">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</latexit><latexit sha1_base64="LCm+fksDR65axCkEoA0SPCgNGAU=">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</latexit>•

カップリング<latexit sha1_base64="ET3j/CINKiYtl3/UkGYJrXV5K70=">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</latexit><latexit sha1_base64="xuv8WF064+LLBw6/EcRoNNzvGFk=">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</latexit><latexit sha1_base64="Zk/T9Ro74qIUNR+rtAFbuqraejk=">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</latexit>

i,j =

•

およびコスト c は行列<latexit sha1_base64="VfhGOP1+Sysy2NcwCMww4RYMfno=">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</latexit>(xi , yj ), ci,j = c(xi , yj )線形計画法を解くことで解が求まる

26

例：1次元の分布の最適輸送
輸送先 ⌫<latexit sha1_base64="Kp1H0N/KPSGrI0wGLAq44FmcycU=">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</latexit>•

コストとして<latexit sha1_base64="3Mp8id2QYpUIazQVNH+auq/tNzY=">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</latexit>

c(x, y) = (x
•

輸送元 µ

2

y) を利用

1次元のヒストグラムを輸送元から輸送先へ
最小コストで動かす<latexit sha1_base64="VfhGOP1+Sysy2NcwCMww4RYMfno=">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</latexit><latexit sha1_base64="lmnL0PZYfcnRQVyF1L6TWliUlUU=">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</latexit>•

によってどこからどこへ動かしたかわかる<latexit sha1_base64="VfhGOP1+Sysy2NcwCMww4RYMfno=">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</latexit>

27

例：経験分布の最適輸送•0 n0{xi }i=1<latexit sha1_base64="BgQ54fVRAAwqg3+IJ/Wh7W4Ik1c=">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</latexit>

輸送コストとして<latexit sha1_base64="QYl29cayZeipVqoEpflXoFm4WtU=">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</latexit>

ci,j = c(xi , x0j ) = kxi

を利用{xi }ni=1x0j k22<latexit sha1_base64="5VXWXOA2FLKuv8SkcRkOsDmfXHs=">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</latexit>•

各標本点上の一様分布を輸送する<latexit sha1_base64="FiXnRgwtK4y3zi3kaX3QO4s3KLI=">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</latexit><latexit sha1_base64="rrcZBg1Y0GN3Zvk6w4IHk64ZOnc=">AAADdnichVLLahRBFL0z7SPGRxIFEQRpHEeDi/F2EBVBCLpxmddkApkxdHUqsUh3ddPdM5K08wP+gAtXBlxI/As3/kAW+YTgMoJZuPBUTRsxIWMV1XVf5/a955ZIQpXlzHuVqnPm7LnzIxdGL166fGVsfOLqYhZ300A2gziM0yXhZzJUWjZzlYdyKUmlH4lQtsTGC+Nv9WSaqVgv5JuJ7ET+ulZrKvBzmFbGr7ejrvvMbQvheu4DV7eVdluv4Khxg+1yTwpeKdSoXDPxROUOtWmVYgqoSxFJ0pRDDsmnDHuZPGJKYOtQAVsKSVm/pD6NAttFlESED+sGvuvQlkurhm5yZhYd4C8hTgqkS3Xe5c98wN94h/f516m5CpvD1LKJWwywMlkZe3dj/ud/URHunF7/RQ1BCOxoSFfFUbYU3jWc4QzkiHliO1dgIrEWw0kwqKa39f5g/ulcvbjL2/wdbHzkPf4KPnTvR/BpVs59QHaTXwP1xrIX2Qo05lXALiwbZkKdUjf8rh4xZbSCavD3y0pPyxTZCpW9t+xkjTVGzRK9mp7N1P5M875l4jiij7fnHX9pJ4XFqYb3qDE1+7A2/bx8hSN0k27TJOp8TNP0kmaoiRrf0jbt0JfqoXPLqTv3BqHVSom5Rv8sh38D0KrCjA==</latexit>n0µ = 1/n 2 W , ⌫ = 1/n 2 W<latexit sha1_base64="NoTLfUhrksJponnjIssvdxhL6wE=">AAADZnichVK9ThtBEB77EgJO+AtCREpjxRAQhbVGiERUKDSUBmKwBJZ1d15gxd3e5e7syD75BVKmSQENkVKgPEYaXoCCNwiiBIkmRb5dX0Bg2dnV3s7OzDc3881YviPCiLGLVNp48nTg2eBQ5vmL4ZHRsfGXW6FXD2xesj3HC8qWGXJHSF6KROTwsh9w07Ucvm0drir7doMHofDkx6jp84pr7kuxJ2wzgqosdyX/lJWz1bEcyzO9st1CIRFylKyiN56apl2qkUc21cklTpIiyA6ZFGLvUIEY+dBVKIYugCS0nVObMsDW4cXhYUJ7iO8+XjuJVuKtYoYabeMvDk4AZJZm2Dk7ZdfsjP1kl+xPz1ixjqFyaeK2OljuV0e/vNq8/S/KxR3RwT2qD8LCdvtUFd9FC2Ddw+nPQASf97pyASZ8rVGc2J1sGq1v15vLGzPxW/adXYGNE3bBfoEP2bixf6zzjSNEV/ElUJ81e67OQKJfMfSWZkN1qJK8Fb+1O6bUK6Yc7O0k016RXJ2h0HdLd1ZpPeTMUauqWXXtXzfnNROPEW3MXuHxpHULWwv5wlJ+YX0xt/IhmcJBek1vaA55vqMVWqMilfSEfaUjOk7/NkaMSWOq45pOJZgJerCM7F+2W74T</latexit>• n 6= n0のときは一対多の対応関係が生じるn0

