Pirmin Lemberger, Ivan Panico, A Primer on Domain Adaptation
機器學習分爲訓練和測試倆步驟, 且往往假設訓練樣本的分佈和測試樣本的分佈是一致的, 但是這種情況在實際中並不一定成立. 作者就prior shift, covratie shift, concept shift, subspace mapping 四種情形給出相應的’解決方案".
x ∈ X ⊂ R p \mathbf{x} \in \mathcal{X} \subset \mathbb{R}^p x ∈ X ⊂ R p y ∈ Y = { ω 1 , … , ω k } y \in \mathcal{Y}=\{\omega_1,\ldots, \omega_k\} y ∈ Y = { ω 1 , … , ω k } S = { ( x 1 , y 1 ) , … ( x m , y m ) } S=\{(\mathbf{x}_1,y_1), \ldots(\mathbf{x_m}, y_m)\} S = { ( x 1 , y 1 ) , … ( x m , y m ) } h ∈ H : X → Y h \in \mathcal{H}:\mathcal{X} \rightarrow \mathcal{Y} h ∈ H : X → Y y ^ = h ( x ) \hat{y}=h(\mathbf{x}) y ^ = h ( x ) ℓ : Y × Y → R \ell: \mathcal{Y} \times \mathcal{Y} \rightarrow \mathbb{R} ℓ : Y × Y → R R [ h ] : = E ( x , y ) ∼ p [ ℓ ( y , h ( x ) ] R[h]:= \mathbb{E}_{(\mathbf{x}, y) \sim p}[\ell(y, h(\mathbf{x})] R [ h ] : = E ( x , y ) ∼ p [ ℓ ( y , h ( x ) ] R ^ [ h ] : = 1 m ∑ i = 1 m [ ℓ ( y i , h ( x i ) ] \hat{R}[h]:= \frac{1}{m} \sum_{i=1}^m [\ell(y_i, h(\mathbf{x}_i)] R ^ [ h ] : = m 1 ∑ i = 1 m [ ℓ ( y i , h ( x i ) ] p S p_S p S p T p_T p T p ^ \hat{p} p ^
p S ( x ∣ y ) = p T ( x ∣ y ) p_S(\mathbf{x}|y)=p_T(\mathbf{x}|y) p S ( x ∣ y ) = p T ( x ∣ y ) p S ( y ) ≠ p T ( y ) p_S(y) \not = p_T(y) p S ( y ) = p T ( y )
假設根據訓練樣本S S S A A A p ^ S ( y ∣ x ) \hat{p}_S(y|\mathbf{x}) p ^ S ( y ∣ x ) p ^ S ( y = ω k ) = m k / ∣ S ∣ \hat{p}_S(y=\omega_k)=m_k/|S| p ^ S ( y = ω k ) = m k / ∣ S ∣ p ^ S ( x ∣ y ) = p ^ T ( x ∣ y ) \hat{p}_S(\mathbf{x}|y)=\hat{p}_T(\mathbf{x}|y) p ^ S ( x ∣ y ) = p ^ T ( x ∣ y ) p ^ T ( ω k ∣ x ) = w ^ ( ω k ) p ^ S ( ω k ∣ x ) ∑ k ′ = 1 K w ^ ( ω k ′ ) p ^ S ( ω k ′ ∣ x ) , w ^ ( ω k ) : = p ^ T ( ω k ) p ^ S ( ω k ) . (9)
\tag{9}
\hat{p}_T(\omega_k|\mathbf{x})= \frac{\hat{w}(\omega_k)\hat{p}_S(\omega_k|\mathbf{x})}{\sum_{k'=1}^K\hat{w}(\omega_{k'})\hat{p}_S(\omega_{k'}|\mathbf{x})}, \hat{w}(\omega_k):=\frac{\hat{p}_T(\omega_k)}{\hat{p}_S(\omega_k)}.
