FY2022/2022年度

High-Dimensional Statistical Modeling Team
Makoto Yamada
⾼次元統計モデリングチーム

⼭⽥ 誠

高次元統計モデリングチームの研究概要

⽬標l 医療, 材料分野において科学的発⾒をサポートする機械学習基盤の構築
l 重要な特徴をデータから精度よく容易に⾒つけられる⼿法の研究開発
l 機械学習研究者以外でも容易に利⽤可能なソフトウェア開発

2022年度主要成果
l 特徴選択l Knockoff filterに基づいた選択的推論 (AISTATS 2022)
l Graph Neural Networkのための解釈⽅法GraphLimeの提案 (IEEE TKDE)

l 最適輸送（⾼次元データの類似度計算）

l ⽊構造Barycenter推定 (AISTATS 2022)
l 1-Wasserstein距離の近似⽅法の提案 (TMLR 2022)l ⾼次元最適輸送⼿法の提案 (ECML 2022)l Word Moverʼs Distanceの再評価 (ICML 2022)l ロバストな最適輸送⽅法の提案 (ICLR 2023)

l 新規の科学的発⾒を容易にする基盤を確⽴し, ヘルスケア, 材料,
農業等の分野で⾰新を⽬指す
l 医療費の削減 (個別化医療, 疾病予測)l 材料発⾒の⼤幅な効率化





1
1
0

 
l へシアン⾏列近似⼿法のimplicit differentiation への適応 (AISTATS 2023)

l 機械学習応⽤

l 細胞核画像分類のための⼿法の



…



l ⾼次元・構造データの学習





0
1

By Guillaume Paumier, from https://commons.wikimedia.org/wiki/File:DNA_microarray.svg

高次元小標本データ

)





 Lasso,#C=?"!:F
69.

10$ %

10

我々のチームでは最適輸送の研究に取り組んでおり特に最適輸送の研究に取り組んでいる。

10'
10&

10'

10%


10#

10$

<latexit sha1_base64="jiWHYSSXHKXxNtKgT1nL2YuFhTA=">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</latexit>

l複数⽊に基づいた⽊構造Wasserstein距離 NeurIPS 2019l不均衡⽊構造最適輸送の研究 NeurIPS 2020, AISTATS 2021
lNeural Architecture Search (NAS)への応⽤ NeurIPS2021
l異なるドメイン間の⽊構造最適輸送 AISTATS 2021
l密度⽐推定に基づいた最適輸送 ECML 2021
l⽊構造データにおけるBarycenter推定 AISTATS 2022l⾼次元データのための最適輸送 ECML 2022
lWord Mover’s Distance (WMD)の再評価 ICML 2022
l⽊構造最適輸送を⽤いたWasserstein距離の近似
TMLR 2022
Yuki Takezawa , Ryoma Sato , Zornitsa Kozareva , Sujith Ravi , Makoto YamadaC=?<D1E)/
0+  , .*;
B4-+**'


10&

lFeature Robust Optimal Transport (FROT)

1,2

1,2

3

4

FROT(µ, ⌫) =Fixed Support Tree-Sliced Wasserstein Barycenter

カーネル選択的推論(AISTATS 2022) A
HSIC(gaussian)

HSIC(gaussian)

`=1

a
x(`)

Feature Robust Optimal Transport (FROT)<latexit sha1_base64="L3BU/IT3q3uES8MG5EgG5Qq4+XQ=">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</latexit>

C`

min

⇧2U (a,b)<latexit sha1_base64="OacnRizJRBX89wDWgRojPha2Xaw=">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

log

L
X
`=1

exp

✓

1
h⇧, C ` i
⌘

◆!

y (`)
<latexit sha1_base64="YQVQlZDTAKyi4yW8oPOIIFbJkMw=">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</latexit>b
Target image

PC

Pearson

任意のメトリックの1-Wasserstein距離をL1距離で近似する⽅法(TWD)を提案。
TWD: W (µ, ⌫) = kdiag(w)B(a b)k ,B = [b , b , . . . , b]. ⽊のパラメータ

lFDRのコントロールが可能
lGithub<latexit sha1_base64="MT/KrGCDukwgax2+13oMNQkcJio=">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</latexit>

Lasso法に基づいた重み学習法を提案.
0.4 w
b := argmin(d(x , y ) w z ) + kwk

Published in Transactions on MachineX
Learning Research (MM/YYYY)<latexit sha1_base64="EgrcICOganc/tvDjb+mfx+vCVtc=">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</latexit>

1−α

(`)

Feature-Robust Optimal
CAM Transport for High-Dimensional Data

⽊構造Wasserstein距離を⽤いた
TR 1-Wasserstein距離の近似 (TMLR 2022)

PC

Test accuracy

DCDC

lK個の特徴を選択
lKnockoﬀ ﬁlterを⽤いて検定を⾏うPeterJackNaylor/knockoff-MMD-HSIC

i=1 j=1(`)↵` c(xi , y j )B

cMMD(gaussian)

lKnockoﬀ ﬁlterを初めてA
cMMD
カーネル法(HSIC, MMD)
Raw
(gaussian)に適⽤.