28

カップリングを用いた
標本点の輸送s<latexit sha1_base64="xPdLzOo97bqiomUK3kUNHq/4fyo=">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</latexit><latexit sha1_base64="sOBFaQNzUOEMnoDi1GFkJwvAmbA=">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</latexit>

s s N
s
s
s
>
• ソースの標本：{(xi , zi )}i=1, z = (z1 , . . . , zN s )<latexit sha1_base64="ZUx5x6jaFCOs0XK0RoTGtpkQ4vk=">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</latexit>tt N• ターゲットの標本：{xj }j=1 ,<latexit sha1_base64="FS26HREsLIg589YYzuxDIs9ACqc=">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</latexit><latexit sha1_base64="FAKU7GSajVVxVAboBkVacifGzN0=">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</latexit>

X

t

t
= [x1

2 (µ, ⌫)•

カップリング：•

重心射像（barycentric mapping）<latexit sha1_base64="ZUx5x6jaFCOs0XK0RoTGtpkQ4vk=">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</latexit>

1.

t Nt
{xj }j=1 の重心にソース標本に付随する<latexit sha1_base64="iaAwO0kBBk8qcVl6UwsMXKcotUw=">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</latexit>

変数を輸送 ((diag( 1))

1

t

s

X ,z )<latexit sha1_base64="ZUx5x6jaFCOs0XK0RoTGtpkQ4vk=">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</latexit>

2.

t Nt
{xj }j=1 上にソース標本の付随する変数の<latexit sha1_base64="HRCYf8jOmzU+rWzj0LNjYM5FdJs=">AAADqnichVLNbhMxEJ50+SnlpylckLisCEVJVSIvIECcKrggTm1K2qBsEtkbN1hdr1e7TlCz2hfgBThwAokD4sI7cOEFQOoJrohjkbhwYOwsBVo12PJ6/M18s+NvzOJQpJqQ3dKMc+z4iZOzp+ZOnzl7br68cH4jVcMk4M1AhSppMZryUES8qYUOeStOOJUs5Jts+77xb454kgoVPdI7Me9IOojElgioRqhXflj1GXNbXb3s+irmCdUqiajkS1lf0EFe9QdUStr1tYpNoFerdbNrXv4X7Bp83E1rvXKF1Ikd7mHDK4wKFGNVLZSugA99UBDAECRwiECjHQKFFGcbPCAQI9aBDLEELWH9HHKYQ+4QozhGUES38TvAU7tAIzybnKllB/iXEFeCTBcWyUfyhuyRD+Qt+Up+HpkrszlMLTu4swmXx735ZxfXf/yXJXHX8OQPawqD4ZRTbpXtZ0vQu4VrugIaY+7YmwtUIraI0SSYVDMaP99bv9tYzK6SV+QbqvGS7JL3qEc0+h68XuONF5jd5I+Q9dSqJ20FEfYrQ5xZNUyHOsXZ6NvfV8qcMqigPy8qPSqTtBUKu49tZw2qsGaOdzV3Nl373c0lq8RBRo5vzzv40g4bG9fr3q36jbWblZV7xSuchUtwGapY521YgQewCk2s8R18gs/wxVl2Gs5jpz0JnSkVnAvwz3D6vwAMdtdg</latexit>t

