Integral representation of shallow neural network …shallow neural network that attains the global...
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数理科学チーム 坂内 健一 Mathematical Science Team Kenichi Bannai 【目的】共同研究を通して、様々な数学理論を機械学習的課題に応用する PERRON-FROBENIUS作用素による時系列データの解析 熱を使ったハイパーグラフの解析 多目的最適化問題のパレートフロントの、ベジエ単体による推定 f : X ! R m 多目的最適化問題： X ⇤ (f )= {x ⇤ 2 X |6 9x 2 Xf (x) <f (x ⇤ )} パレート解集合 パレートフロント（パレート解集合の像） fX ⇤ (f ) := f (X ⇤ (f )) ⇢ R m X*( f , f , f ) 1 2 3 X*( f ) X*( f ) 1 X*( f ) 2 3 X*( f , f ) X*( f , f ) X*( f , f ) 2 2 1 1 3 3 f f f f f f f (b) Pareto set images fX ∗ ( f )( J ⊆ I ) Smale73などにより、f が凸連続な場合などには、 位相幾何的な「単体」の構造が入ることが期待されている。 本研究では、CGなどに用いられるベジエ曲線の高次元版である 「ベジエ単体」を用いて、パレートフロントに誘導される単体の構造を 低次元から高速・帰納的に近似する「骨格近似法」を提案した。 ベジエ単体 P d ：制御点 B (t) := X |d|=D ✓ N d ◆ t d P d d =(d i ) 2 N m |d| = X d i = D t 2 Δ m ⇢ R m の最小値を調べる 富士通、AIP杉山氏との共同研究 三内、田中、坂内が担当 浅いNNの積分表示 Kobayashi, et. al, Bezier Simplex Fitting: Describing Pareto Fronts of Simplicial Problems with Small Samples in Multi-objective Optimization, AAAI-19 ハイパーグラフの例 人間関係（グループ） 商品購入 同僚 近所 同級生 枝：一度の購入で買われた商品 AIP河原チーム、NTT研究所との共同研究。石川、池田が担当 M. Ikeda, A. Miyauchi, Y. Takai, and Y. Yoshida. Finding Cheeger Cuts in Hypergraphs via Heat Equation, arXiv:1809.04396 本研究では、F. Chungにより定義されたハイパーグラフの熱の 概念を、ハイパーグラフのコミュニティ検索に応用した。 ハイパーグラフの熱 （Chan et al.）の伝わり方 本研究では、時系列データのモデルである非線形力学系に対して、対応するRKHS上の Perrin-Frobenius作用素を用いて、力学系間の角度（metric）を定義した。 Ishikawa, K. Fujii, M. Ikeda, Y. Hashimoto, Y. Kawahara, Metric on nonlinear dynamical systems with Perron-Frobenius operators Proc. of NeurIPS 2018 NII吉田悠一氏、AIP宮内氏と共同研究。池田、高井が担当 S.Sonoda, I.Ishikawa, M.Ikeda, K.Hagihara, Y.Sawano, T.Matsubara, N.Murata, Integral representation of shallow neural network that attains the global minimum,. arXiv:1805.07517, 2018 AIP園田氏などとの共同研究。石川、池田、萩原が担当 g (x)= p ÿ j =1 c j ‡ (a j · x ≠ b j ) S [“ ](x)= ⁄ R m ◊R “ (a, b)‡ (a · x ≠ b)μ(a)dadb 本研究では、浅いNeural Networkの backpropagationに よる学習を、リジレット変換で解釈した。 離散化 連続化 S [R[f ]] = f R[f ] のリジレット変換 は、 をみたす関数 f f : M ! M 理研研究奨励賞受賞 力学系 φ : M ! H k RKHSで捉える K f : H k ! H k Perron-Frobenius作用素 D =(f,h,x) ：力学系 ：初期値 ：observable f h x 間の角度（metric）を定義
Transcript of Integral representation of shallow neural network …shallow neural network that attains the global...