Algorithm 1: PSD for the FS-TWB.
1: Input: Probability measures a1 , a2 , . . . , aN , andstep size 0 < 1 and 0 < 2  1.
2: Output: The FS-TWB.
3: Let a(0) 2 A.
4: abest
a(0)
best
5: f
f (a(0) )(a) MNIST(b) AMAZON(c) AGNews
which we refer to as the fixed support tree-Wasserstein
6:
for
k
=
0,
1,
.
.
.
,
K
do
Figure 2:(FS-TWB).
Objective function
value for Eq.
(3) with
The results are averages for all
barycenterIn the FS-TWB
problem,
we the entropic regularization.
(k)
7:
Let
g
be
an
any
subgradientof f ata(k) .
To computethe loss atmeasures
each barycenter,
Sinkhorn algorithmwith the same
parameters
onlycategories.
need to consider
the probability
on Vleaf . we use the(k)18:
as the IBP.
FS-WB
(IBPthe+ objective1e-3) denotes
the barycenternormalized
the sum is one after all(k+1) 2such
kg(k) kthat
2
Combining
Eq. The(5) with
Eq. (2),function(k+1)(k)(k) (k)probabilities in the FS-WB (IBP) less than 0.001 are set 9:
to zero.
aprojA (ag )is rewritten as follows:
(k+1)
(k+1)
10:
f
f (a
)
✓
◆
1
best
(k+1)
)
D
SWB on all datasets. (IThe D
reasonisthat,because
a
1
2
11:
if f
>f
then
B = wv
,
(6)
best
(k+1)
I than a chain, a tree 12:tree has more degrees
of freedomaa
can approximate theX
Noriginal space better than a chain. 13:
f bestf (k+1)
1
Next, wefcompare
andthe FS-TSWB.(7)On 14:
end if(a) = the FS-WB
kBa Bai k1 ,
all datasets, theNlossatthe
FS-TSWB
decreases
asi=1
15: end forthe number of sampled trees increases. In particular, 16: return abest
where[ai ]k = µas
). We
i (v
k = µ(vtrees
on MNIST,
thesampledincreases,|Vinnumber|+k ) andof[a]|Vin |+k
|V
|
leaf
the A
loss=at{atheFS-TSWB
becomesthan thedefine2 R| kak
The FS-TWB
1 = 1}. smaller
+
loss
at
the
FS-WB
obtained
by
the
IBP.
Becausethere
problem can then be formulated as follows:needs to be computed only once per iteration. Because
are many pixels on which the probability is zero in all
B is a sparse matrix that has at most D|Vleaf | non-zeroimages, the probability
on manythea 2 argminf (a). pixels is zero in(8)(k)
Figure 3: Visualizationof theFS-WB,
theTherefore,
FS-SWB,is computedin O(D|V
leaf |).
a2A in practice, the probability elements, Ba
optimal FS-WB. However,(k)
FS-TSWBMNIST.
g andisthecomputedin on
O(N|V |+D|Vleaf |). Because theon these pixels are not zero in the FS-WB obtained byinternal nodes that have only one child node can be
3.2theOptimization
IBP. Indeed, theMethod
result shows that, in the FS-WB
4.5 Time Consumption
abbreviated,
we can assume |V |< 2|Vleaf | without aobtained by the IBP, the loss decreases
by setting the
1.00
of generality.
Then,the time
complexity
required
Theprobability
objective function
is a nondi↵erentiable
convexIn thissection, we
evaluate
the time
consumptionofbelowTR
thef thresholdto zero. On theother lack
Pearson(k)B
tocompute
g
is
O((N
+
D)|V
|).
leaf
functionandLipschitz
continuous,
andthefeasible
the FS-TSWB. Table 1 shows the time required tohand, in the FS-TSWB obtained by the FastPSD, the
1.00
regionA
isconvex.
Therefore,
the
FS-TWBproblemissolve the FS-WB
andvalue.
FS-TSWB
by using
probability on these pixels is zero by the projection Objectivefunction
Next,problems
we describe
the0.75
a convex
optimization
problem,which
be solved
the complexity
IBP and the
FastPSD
respectively.
theonto thesimplex per
iteration.
Ascan
a result,
the byloss timerequired
to computef (a(k)When). Considnumber
is one, the FS-TSWB canusing
projected subgradient(PSD)is(Boyd(k)
at the FS-TSWB
obtained by descent
the FastPSD
smaller ering
thatofBasampled