p ^ T ( ω k ∣ x ) = ∑ k ′ = 1 K w ^ ( ω k ′ ) p ^ S ( ω k ′ ∣ x ) w ^ ( ω k ) p ^ S ( ω k ∣ x ) , w ^ ( ω k ) : = p ^ S ( ω k ) p ^ T ( ω k ) . ( 9 )
倘若我們知道p ^ T ( ω k ) , k = 1 , … , K \hat{p}_T(\omega_k), k=1,\ldots, K p ^ T ( ω k ) , k = 1 , … , K
假設, 我們的目標數據集的樣本數據爲x 1 ′ , … , x m ′ \mathbf{x}_1', \ldots, \mathbf{x}_m' x 1 ′ , … , x m ′ p ^ T ( ω k ∣ x ′ ) \hat{p}_T(\omega_k|\mathbf{x}') p ^ T ( ω k ∣ x ′ ) p ^ T ( ω k ) = ∑ i = 1 m p ^ T ( ω k , x i ′ ) = 1 m ∑ i = 1 m p ^ T ( ω k ∣ x i ′ ) , (10)
\tag{10}
\hat{p}_T(\omega_k)=\sum_{i=1}^m \hat{p}_T(\omega_k,\mathbf{x}_i')=\frac{1}{m} \sum_{i=1}^m \hat{p}_T(\omega_k|\mathbf{x}_i'),
p ^ T ( ω k ) = i = 1 ∑ m p ^ T ( ω k , x i ′ ) = m 1 i = 1 ∑ m p ^ T ( ω k ∣ x i ′ ) , ( 1 0 ) p ( x i ′ ) = 1 m p(\mathbf{x}_i')=\frac{1}{m} p ( x i ′ ) = m 1
於是, 很自然地, 我們可以利用交替迭代求解p ^ T ( s ) ( ω k ∣ x ′ ) = w ^ ( ω k ) p ^ S ( ω k ∣ x ′ ) ∑ k ′ = 1 K w ^ ( ω k ′ ) p ^ S ( ω k ′ ∣ x ′ ) , w ^ ( ω k ) : = p ^ T ( s ) ( ω k ) p ^ S ( ω k ) . p ^ T ( s + 1 ) ( ω k ) = 1 m ∑ i = 1 m p ^ T ( s ) ( ω k ∣ x i ′ ) . (11)
\tag{11}
\hat{p}_T^{(s)}(\omega_k|\mathbf{x}')= \frac{\hat{w}(\omega_k)\hat{p}_S(\omega_k|\mathbf{x}')}{\sum_{k'=1}^K\hat{w}(\omega_{k'})\hat{p}_S(\omega_{k'}|\mathbf{x}')}, \hat{w}(\omega_k):=\frac{\hat{p}_T^{(s)}(\omega_k)}{\hat{p}_S(\omega_k)}. \\
\hat{p}_T^{(s+1)}(\omega_k)=\frac{1}{m} \sum_{i=1}^m \hat{p}_T^{(s)}(\omega_k|\mathbf{x}_i').
p ^ T ( s ) ( ω k ∣ x ′ ) = ∑ k ′ = 1 K w ^ ( ω k ′ ) p ^ S ( ω k ′ ∣ x ′ ) w ^ ( ω k ) p ^ S ( ω k ∣ x ′ ) , w ^ ( ω k ) : = p ^ S ( ω k ) p ^ T ( s ) ( ω k ) . p ^ T ( s + 1 ) ( ω k ) = m 1 i = 1 ∑ m p ^ T ( s ) ( ω k ∣ x i ′ ) . ( 1 1 )
注: 在實際中, 由於各種因素, 這麼做反而畫蛇添足, 起到反效果, 我們可以通過假設檢驗來判斷是否接受.
其趨向於χ ( K − 1 ) 2 \chi^2_{(K-1)} χ ( K − 1 ) 2
p S ( y ∣ x ) = p T ( y ∣ x ) p_S(y|\mathbf{x})=p_T(y|\mathbf{x}) p S ( y ∣ x ) = p T ( y ∣ x ) p S ( x ) ≠ p T ( x ) p_S(\mathbf{x})\not = p_T(\mathbf{x}) p S ( x ) = p T ( x )
A covariate shift typically occurs when the cost or the difficulty of picking an observation with given features x strongly impacts the probability of selecting an observation (x, y) thus making it practically impossible to replicate the target feature distribution p T ( x ) p_T(\mathbf{x}) p T ( x )
我們所希望最小化,R T [ h ] : = E p T [ ℓ ( h ( x ) ) , y ) ] = E p S [ w ( x ) ℓ ( h ( x ) ) , y ) ] . (14,15)
\tag{14,15}
R_T[h]:= \mathbb{E}_{p_T}[\ell(h(\mathbf{x})),y)] =\mathbb{E}_{p_S}[w(\mathbf{x})\ell(h(\mathbf{x})),y)].