数理科学チーム 坂内 健一 Mathematical Science Team
Kenichi Bannai
【目的】共同研究を通して、様々な数学理論を機械学習的課題に応用する
PERRON-FROBENIUS作用素による時系列データの解析
熱を使ったハイパーグラフの解析
多目的最適化問題のパレートフロントの、ベジエ単体による推定f : X ! Rm多目的最適化問題：
X⇤(f) = {x⇤ 2 X | 6 9x 2 X f(x) < f(x⇤)}パレート解集合
パレートフロント（パレート解集合の像）fX⇤(f) := f(X⇤(f)) ⇢ Rm
X*( f , f , f )1 2 3
X*( f )
X*( f )1
X*( f )2
3
X*( f , f )
X*( f , f )
X*( f , f )2
21
13 3
(a) Pareto sets X∗( f J ) ( J ⊆ I)
X*( f , f , f )1 2 3
X*( f )
X*( f )1
X*( f )2
3
X*( f , f )
X*( f , f )
X*( f , f )
2
21
1
3
3
f
f
f
f
f
f
f
(b) Pareto set images f X∗( f J ) ( J ⊆ I)
Fig. 1: The face relation of Pareto sets (their images) of a simplicial problem f ! ( f1 , f2 , f3).
ここで !Ni" は 2項係数を表す．ベジェ曲線を規定
する点 P0 , P1 , . . . , PN を制御点という．B(t)について B(0) ! P0 , B(1) ! PN が成り立ち， B(t) は0 ≤ t ≤ 1の範囲で 2つの制御点 P0 , PN を端点とする曲線を描く．続いて Borges and Pastva 1) によるベジェ曲線
を用いた曲線近似法について述べる．ここでは入力として与えられたサンプル集合 x1 , x2 , . . . , xn ∈ Rd
を通る N 次ベジェ曲線を考える．Borges andPastva 1) はベジェ曲線の当てはめのために以下の最適化問題を解くことを考案した：
minn#
j!1∥B(t j) − xj ∥2. (2)
ここで t j はサンプル xj に対応するベジエ曲線の媒介変数であり，問題 (2)の目的関数は各サンプル xj とベジエ曲線上の点 B(t j) との残差 2乗和である．いま各サンプル xj に対応する媒介変数 t j を固定
すると，ベジェ曲線 B(t j)は制御点 P0 , P1 , . . . , PN
に関して線形関数となる．したがって媒介変数t j を固定したもとで，問題 (2)の目的関数を最小化する制御点は線形方程式を解けば得られる．しかし実際は各サンプルに対応する媒介変数 t j
は未知であり，ベジェ曲線を用いた近似では t j
の調整も合わせて行う必要がある．そこで Borges and Pastva 1) は媒介変数の割当
てと制御点の調節を交互に繰り返すベジエ曲線近似法を考案した．まず，制御点の初期点を適当に与える．続いてその制御点を固定したもとで各サンプルに対する媒介変数 t j の割当てを行う．Borges and Pastva 1) では点 xj からベジエ曲線に垂線をおろし，その垂線の足に対応する媒
介変数を t j の値とした．ベジェ曲線 B(t)が点 xj
の垂線の足であるとき，$∂∂t
B(t), B(t) − xj
%! 0 (3)
を満たす．そこで各サンプル xj について非線形方程式 (3)をNewton法で解き，その解を t j の値とする．続いて，割当てられた t j を固定したもとで問題 (2)を解き，制御点の調節を行う．先の延べた通り t jを固定した問題 (2)では，目的関数を最小化する制御点は線形方程式を解けば得られる．そして得られた制御点を固定したもとで再度媒介変数の割当てを行う．このように媒介変数の割当てと制御点の調節を適当な終了条件が満たされるまで繰り返してベジエ曲線の当てはめを行う．Borges and Pastvaのアルゴリズムを Algorithm 1にまとめる．
注意 1 Algorithm 1ではベジエ曲線のすべての制御点を同時に調節する．一方，事前にいくつか制御点を固定したもとで残りの制御点を調節する場合でも，Algorithm 1と同様，調節対象の制御点の調節と媒介変数の割当てを繰り返してベジェ曲線の当てはめを行うことができる．
3 ベジエ単体とその当てはめ法の提案本章では，2章のベジエ曲線を任意次元に一般
化したベジエ単体を定義し，その当てはめ法について述べる．3.1 ベジエ単体ベジエ曲線を一般の単体に拡張させて M + 1
次元の N 次ベジエ単体を定義する．まず非負整数を成分に持つ M 次元ベクトル全体の集合を
Smale73などにより、f が凸連続な場合などには、
位相幾何的な「単体」の構造が入ることが期待されている。
本研究では、CGなどに用いられるベジエ曲線の高次元版である
「ベジエ単体」を用いて、パレートフロントに誘導される単体の構造を 低次元から高速・帰納的に近似する「骨格近似法」を提案した。
ベジエ単体 Pd：制御点
B(t) :=X
|d|=D
✓N
d
◆tdPd
d = (di) 2 Nm
|d| =X
di = D
t 2 �m ⇢ Rm
の最小値を調べる
富士通、AIP杉山氏との共同研究 三内、田中、坂内が担当
浅いNNの積分表示
Kobayashi, et. al, Bezier Simplex Fitting: Describing Pareto Fronts of Simplicial Problems with Small Samples in Multi-objective Optimization, AAAI-19
ハイパーグラフの例人間関係（グループ） 商品購入
同僚
近所 同級生
枝：一度の購入で買われた商品
AIP河原チーム、NTT研究所との共同研究。石川、池田が担当
M. Ikeda, A. Miyauchi, Y. Takai, and Y. Yoshida. Finding Cheeger Cuts in Hypergraphs via Heat Equation, arXiv:1809.04396
本研究では、F. Chungにより定義されたハイパーグラフの熱の
概念を、ハイパーグラフのコミュニティ検索に応用した。
ハイパーグラフの熱 （Chan et al.）の伝わり方
本研究では、時系列データのモデルである非線形力学系に対して、対応するRKHS上の