Batreesi is computed when computing
0.75
besolvedfaster
than
the
FS-WB
on all datasets. In
et al.,
2003).
In other
PSD converges
to an
than
the lossat thewords,
FS-WBtheobtainedby the IBP.
the subgradient, the time complexity required to comparticular, on AGNews, using the FastPSD, the FSarbitrarily close approximation to the0.50
global minimum
pute the objective function value is O(N |Vleaf |). In
TSWB can be solved approximately 125 times fastervalue of the FS-TWB problem. Algorithm 1 shows thesummary, the time complexity for each iteration of thethan the FS-WB. Comparing the time consumption
PSD
the FS-TWB problem.
In the following, we
4.4for0.50
Visualizationof Barycenters
PSD is O(|Vleaf |(log(|Vleaf |) + N + D)), which is fasterof the FS-TSWB when the number of trees increases,
describe each modules of this algorithm
in detail.
0.25
than the IBP in terms of the number of supports |Vleaf |.
the time consumption of the FS-TSWB increases inIn this section,
weshow a visualizationof thebarycenProjectionontoasimplex.
The
functionproj
inA
proportion to the number of sampled trees. Then, there
Tprojection
1
ters. 0.25
Fig.1 3isshows
the FS-WB,
FS-SWB,the
Algorithm
theof thea givenvectorand
x 2
is the trade o↵ between the performance and the time
onsimplex
MNIST.
the FS-SWB
and 3.3 Fast Projected Subgradient Descentleaf |
0.00
R|VFS-TSWB
onto the
A,Comparing
which is defined
as follows:
consumption.
the FS-TSWB, the FS-TSWB is closer to the FS-WB
DCleaf
HSIC
PC
Pearson1 cMMD2(9)an The
N
than 0.00
the FS-SWB.
FS-SWB,have
In addition,
we ofevaluatethearetime
consumption
morebottlenecks
the PSD
two
parts: the in
partto
projA (x)In=theargminkx someak22 . pixels(gaussian)(gaussian)DC
Pearsona2A
unnaturally highcMMD
probability.
In particular,
theHSIC
pixels computedetailsPC
on
Fig. 4g(k)
shows
time
theMNIST.
subgradientandthetheTRpartconsumption
to compute(k)(gaussian)
Association
measurein the area indicatedby the blue stars in Fig.(gaussian)
3 have a thewhenvaryingthenumberofimagesandwhenvarying
objective function value f (a ). In this section,
weThis can be solved using the algorithm proposed byhigh probability even if the number of chains
increases. propose
themeasure
numberofsupports
byresizing
theimage.
The
Association
algorithm
time complexity.
0.1 results
0.2the0.3 to reduce
0.4 these0.5
0.6
0.7
Duchi et al. (2008) in O(|Vleaf |log(|Vleaf |)).
α
By contrast, in the FS-TSWB, the probabilityon theshow that the time consumption of the IBP and
2with
0.15Subgradient
0.35
0.55
0.65to 0.75
0.1 by the0.2blue stars
0.3is properly
0.4
0.50.25 0.6of>
0.7 0.45
0.8
f . linearlyFirst,
weshow
the0.9algorithm
0.5
αf .indicatedpixels in the ofarea
the PSD increases
respect
to the numOneof the 0.25
subgradientsof f 0.45
at
Subgradient
0.15
0.35
0.55
0.65
0.75
0.85
i
j samples.
i,j forthecomputing
1 oftime complexity
the subgraeven if theasnumber
ber ofthe
By contrast,
time consumptiona(k)zerois calculated
follows:of trees is one. Moreover, the reduce
(k)
(k)
(k)
d
gP . Weincreases
define bwith=O(log(NBa , )).
bi =i andresults showw2Rthat increasing the number of thethe FastPSD
AsBaa result,! trees can dient
N
+
N
1
(k)
(k)
(k)
>
(k)
(i,j)2⌦sign(bbithe). (i.e.,
g
=
).
1 > X
make(k)the FS-TSWB
smoother.(k)Appendix F includes z the=timei=1
consumption
of
FastPSDis almostN B z theg=Bsign(Ba
Ba
)
,
(k)
i
Then,
j-th
elementof zof samples
is computed
as follows:
the remaining
sametheeven
if thenumberincreases.
Next,
N visualization of the barycenters.