重心を輸送 (X , diag(

>

1))

1 > s

z )

t
>
· · · xN t ]ニューラルネットワークを用いた場合の実験•

複雑なモデルを用いると誤ったラベルへの過剰適合が起こる•

交互最適化によるラベル
予測精度の改善とモデル
学習が難しい

29

30

パラメータ ↵ の影響<latexit sha1_base64="3MWmLKRWEbpNnXHZtbxseFEfJiM=">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</latexit><latexit sha1_base64="fRZVMwhN3bv6EY9RI4/9EC1bhCc=">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</latexit>↵ = 0.01<latexit sha1_base64="hJwUgqs+p7m+kqNKmrHDZ7SN/cI=">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</latexit><latexit sha1_base64="5UQexZfHfiJ93P9IlUtWskes6E0=">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</latexit>↵ = 0.1<latexit sha1_base64="QjKhpz6TFTelzDosnhetsskB+sk=">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</latexit>↵=1↵ = 10テスト誤差の評価

32

輸送に関する誤差の評価<latexit sha1_base64="fnUw72CUboT/aZsqPG0nWG9Hzj8=">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</latexit>

仮説集合 F に属する f について
ターゲットドメインでの誤差を評価する
ceerrT (f ) = E(xc ,xe ,y)⇠PT L(y, f (x , x ))ce0errS (f ) = E(xc ,xe ,y)⇠PS0 L(y, f (x , x ))<latexit sha1_base64="MwW9E68eQc6gfZXfhNwT2JGyKAQ=">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</latexit>•<latexit sha1_base64="VYCtXNFTU24I/4in3qPCtKGvoS4=">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</latexit><latexit sha1_base64="zjVI0Uy2iEdbFUJOjgqab48XoSo=">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</latexit><latexit sha1_base64="j/vPck+AH8/NW3U8rpcP+Xdd3iA=">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</latexit>•

予測誤差関数 L は対称で三角不等式を満たし
k-リプシッツ連続 |L(a, b)<latexit sha1_base64="CzI66soAhE1aEt0cXkrgTJv3EuU=">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</latexit>

L(a, c)|  k|b

c|

⇤
0
err
(f
)
+
err
(f
)
f
•
S
T
を最小化するモデル
が有界
⇤
c
e
⇤
c
e
|f (xs , xs ) f (xt , xt )|  M
f⇤• モデル f と PS 0 からPT への最適輸送計画 ⇧⇤ が⇤<latexit sha1_base64="Vb1wNC/zT5SXJImD4v0FGaWjSkc=">AAADyHichVJNSxxBEK11TDTmwzVegl4ki4kxsPSKRPEk8SLmoq6rgm6WmbHXNM4XPbMr6zAQcswf8OBJQSHkEP+Dl/yBHPwJIUcFLx583TvZEMXdHma6qrrem6pXbQWOCCPGzjNdRveDhz29j/oeP3n6rD878Hw19GvS5iXbd3y5bpkhd4THS5GIHL4eSG66lsPXrJ05db5W5zIUvrcSNQJeds1tT1SFbUYIVbJDm1zKSlx8nYxVP46/eavdFW1XsjmWZ3qN3DUKqZGjdC36A5kPtElb5JNNNXKJk0cRbIdMCvFsUIEYBYiVKUZMwhL6nFNCfcDWkMWRYSK6g+82vI006sFXnKFG2/iLg1cCOUKj7Bf7xi7YT/ad/WbX93LFmkPV0sBuNbE8qPR/fVG86ohysUf06R+qDcLC47bpKm6xSZxW8bZXIELOtO5cQIlAR5QmdrOa+t7+RXFmeTR+xY7YH6hxyM7ZGfTw6pf28RJfPgC74veA2tXquboCD/OKEbe0GmpC5dRX+m61lFJeTDmcJ2ml9zG5ukKh9z09WRX1UTNHr6pnNbW/0xzXStxGJB2qVUyyA3Pcykpwkwu37+1dY3UiX3iXn1yazM2+T+90Lw3TSxpD11M0S/O0SCVU85lO6AedGgtGYOwajWZqVybFDNJ/y/hyAwES1TQ=</latexit>⇤<latexit sha1_base64="JjyTUFk5CvlWxsl31CqkAI+BGrw=">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</latexit><latexit 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sha1_base64="WAN5EuONLcDQha9cz31zqYs8HbU=">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</latexit><latexit 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sha1_base64="I+JVGJ9CowtlFIMkEDWOW4mpK8k=">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</latexit><latexit 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⇤
c
e
⇤
c
e
c
e
c
e
Pr
[|f
(x
,
x
)
f
(x
,
x
)|
>
d((x
,
x
),
(x
,
x
))]