Perrin-Frobenius作用素を用いて、力学系間の角度（metric）を定義した。
Ishikawa, K. Fujii, M. Ikeda, Y. Hashimoto, Y. Kawahara, Metric on nonlinear dynamical systems with Perron-Frobenius operators Proc. of NeurIPS 2018
NII吉田悠一氏、AIP宮内氏と共同研究。池田、高井が担当
S.Sonoda, I.Ishikawa, M.Ikeda, K.Hagihara, Y.Sawano, T.Matsubara, N.Murata, Integral representation of shallow neural network that attains the global minimum,. arXiv:1805.07517, 2018
AIP園田氏などとの共同研究。石川、池田、萩原が担当
Integral representation of shallow neural network
that attains the global minimum
Sho Sonoda1, Isao Ishikawa
1,2, Masahiro Ikeda
1,2, Kei Hagihara
1,2,
Yoshihiro Sawano3, Takuo Matsubara
4, and Noboru Murata
4.
1 Advanced Intelligence Project, RIKEN, 2 Department of Mathematics, Faculty of Science and Technology, Keio University,3 Department of Mathematics and Information Science, Tokyo Metropolitan University,
and 4 School of Advanced Science and Engineering, Waseda University.
Abstract
We consider the supervised learning problem with shallow neural networks. According to our unpublished experiments conducted several years prior to this study, we hadnoticed an interesting similarity between the distribution of hidden parameters after backprobagation (BP) training, and the ridgelet spectrum of the same dataset. Therefore,we conjectured that the distribution is expressed as a version of ridgelet transform, but it was not proven until this study. One di�culty is that both the local minimizers andthe ridgelet transforms have an infinite number of varieties, and no relations are known between them. By using the integral representation, we reformulate the BP trainingas a strong-convex optimization problem and find a global minimizer. Finally, by developing ridgelet analysis on a reproducing kernel Hilbert space (RKHS), we write theminimizer explicitly and succeed to prove the conjecture. The modified ridgelet transform has an explicit expression that can be computed by numerical integration, whichsuggests that we can obtain the global minimizer of BP, without BP.
Motivation
Hidden parameters after BP training
argmin(aj ,bj ,cj )
nÿ
i=1
|f(xi) ≠ g(xi)|2.
Here, g is a neural network
g(x) =pÿ
j=1
cj ‡(aj · x ≠ bj)
with activation function ‡.
?
=
Spectrum of the ridgelet transform of f
R[f ](a, b)
:=⁄
Rm
f(x)fl(a · x ≠ b)dx
¥nÿ
i=1
f(xi)fl(a · xi ≠ b)�x,
with a ridgelet function fl.
Integral representation of neural network
g(x) =pÿ
j=1
cj‡(aj · x ≠ bj)
Continuum limit Discretization
S[“](x) =⁄
Rm◊R“(a, b)‡(a · x ≠ b)µ(a)dadb
The ridgelet transform
The ridgelet transform R is a solution operator of S[“] = f . Namely,S[R[f ]] = f. (1)
For a fixed S, there are an infinite number of di�erent ridgelet function fl.
BP training in the integral representation
a, b, c
L
Minimizenÿ
i=1
|f(xi) ≠ g(xi)|2 w.r.t. (aj , bj , cj)
(nonlinear optimization, many local minima)
γ
L
Minimize Îf ≠ S[“]Î2 + —ΓÎ2 w.r.t. “
(strong convex optimization, unique global minimum!)