l提案法は凸最適化かつProjected gradient descentを使って⾼速に計算可能︕

l ⾮線形モデルは複雑 （誰も利⽤しない）
l ⾮凸最適化が利⽤される（最適化が難しい）⇧2U (µ,⌫) ↵2⌃L

⇡ij

CAM

lTree Wasserstein distanceを利⽤したBarycenterの推定
⽅法を提案

チャレンジ

Raw

Source image

Then, given a set of probability measures {µi |µi 2
P (Vleaf )}N
i=1 , the tree-Wasserstein barycenter on Vleaf
is defined as follows:!
N
1 XµdT 2 argmin
WdT (µ, µi ) ,(5)µ2P (Vleaf ) N
i=1

l 少数特徴で⾼い予測性能l 特徴の信頼度がわかる （仮説検定）

max

L
X

1,2

⽊構造重⼼推定(AISTATS 2022)

解釈性

min

n X
m
X

Test accuracy

(




10#

Feature Robust Optimal Transportの提案 (ECML 2022)

最適輸送の研究開発

35-,/Lasso(Tibshirani, 1996, 728 30000)

 #"! (>A, @()(SNPs, $ &%, etc.)

主要成果２：最適輸送 (続き)主要成果２: 最適輸送

主要成果１：高次元非線形特徴選択

0.0

1−α

0.1

0.0

0.2

0.3

0.1

0.4

0.2

0.5

0.3

TR

0.8
0.85

0.9

Fig. 7: Qualitative
examplesFor eachBaseline(CVPRimage,2020) the red
Proposedmethod sampled from SPair-71k.
points represent the keypoints to be matched while the green points denote theprediction and the blue ones denote the ground truth. Figures in left panels are
zi,j = bi +i=1bj 2bi bj
N⇣⌘
X
Table 4: Document
results
theHereTwitter, BBCSport,
and Amazon datasets.
We usedtheresults
ten and the in right panels are Sinkhorn results, respectively.
whereclassification
sign is the element-wisesignum forfunction.
[z(k) ]j =sign [b(k) ]j [bi ]j .
(10)
FROT
we describe the time complexity required to comnearest neighbor after,classifiers.
we set i=1
the regularization parameter with 10≠2 andputeg(k)i. ,For
i,は1-Wasserstein距離の任意のメトリック.
Bai theneedsSinkhorn
to be computedalgorithm,
onlyd(xyiall)For
once before withstarting theiterations.
In addition, Ba(k)[z(k) ]j depends five
only ontimes
the number
the maximum iteration
100,
respectively.
We Fromran Eq.
the(10),algorithms
by changing the randomlGraph Neural Network (GNN)の解釈⼿法であるseed and report 1-Wasserstein距離
the averaged classification
accuracy. For the proposed method, we set the regularization- 高次元非線形特徴選択手法の提案(WMD)よりも数百倍⾼速かつ同等≠3parameter as ⁄ =10.
GraphLIMEを提案- 最適輸送における重心(Barycenter)の超高速計算技術をの性能を得る⽅法を提案︕

GraphLIMEの提案 (TKDE 2022)<latexit sha1_base64="Sc09GWjf86oVhK4x1qw53kl/UL4=">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</latexit><latexit sha1_base64="rZ67Cte4MPOlOIZl73zAQ8hCppo=">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</latexit>

lアイディア: HSIC Lasso法をGNN解釈に適応
1
min
L̄
d
2
↵2R+
<latexit sha1_base64="Soo4iAJcm4lM5DC8VDvWW4dEfwQ=">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</latexit>

d
X

k=1

(a)

pyHSICLasso

2

↵k K̄

(k)
F

github.com/rikenaip/pyHSICLasso$ pip install
pyHSICLasso+ k↵k1

Neural_network

Explained node
Sampling node

Learning

Original feature

Neural(b)

Neural(c)

Powerful
Deep

Feature as explanation
Prediction from GNN

Deep

1: Sample neighbors of the explained red node; (b) Obtain finite numbers of features as explanations (the green part of the features

Methods
WMD (Sinkhorn)
QuadTreeqTWD
Sliced-QuadTree
Sliced-qTWD
ClusterTree
cTWD
Sliced-ClusterTree
Sliced-cTWD

Twitter

BBCSport

Amazon0.675 ± 0.033

0.973 ± 0.015

0.903 ± 0.006

0.701 ± 0.027
0.691 ± 0.028
0.694 ± 0.017
0.697 ± 0.027
0.683 ± 0.019
0.699 ± 0.032
0.694 ± 0.010
0.700 ± 0.021

0.970 ± 0.016
0.967 ± 0.014
0.970 ± 0.020
0.967 ± 0.014
0.901 ± 0.056
0.962 ± 0.016
0.929 ± 0.037
0.970 ± 0.018

本年度まとめ開発- 高次元データからの最適輸送手法を提案

0.865 ± 0.001
0.847 ± 0.036

お知らせ

0.873 ± 0.010
0.878 ± 0.006

高次元統計モデリングチームは2023/3月末で閉鎖します。
これまで6年間サポートいただきありがとうございました。
今後は研究の場を沖縄科学技術大学院大学に移し引き続き
機械学習の研究活動を続ける予定です。上記研究テーマにご興味あれば是非 mlds@oist.jp までご連絡ください！

0.877 ± 0.001
0.887 ± 0.011

0.900 ± 0.011
0.905 ± 0.010