(
)
s
s
t
t
s
s
t
t
c
e
c
e
⇤

((xs ,xs ),(xt ,xt ))⇠⇧

33

輸送に関する誤差の評価
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1<latexit sha1_base64="thNCBlHoutpxf1wV3hR3Ma/H+T8=">AAAEmHichVLLbtNAFL0uAUp4tIUNgo1F1DZpUTWpKkCsKlgAQqA0SR9SJ7VsZ5xY9SvjSVBrjWDND7BgBRILxGew4QdY9BOAZZHYsODO1A3QqO1Y9tw5d87xvWfGSQI/FYTsGWNnCmfPnR+/ULx46fKVicmpq2tp3OcuW3XjIOYbjp2ywI/YqvBFwDYSzuzQCdi6s/1Q5dcHjKd+HDXFTsJaod2JfM93bYGQNfm9aOKgjHOrWfYqMzRgPXPdqpZp1xYZDW3Rde3ArEmrcXsEam55FXOepj2OuMdtN1uUmTsraRB3UMgT5SFK2ywQtqTc73RFRZr/pquY1hrPrYaU5rw5AjflIZNSXfAM7fX6dht/rirPGrOy7G3NVeYPG1HxtvmMJl2/TAO0o21XrMkSWSB6mKNBNQ9KkI9aPGU8BQptiMGFPoTAIAKBcQA2pPhsQhUIJIi1IEOMY+TrPAMJReT2cRfDHTai2/jt4GozRyNcK81Us138S4AvR6YJ0+Qr+Uj2yRfyiXwjv4/VyrSGqmUHZ+eAyxJr4vX1xq9TWSHOArp/WScwHHzCE7rKhmocsx6+JzsgcM893bmPTiQaUZ64B9UMdt/sN+7Xp7MZ8p78QDfekT3yGf2IBj/dDyus/hbVlX6ErBfavVBXEOF5ZYg72g11Qq18rfxtD51SqwxKmJd5pccphbpCX8+7+mQVGmPNDHtVPatTOzzNOe3EUYY8pVqlxE9Rzoa7JN7k6tF7OxqsLS5U7ywsrSyVlh/kd3ocbsItKGPXd2EZHkMNVsE1asbAeGm8KtwoLBceFZ4cbB0zcs41+G8U6n8A3J8ehg==</latexit>ferrT (f )  W1 (P̂S , P̂T ) +

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f
c
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分布 PT (x , x , f (x , x )) 間のWasserstein距離
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• 輸送コストは<latexit sha1_base64="x0LnCTrB3/k85MkCXn5l4CO8VRo=">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</latexit><latexit sha1_base64="Vb1wNC/zT5SXJImD4v0FGaWjSkc=">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</latexit>•err (f ) + errT (f ) を最小化するモデル f ⇤ による誤差
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L の連続性，仮説 f の上界および（確率的な）連続性

によって決まる定数34

先行研究の評価の利点と欠点
•

利点
•

•

推定に用いる最適輸送問題の距離が上界に出現

欠点<latexit sha1_base64="OJcKON279RS69qXfkWSWuGMOioc=">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</latexit>