Problem formulation
Let F be a space of data f : Rm æ R; and G = L2(µ(a)dadb) be the space of
coe�cient “. Given f œ F and — > 0, minimizeL[“] := Îf ≠ S[“]Î2
F + —ΓÎ2G , (2)
subject to “ œ G.
RKHS associated with neural networks
We define the kernel function associated with G as follows
k(x, y) :=⁄
Rm◊R‡(a · x ≠ b)fl(a · y ≠ b)µ(a)dadb, (3)
where fl is a function.Let
q(a) := (2fi)m≠1⁄
R
‚‡(’)‚fl(’)|’|m
µ
1a
’
2d’. (4)
Then,
k(x, y) = (2fi)≠m
⁄
Rm
q(a)eia·(x≠y)da. (5)
If q is a probability density function, then according to Bochner’s theorem, k is ashift invariant positive definite kernel (= reproducing kernel). The correspondingRKHS, say Hk, reflects the approximation ability of neural network S[“].For example, when ‡, fl and µ are Gaussian,
k(x, y) = (1 + |x ≠ y|)≠ m+12 . (6)
Ridgelet transform on RKHS
The ridgelet transform, or the solution operator of S[“] = f for a given f œ Hk, isgiven by
R[f ](a, b) := 12fi
⁄
R
‚f(’a)‚fl(’)q(’a)
ei’bd’. (7)
Main results (arXiv:1805.07517)
Let F = Hk. Then, the global minimizer of BP training is given by the ridgelettransform
argmin“œG
L[“] = 11 + —
R[f ]. (8)
Sketch of proof
Because S : G æ Hk is bounded, according to the Tikhonov regularization theoryfor operators, the global minimizer is given by
argmin“œG
L[“] = (— + Sú
S)≠1S
úf. (9)
We showed that (— + Sú
S)≠1S
ú = (— + 1)≠1R by calculating S
ú and — + Sú
S.
[1] S.Sonoda, I.Ishikawa, M.Ikeda, K.Hagihara, Y.Sawano, T.Matsubara, N.Murata, arXiv:1805.07517, 2018.
Integral representation of shallow neural network
that attains the global minimum
Sho Sonoda1, Isao Ishikawa
1,2, Masahiro Ikeda
1,2, Kei Hagihara
1,2,
Yoshihiro Sawano3, Takuo Matsubara
4, and Noboru Murata
4.
1 Advanced Intelligence Project, RIKEN, 2 Department of Mathematics, Faculty of Science and Technology, Keio University,3 Department of Mathematics and Information Science, Tokyo Metropolitan University,
and 4 School of Advanced Science and Engineering, Waseda University.
Abstract
We consider the supervised learning problem with shallow neural networks. According to our unpublished experiments conducted several years prior to this study, we hadnoticed an interesting similarity between the distribution of hidden parameters after backprobagation (BP) training, and the ridgelet spectrum of the same dataset. Therefore,we conjectured that the distribution is expressed as a version of ridgelet transform, but it was not proven until this study. One di�culty is that both the local minimizers andthe ridgelet transforms have an infinite number of varieties, and no relations are known between them. By using the integral representation, we reformulate the BP trainingas a strong-convex optimization problem and find a global minimizer. Finally, by developing ridgelet analysis on a reproducing kernel Hilbert space (RKHS), we write theminimizer explicitly and succeed to prove the conjecture. The modified ridgelet transform has an explicit expression that can be computed by numerical integration, whichsuggests that we can obtain the global minimizer of BP, without BP.
Motivation
Hidden parameters after BP training
argmin(aj ,bj ,cj )
nÿ
i=1
|f(xi) ≠ g(xi)|2.
Here, g is a neural network
g(x) =pÿ
j=1
cj ‡(aj · x ≠ bj)
with activation function ‡.
?
=
Spectrum of the ridgelet transform of f
R[f ](a, b)
:=⁄
Rm
f(x)fl(a · x ≠ b)dx
¥nÿ
i=1
f(xi)fl(a · xi ≠ b)�x,
with a ridgelet function fl.
Integral representation of neural network
g(x) =pÿ
j=1
cj‡(aj · x ≠ bj)
Continuum limit Discretization
S[“](x) =⁄
Rm◊R“(a, b)‡(a · x ≠ b)µ(a)dadb
The ridgelet transform
The ridgelet transform R is a solution operator of S[“] = f . Namely,S[R[f ]] = f. (1)
For a fixed S, there are an infinite number of di�erent ridgelet function fl.