R T [ h ] : = E p T [ ℓ ( h ( x ) ) , y ) ] = E p S [ w ( x ) ℓ ( h ( x ) ) , y ) ] . ( 1 4 , 1 5 ) w ( x ) = p T ( x ) / p S ( x ) w(\mathbf{x})=p_T(\mathbf{x})/p_S(\mathbf{x}) w ( x ) = p T ( x ) / p S ( x ) w ^ ( x ) \hat{w}(\mathbf{x}) w ^ ( x ) R ^ S , w [ h ] : = 1 m ∑ i = 1 m w ( x i ) ℓ ( h ( x i ) , y i ) . (16)
\tag{16}
\hat{R}_{S, w} [h]:= \frac{1}{m} \sum_{i=1}^m w(\mathbf{x}_i) \ell(h(\mathbf{x}_i),y_i).
R ^ S , w [ h ] : = m 1 i = 1 ∑ m w ( x i ) ℓ ( h ( x i ) , y i ) . ( 1 6 )
注: 以下情況不適合用(16):
p S ( x i ) = 0 p_S(\mathbf{x}_i)=0 p S ( x i ) = 0 p T ( x ) i ≠ 0 p_T(\mathbf{x})_i \not=0 p T ( x ) i = 0 p S , p T p_S, p_T p S , p T w w w
即p S p_S p S p T p_T p T 1 − δ 1-\delta 1 − δ
w ^ \hat{w} w ^ 顯然, 解決(16)的關鍵在於w ^ : = p ^ T ( x ) / p ^ S ( x ) \hat{w}:=\hat{p}_T(\mathbf{x})/\hat{p}_S(\mathbf{x}) w ^ : = p ^ T ( x ) / p ^ S ( x )
一個策略是直接估計w ^ \hat{w} w ^ p ^ T , p ^ S \hat{p}_T, \hat{p}_S p ^ T , p ^ S
期望均方誤差E p S [ ( w ^ − p T / p S ) 2 ] \mathbb{E}_{p_S}[(\hat{w}-p_T/p_S)^2] E p S [ ( w ^ − p T / p S ) 2 ]
KL散度K L ( p T ∥ w ^ p S ) \mathbf{KL}(p_T \| \hat{w}p_S) K L ( p T ∥ w ^ p S )
最大平均差異(maximum mean discrepancy, MMD).
選擇kernel K ( x , y ) K(\mathbf{x}, \mathbf{y}) K ( x , y ) x \mathbf{x} x x → Φ x \mathbf{x} \rightarrow \Phi_{\mathbf{x}} x → Φ x ⟨ Φ x , Φ y ⟩ = K ( x , y ) \langle \Phi_{\mathbf{x}}, \Phi_{\mathbf{y}} \rangle=K(\mathbf{x}, \mathbf{y}) ⟨ Φ x , Φ y ⟩ = K ( x , y ) ( M M D [ α , β ] ) 2 : = ∥ E x ∼ α [ Φ x ] − E x ∼ β [ Φ x ] ∥ 2 = ∥ E x ∼ α [ Φ x ] ∥ 2 − 2 ⟨ E x ∼ α [ Φ x ] , E x ∼ β [ Φ x ] ⟩ + ∥ E x ∼ β [ Φ x ] ∥ 2 .
(\mathrm{MMD}[\alpha, \beta])^2:=\|\mathbb{E}_{\mathbf{x} \sim \alpha} [\Phi_{\mathbf{x}}]-\mathbb{E}_{\mathbf{x} \sim \beta} [\Phi_{\mathbf{x}}]\|^2= \|\mathbb{E}_{\mathbf{x} \sim \alpha} [\Phi_{\mathbf{x}}]\|^2-2\langle \mathbb{E}_{\mathbf{x} \sim \alpha} [\Phi_{\mathbf{x}}],\mathbb{E}_{\mathbf{x} \sim \beta} [\Phi_{\mathbf{x}}] \rangle+ \|\mathbb{E}_{\mathbf{x} \sim \beta} [\Phi_{\mathbf{x}}]\|^2.
( M M D [ α , β ] ) 2 : = ∥ E x ∼ α [ Φ x ] − E x ∼ β [ Φ x ] ∥ 2 = ∥ E x ∼ α [ Φ x ] ∥ 2 − 2 ⟨ E x ∼ α [ Φ x ] , E x ∼ β [ Φ x ] ⟩ + ∥ E x ∼ β [ Φ x ] ∥ 2 .