モデル f の学習が考慮されていない（モデルは任意）• ラベル輸送可能性とは別に，ソース分布とターゲットの推定
分布の距離が離れると上界の評価が過大になる• 実際には考慮しないソース，ターゲット両方の•<latexit sha1_base64="qrFSSdwiSfEfhM2KOQ+BkbDM/10=">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</latexit>⇤

誤差の和を最小化するモデル f の評価が上界に含まれる<latexit sha1_base64="OJcKON279RS69qXfkWSWuGMOioc=">AAAEDXichVO/TxRBFP6ORUH8AWhjYkO8YIwxlzlCwFgRbSg58IAELmR3mTsm7K/s7p2Bzf0DWthoKKwkoTDWdsTGQv8BC/4EY4mJjYXfzK1nhHA7k9158977vn3zvR0n8lSSCnFSGrKGL10eGb0ydvXa9RvjE5M3V5OwHbuy7oZeGK87diI9Fch6qlJPrkextH3Hk2vO7lMdX+vIOFFh8Czdi2TDt1uBairXTumqNbcmyqIizJg6b1Rzo4x8LIWTpQyb2EYIF234kAiQ0vZgI+HcQBUCEX0NZPTFtJSJS3QxRmybWZIZNr27fLe428i9AfeaMzFol1/x+MRETmFafBPvxan4Kj6I7+L3hVyZ4dC17HF1elgZbY2/uL3yqxDlc02x8w81AOFw+gNOlfXZYkabfAYrkDLnkTm5ohKR8WhN3F41nf2D05XHy9PZPXEoflCNd+JEfKYeQeene1STy2/JrvkDop4b9XxTQcB+6a45Rg3doUa+1/pu95XSuwxlxrt5pRcx+aZCZdZ901ntDVmz5Fn1mXXX/nbzgVHiLKJbUK3+d1o5rpi/l72Jh31EEb9miguYs35WlzelevZenDdWZyrVucpsbba88CS/M6O4g7u4T1XnsYBFLKHOaiRe4jXeWK+sj9ax9amXOlTKMbfw37C+/AGb5uD8</latexit>→モデル f の学習を考慮した上界を導出

35

予測モデルのテスト誤差上界<latexit sha1_base64="kZcuVxPndPK72D6dVFCM5u9iNmk=">AAAEFHichVM9TxtBEB1zSfhMsJMGKQ3CAqEostaRRSIqFBo6vmKwhC3r7licle9Ld2sjOPkPpKSAAhoipYjyF9KhSKnSpeAnIEoipaHg7fpiFBC+Xd3t7Oy8d7NvbqzAEZFk7DwzYDx6/GRwaHhkdOzps/Fs7vlG5LdCm5dt3/HDimVG3BEeL0shHV4JQm66lsM3reaiOt9s8zASvvdB7gW85poNT+wI25RwVarLLm+YdVnP5lmB6TF53ygmRp6SseLnMjFVaZt8sqlFLnHySMJ2yKQIc4uKxCiAr0YxfCEsoc85dWgE2BaiOCJMeJt4N7DbSrwe9ooz0mgbX3HwhEBO0jT7zb6yK/aTfWMX7PpBrlhzqFz2sFpdLA/q458m1v+molyskj7eovogLEy3z63iHluI0x08/RWQiHmnby6gRKA9ShO7m017/+hqfX5tOp5hn9kl1Dhl5+wMenjtP/aXVb52DHbF7wG1q9VzdQYe6qWqZmk1VIVqyV7pu91TSu1iyuO8k2T6EJOrMxR63deVVV4fOXPcVd1ZVe1fNV9pJe4iOinZqn+nkeDS+bvRVXrdQ6TxK6YwhTnuRXXQKcW7fXHf2HhTKM4VSqul/ML7pGeG6CVN0SxUfUsLtEQrVNb9cEDHdGIcGt+NM+NHN3Qgk2Be0H/D+HUDld/j9A==</latexit>

ターゲット変数の空間 ⌦t は正定値カーネル K を備え
H を K に対応する再生核ヒルベルト空間であるとする<latexit sha1_base64="tCYkfyDAJ/Ju53eCCVj9ah6OsSs=">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</latexit>•<latexit sha1_base64="GsCSVyjq6l/wC8rCxI/E06NffzU=">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</latexit><latexit sha1_base64="tCYkfyDAJ/Ju53eCCVj9ah6OsSs=">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</latexit>