BP training in the integral representation
a, b, c
L
Minimizenÿ
i=1
|f(xi) ≠ g(xi)|2 w.r.t. (aj , bj , cj)
(nonlinear optimization, many local minima)
γ
L
Minimize Îf ≠ S[“]Î2 + —ΓÎ2 w.r.t. “
(strong convex optimization, unique global minimum!)
Problem formulation
Let F be a space of data f : Rm æ R; and G = L2(µ(a)dadb) be the space of
coe�cient “. Given f œ F and — > 0, minimizeL[“] := Îf ≠ S[“]Î2
F + —ΓÎ2G , (2)
subject to “ œ G.
RKHS associated with neural networks
We define the kernel function associated with G as follows
k(x, y) :=⁄
Rm◊R‡(a · x ≠ b)fl(a · y ≠ b)µ(a)dadb, (3)
where fl is a function.Let
q(a) := (2fi)m≠1⁄
R
‚‡(’)‚fl(’)|’|m
µ
1a
’
2d’. (4)
Then,
k(x, y) = (2fi)≠m
⁄
Rm
q(a)eia·(x≠y)da. (5)
If q is a probability density function, then according to Bochner’s theorem, k is ashift invariant positive definite kernel (= reproducing kernel). The correspondingRKHS, say Hk, reflects the approximation ability of neural network S[“].For example, when ‡, fl and µ are Gaussian,
k(x, y) = (1 + |x ≠ y|)≠ m+12 . (6)
Ridgelet transform on RKHS
The ridgelet transform, or the solution operator of S[“] = f for a given f œ Hk, isgiven by
R[f ](a, b) := 12fi
⁄
R
‚f(’a)‚fl(’)q(’a)
ei’bd’. (7)
Main results (arXiv:1805.07517)
Let F = Hk. Then, the global minimizer of BP training is given by the ridgelettransform
argmin“œG
L[“] = 11 + —
R[f ]. (8)
Sketch of proof
Because S : G æ Hk is bounded, according to the Tikhonov regularization theoryfor operators, the global minimizer is given by
argmin“œG
L[“] = (— + Sú
S)≠1S
úf. (9)
We showed that (— + Sú
S)≠1S
ú = (— + 1)≠1R by calculating S
ú and — + Sú
S.
[1] S.Sonoda, I.Ishikawa, M.Ikeda, K.Hagihara, Y.Sawano, T.Matsubara, N.Murata, arXiv:1805.07517, 2018.
本研究では、浅いNeural Networkの backpropagationに
よる学習を、リジレット変換で解釈した。
離散化連続化
S[R[f ]] = f<latexit sha1_base64="rLLqbjihp04g8fwvDr34zEkMuXs=">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</latexit><latexit 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のリジレット変換 は、 をみたす関数f<latexit sha1_base64="LLagMHebbSaRgSA+s0QanM4n3EU=">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</latexit><latexit sha1_base64="LLagMHebbSaRgSA+s0QanM4n3EU=">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</latexit><latexit sha1_base64="LLagMHebbSaRgSA+s0QanM4n3EU=">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</latexit><latexit sha1_base64="LLagMHebbSaRgSA+s0QanM4n3EU=">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</latexit>
f : M ! M<latexit sha1_base64="ObCmBOoIjE5R/t7HasKbSsSYD8c=">AAACjXicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQtopMWADMmLyU0syUhOzKn2rY0pykzPKEksKsovRxKNF1A20DMAAwVMhiGUocwABQH5AssZYhhSGPIZkhlKGXIZUhnyGEqA7ByGRIZiIIxmMGQwYCgAisUyVAPFioCsTLB8KkMtAxdQbylQVSpQRSJQNBtIpgN50VDRPCAfZGYxWHcy0JYcIC4C6lRgUDW4arDS4LPBCYPVBi8N/uA0qxpsBsgtlUA6CaI3tSCev0si+DtBXblAuoQhA6ELr5tLGNIYLMBuzQS6vQAsAvJFMkR/WdX0z8FWQarVagaLDF4D3b/Q4KbBYaAP8sq+JC8NTA2azcAFjABD9ODGZIQZ6Rka6BkGmig7OEGjgoNBmkGJQQMY3uYMDgweDAEMoUB7+xi2MOxl2MfEz2TKZMNkB1HKxAjVI8yAApjcAVIWmrA=</latexit><latexit sha1_base64="ObCmBOoIjE5R/t7HasKbSsSYD8c=">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</latexit><latexit sha1_base64="ObCmBOoIjE5R/t7HasKbSsSYD8c=">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</latexit><latexit sha1_base64="ObCmBOoIjE5R/t7HasKbSsSYD8c=">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</latexit>