則令α = w ^ p ^ S , β = p ^ T \alpha=\hat{w}\hat{p}_S, \beta=\hat{p}_T α = w ^ p ^ S , β = p ^ T ( M M D [ p ^ T , w ^ p ^ S ] ) 2 = 1 m S 2 ( 1 2 w ^ T K w ^ − k T w ^ ) + c o n s t , (21)
\tag{21}
(\mathrm{MMD}[\hat{p}_T, \hat{w} \hat{p}_S])^2 = \frac{1}{m_S^2} (\frac{1}{2} \hat{w}^TK \hat{w} - k^T\hat{w}) +\mathrm{const},
( M M D [ p ^ T , w ^ p ^ S ] ) 2 = m S 2 1 ( 2 1 w ^ T K w ^ − k T w ^ ) + c o n s t , ( 2 1 ) w ^ : = ( w ^ ( x 1 ) , … , w ^ ( x m S ) ) T \hat{w}:=(\hat{w}(\mathbf{x}_1),\ldots, \hat{w}(\mathbf{x}_{m_S}))^T w ^ : = ( w ^ ( x 1 ) , … , w ^ ( x m S ) ) T K i j : = 2 K ( x i , x k ) K_{ij}:=2K(\mathbf{x}_i,\mathbf{x}_k) K i j : = 2 K ( x i , x k ) k i : = 2 m S m T ∑ j = 1 m T K ( x i , x j ) k_i:=\frac{2m_S}{m_T} \sum_{j=1}^{m_T} K(\mathbf{x}_i,\mathbf{x}_j) k i : = m T 2 m S ∑ j = 1 m T K ( x i , x j )
在實際中, 求解下面的優化問題min w 1 2 w ^ T K w ^ − k T w ^ s . t . w ^ ( x i ) ∈ [ 0 , B ] , ∣ 1 m S ∑ i = 1 m S w ^ ( x i ) − 1 ∣ ≤ ϵ .
\begin{array}{rc}
\min_w & \frac{1}{2} \hat{w}^T K\hat{w} - k^T\hat{w} \\
\mathrm{s.t.} & \hat{w}(\mathbf{x}_i) \in [0,B], \\
& |\frac{1}{m_S} \sum_{i=1}^{m_S} \hat{w}(\mathbf{x}_i) -1| \le \epsilon.
\end{array}
min w s . t . 2 1 w ^ T K w ^ − k T w ^ w ^ ( x i ) ∈ [ 0 , B ] , ∣ m S 1 ∑ i = 1 m S w ^ ( x i ) − 1 ∣ ≤ ϵ . p ^ S , p ^ T \hat{p}_S,\hat{p}_T p ^ S , p ^ T
p S ( y ∣ x ) ≠ p T ( y ∣ x ) p_S(y|\mathbf{x})\not= p_T(y|\mathbf{x}) p S ( y ∣ x ) = p T ( y ∣ x ) p S ( x ) = p T ( x ) p_S(\mathbf{x})=p_T(\mathbf{x}) p S ( x ) = p T ( x ) p p p
週期性地利用新數據重新訓練模型;
保留部分舊數據, 結合新數據訓練;
加入權重;
引入有效的迭代機制;
檢測偏移, 並作出反應.