2
sup
K(x,
x)

⇤
• カーネルが有界
x2⌦t<latexit sha1_base64="qMO61nVzVuUUVQJ3Ci2MaFtnjf4=">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</latexit>

モデル集合 F = {f 2 H | kf kH  a}<latexit sha1_base64="+BfvMoVoDu0oFcFYFDrD1rzhehc=">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</latexit>•<latexit sha1_base64="VSPjkipvVMKDGzVa5KK77IAJM6I=">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</latexit>•

損失関数 L は以下を満たす

対称性 L(y1 , y2 ) = L(y2 , y1 )• 三角不等式 L(y1 , y2 ) + L(y2 , y3 )<latexit sha1_base64="uWEP4g/spRWcPP02EJcmV3FCen8=">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</latexit>•<latexit sha1_base64="MO/ASK7TYd3lEu4wOjwIqxNslqM=">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</latexit>

L(y1 , y3 )リプシッツ連続性 |L(y1 , y2 ) L(y1 , y3 )|  k|y2
• 有界性 L0 = supy2C L(0, y) < 1<latexit sha1_base64="NN3Zs+9M2pNBpuGBsLT0VHym4tU=">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</latexit>•<latexit sha1_base64="YKu2NKLX4TuggQlUSlL9vlBtP8g=">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</latexit>

y3 |

36

予測モデルのテスト誤差上界
c
e
c
e
x
,
x
,
x
,
x
• 最適なモデル f0 2 F0 は任意の 1122 に対して<latexit sha1_base64="iTSOThiIwFcTTyXkJX9n4/DhKi0=">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</latexit><latexit 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sha1_base64="XNTSuDbqXbAIn4f8Rat8YL1LuqQ=">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</latexit>⇡ ⇤ は -確率的転移リプシッツ性をみたす<latexit 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提案法では最適輸送により推定される分布• PT ：真のターゲット分布<latexit sha1_base64="1Z45NkJGNtLSySOMVpcQYSJllNo=">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</latexit>

37

予測モデルのテスト誤差上界•前ページまでの仮定のもと<latexit sha1_base64="jm4T/ibzW2vBzKmO3m5QimgC5Ko=">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</latexit>fˆs ⌘ arg minf 2F err
c T̂ (f ) とする

任意の > 0 および ↵ = k のもと任意の 2 (0, 1)
に対して少なくとも確率 1
で<latexit sha1_base64="LHelSdDtQopSsAkM9RFT6RnZPrY=">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</latexit><latexit sha1_base64="/EsVJ9jjTmtVVBRbkWEEVzntU24=">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</latexit><latexit sha1_base64="CwJJLSsQZqn1hspnKbHTdq8zzHQ=">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</latexit><latexit sha1_base64="6cPwGwtOSI2RkrQCKI56InxQKWM=">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</latexit><latexit sha1_base64="Fu6gCB6EqO2Q0miWphtf98BBg20=">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</latexit>

2ka⇤
ˆ
ˆ
errT (y, fs )  err
c T̂ (fs , y) + p+ (L0 + ka⇤)Ntsln(1/ )Nt

+ W (PT̂ , PT ) + 2 errT (y, f0 ) + kM ( )<latexit sha1_base64="H8FfftUw1AHY1yK+x7VlBJKhdfQ=">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</latexit>

•

c

e

err
c T̂ (f ) = E(xc ,xe ,y)⇠PT̂ L(y, f (x , x ))

38

予測モデルのテスト誤差上界<latexit sha1_base64="Fu6gCB6EqO2Q0miWphtf98BBg20=">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</latexit>

2ka⇤
errT (y, fˆs )  err
c T̂ (fˆs , y) + p+ (L0 + ka⇤)Ntsln(1/ )Nt

+ W (PT̂ , PT ) + 2 errT (y, f0 ) + kM ( )•利点
推定したテスト分布で学習したモデルに対する誤差の評価を行うことで上界がよりタイトになる• ソース，ターゲット共通のモデルの評価は不要
•

•

欠点
•

ターゲット分布の推定方法についての評価は行わない•

Wasserstein距離は推定で考える輸送の距離とは異なる