理研研究奨励賞受賞
力学系
� : M ! Hk<latexit sha1_base64="XLOhuLBr/33Q3O9oB6aQ1LgmWYw=">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</latexit><latexit sha1_base64="XLOhuLBr/33Q3O9oB6aQ1LgmWYw=">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</latexit><latexit sha1_base64="XLOhuLBr/33Q3O9oB6aQ1LgmWYw=">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</latexit><latexit sha1_base64="XLOhuLBr/33Q3O9oB6aQ1LgmWYw=">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</latexit>
RKHSで捉える
Kf : Hk ! Hk<latexit sha1_base64="1KmUY/7VvUhhGWPmWDc91w326Os=">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</latexit><latexit sha1_base64="1KmUY/7VvUhhGWPmWDc91w326Os=">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</latexit><latexit sha1_base64="1KmUY/7VvUhhGWPmWDc91w326Os=">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</latexit><latexit sha1_base64="1KmUY/7VvUhhGWPmWDc91w326Os=">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</latexit>
Perron-Frobenius作用素
D = (f, h, x)<latexit sha1_base64="SWECk4u6Kne202ajFnI43APiGqc=">AAACbHichVG7SgNBFD1Z3/GR+CiEIIhBUQjhrgiKIAS1sEzU+EBFdtdJXNwXu5ugBn/A1sJCLRRExM+w8Qcs/AQRbBRsLLzZLIiKeoeZOXPmnjtnZlTH0D2f6CEi1dU3NDY1t0Rb29o7YvHOriXPLrmayGu2YbsrquIJQ7dE3td9Q6w4rlBM1RDL6s5MdX+5LFxPt61Ff88RG6ZStPSCrik+U6uzU8OF1HZqd2QznqQ0BdH/E8ghSCKMrB2/wjq2YENDCSYELPiMDSjwuK1BBsFhbgMV5lxGerAvcIAoa0ucJThDYXaHxyKv1kLW4nW1pheoNT7F4O6ysh+DdE/X9EJ3dEOP9P5rrUpQo+plj2e1phXOZuywd+HtX5XJs4/tT9Wfnn0UMBF41dm7EzDVW2g1fXn/+GVhcn6wMkQX9MT+z+mBbvkGVvlVu8yJ+RNE+QPk78/9EyyNpmVKy7mxZGY6/IpmJDCAYX7vcWQwhyzyfK6JI5ziLPIs9UgJqa+WKkVCTTe+hDT0ARtPjEA=</latexit><latexit sha1_base64="SWECk4u6Kne202ajFnI43APiGqc=">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</latexit><latexit sha1_base64="SWECk4u6Kne202ajFnI43APiGqc=">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</latexit><latexit sha1_base64="SWECk4u6Kne202ajFnI43APiGqc=">AAACbHichVG7SgNBFD1Z3/GR+CiEIIhBUQjhrgiKIAS1sEzU+EBFdtdJXNwXu5ugBn/A1sJCLRRExM+w8Qcs/AQRbBRsLLzZLIiKeoeZOXPmnjtnZlTH0D2f6CEi1dU3NDY1t0Rb29o7YvHOriXPLrmayGu2YbsrquIJQ7dE3td9Q6w4rlBM1RDL6s5MdX+5LFxPt61Ff88RG6ZStPSCrik+U6uzU8OF1HZqd2QznqQ0BdH/E8ghSCKMrB2/wjq2YENDCSYELPiMDSjwuK1BBsFhbgMV5lxGerAvcIAoa0ucJThDYXaHxyKv1kLW4nW1pheoNT7F4O6ysh+DdE/X9EJ3dEOP9P5rrUpQo+plj2e1phXOZuywd+HtX5XJs4/tT9Wfnn0UMBF41dm7EzDVW2g1fXn/+GVhcn6wMkQX9MT+z+mBbvkGVvlVu8yJ+RNE+QPk78/9EyyNpmVKy7mxZGY6/IpmJDCAYX7vcWQwhyzyfK6JI5ziLPIs9UgJqa+WKkVCTTe+hDT0ARtPjEA=</latexit>
：力学系 ：初期値 ：observable
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間の角度（metric）を定義