訓練數據爲x \mathbf{x} x x ′ = T ( x ) \mathbf{x}'=T(\mathbf{x}) x ′ = T ( x ) p T ( T ( x ) , y ) = p S ( x , y ) p_T(T(\mathbf{x}), y) = p_S(\mathbf{x},y) p T ( T ( x ) , y ) = p S ( x , y ) T T T
我們現在的目標是找到一個有關
以離散情形爲例, 介紹,α : = ∑ i = 1 m α i δ z i ,
\alpha := \sum_{i=1}^m \alpha_i \delta_{\mathbf{z}_i},
α : = i = 1 ∑ m α i δ z i , δ z \delta_{\mathbf{z}} δ z T α : = ∑ i = 1 m α i δ T ( z i ) ,
T \alpha := \sum_{i=1}^m \alpha_i \delta_{T(\mathbf{z}_i)},
T α : = i = 1 ∑ m α i δ T ( z i ) , arg min T , T α = β E z ∼ α [ c ( z , T ( z ) ) ] ,
\arg \min_{T, T\alpha = \beta} \mathbb{E}_{\mathbf{z} \sim \alpha} [c(\mathbf{z}, T(\mathbf{z}))],
arg T , T α = β min E z ∼ α [ c ( z , T ( z ) ) ] , c ( ⋅ , ⋅ ) c(\cdot, \cdot) c ( ⋅ , ⋅ )
但是呢, 這種方式找T T T γ : = ∑ i , j γ i j δ z i , z j ′ (30)
\tag{30}
\gamma := \sum_{i,j} \gamma_{ij} \delta_{\mathbf{z}_i,\mathbf{z}_j'}
γ : = i , j ∑ γ i j δ z i , z j ′ ( 3 0 ) E ( z , z ′ ) ∼ γ [ c ( z , z ′ ) ] : = ∑ i , j γ i , j c ( z i , z j ) , (33)
\tag{33}
\mathbb{E}_{(\mathbf{z},\mathbf{z}') \sim \gamma}[c(\mathbf{z},\mathbf{z}')]:=\sum_{i,j} \gamma_{i,j}c(\mathbf{z}_i,\mathbf{z}_j),
E ( z , z ′ ) ∼ γ [ c ( z , z ′ ) ] : = i , j ∑ γ i , j c ( z i , z j ) , ( 3 3 ) L c ( α , β ) : = min γ ∈ U ( α , β ) E ( z , z ′ ) ∼ γ [ c ( z , z ′ ) ] (34)
\tag{34}
\mathcal{L}_c (\alpha, \beta) := \min_{\gamma \in U(\alpha, \beta)} \mathbb{E}_{(\mathbf{z}, \mathbf{z}') \sim \gamma}[c(\mathbf{z}, \mathbf{z}')]
L c ( α , β ) : = γ ∈ U ( α , β ) min E ( z , z ′ ) ∼ γ [ c ( z , z ′ ) ] ( 3 4 ) α \alpha α β \beta β U ( α , β ) : = { γ : ∑ j = 1 s γ i j = α i , ∑ i = 1 r γ i j = β j } ,
U(\alpha, \beta):=\{ \gamma: \sum_{j=1}^s \gamma_{ij} =\alpha_i, \sum_{i=1}^r \gamma_{ij} = \beta_j\},
U ( α , β ) : = { γ : j = 1 ∑ s γ i j = α i , i = 1 ∑ r γ i j = β j } , α , β \alpha, \beta α , β γ \gamma γ
而Wasserstein distance實際上就是W p ( α , β ) : = [ L d p ( α , β ) ] 1 / p , c ( z , z ′ ) = [ d ( z , z ′ ) ] p , p ≥ 1. (35)
\tag{35}
W_p(\alpha,\beta) := [\mathcal{L}_{d^p} (\alpha, \beta)]^{1/p}, c(\mathbf{z},\mathbf{z}')=[d(\mathbf{z},\mathbf{z}')]^p, p\ge1.
W p ( α , β ) : = [ L d p ( α , β ) ] 1 / p , c ( z , z ′ ) = [ d ( z , z ′ ) ] p , p ≥ 1 . ( 3 5 ) d d d
策略一:α = p ^ S ( x ) , β = p ^ T ( x ′ ) \alpha=\hat{p}_S(\mathbf{x}), \beta=\hat{p}_T(\mathbf{x}') α = p ^ S ( x ) , β = p ^ T ( x ′ ) γ \gamma γ γ \gamma γ S S S p ^ T \hat{p}_T p ^ T γ \gamma γ
注:爲了防止( x i , y i ) , ( x j , y j ) , y i ≠ y j (\mathbf{x}_i,y_i),(\mathbf{x}_j,y_j), y_i \not =y_j ( x i , y i ) , ( x j , y j ) , y i = y j
策略二:α = p ^ S ( x , y ) , β = p ^ T ( x ′ , y ′ ) \alpha=\hat{p}_S(\mathbf{x},y), \beta=\hat{p}_T (\mathbf{x}',y') α = p ^ S ( x , y ) , β = p ^ T ( x ′ , y ′ ) y ′ y' y ′ h ( x ′ ) h(\mathbf{x}') h ( x ′ ) p ^ T h ( x ′ , y ′ ) : = p ^ T ( x ′ ) δ y ′ = h ( x ′ ) ,
\hat{p}_T^h(\mathbf{x}',y'):= \hat{p}_T(\mathbf{x}') \delta_{y'=h(\mathbf{x}')},
p ^ T h ( x ′ , y ′ ) : = p ^ T ( x ′ ) δ y ′ = h ( x ′ ) , h O T : = arg min h ∈ H W 1 ( p ^ S , p ^ T h ) , (37)
\tag{37}
h_{OT} := \arg \min_{h \in \mathcal{H}} W_1(\hat{p}_S, \hat{p}_T^h),
h O T : = arg h ∈ H min W 1 ( p ^ S , p ^ T h ) , ( 3 7 )
即h O T = arg min h ∈ H min γ ∈ U ( p ^ S , p ^ T h ) ∑ i , j γ i j d ( ( x i , y i ) , ( x j ′ , h ( x j ′ ) ) ) . (38)
\tag{38}
h_{OT} = \arg \min_{h \in \mathcal{H}} \min_{\gamma \in U(\hat{p}_S, \hat{p}_T^h)} \sum_{i,j} \gamma_{ij} d((\mathbf{x}_i,y_i),(\mathbf{x}_j', h(\mathbf{x}_j'))).
h O T = arg h ∈ H min γ ∈ U ( p ^ S , p ^ T h ) min i , j ∑ γ i j d ( ( x i , y i ) , ( x j ′ , h ( x j ′ ) ) ) . ( 3 8 ) d ( ( x , y ) , ( x ′ , y ′ ) ) : = λ ρ ( x , x ′ ) + ℓ ( y , y ′ ) .
d((\mathbf{x},y),(\mathbf{x}', y')) := \lambda \rho(\mathbf{x},\mathbf{x}') + \ell(y,y').
d ( ( x , y ) , ( x ′ , y ′ ) ) : = λ ρ ( x , x ′ ) + ℓ ( y , y ′ ) . h O T = arg min h ∈ H min γ ∈ U ( p ^ S , p ^ T h ) ( ∑ i , j γ i j [ λ ρ ( x i , x j ′ ) + ℓ ( y i , h ( x j ′ ) ) ] + ϵ r e g [ h ] ) . (39)
\tag{39}
h_{OT} = \arg \min_{h \in \mathcal{H}} \min_{\gamma \in U(\hat{p}_S, \hat{p}_T^h)} \big(\sum_{i,j} \gamma_{ij} [ \lambda \rho(\mathbf{x}_i,\mathbf{x}_j') + \ell(y_i,h(\mathbf{x}_j'))] + \epsilon \mathrm{reg}[h] \big).
h O T = arg h ∈ H min γ ∈ U ( p ^ S , p ^ T h ) min ( i , j ∑ γ i j [ λ ρ ( x i , x j ′ ) + ℓ ( y i , h ( x j ′ ) ) ] + ϵ r e g [ h ] ) . ( 3 9 )
考慮聯合概率p θ ( x 1 , … , x m ; z 1 , … , z m ) p_{\theta}(\mathbf{x}_1, \ldots, \mathbf{x}_m; \mathbf{z}_1,\ldots, \mathbf{z}_m) p θ ( x 1 , … , x m ; z 1 , … , z m ) z i , i = 1 , … , m \mathbf{z}_i,i=1,\ldots, m z i , i = 1 , … , m x i , i = 1 , … , m \mathbf{x}_i, i=1,\ldots,m x i , i = 1 , … , m
E步: E z [ log p θ ( x 1 , … , x m ; z 1 , … , z m ) ] \mathbb{E}_{\mathbf{z}}[\log p_{\theta}(\mathbf{x}_1, \ldots, \mathbf{x}_m; \mathbf{z}_1,\ldots, \mathbf{z}_m)] E z [ log p θ ( x 1 , … , x m ; z 1 , … , z m ) ]
Prior shift中, θ : = [ p T ( ω 1 ) , … , p T ( ω K ) ] T \theta:= [p_T(\omega_1), \ldots, p_T(\omega_K)]^{\mathrm{T}} θ : = [ p T ( ω 1 ) , … , p T ( ω K ) ] T z i : = ( z i 1 , … , z i K ) \mathbf{z}_i:=(z_{i1},\ldots, z_{iK}) z i : = ( z i 1 , … , z i K ) y i ∈ { ω 1 , … , ω K } y_i \in \{\omega_1,\ldots, \omega_K\} y i ∈ { ω 1 , … , ω K }
θ k \theta_k θ k ∑ k θ k = 1 \sum_k \theta_k=1 ∑ k θ k = 1
