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𝐹 𝑥 𝑦
𝑦∗ = argmax𝑦 𝐹(𝑥, 𝑦;𝑊)
𝐹(𝑥, 𝑦;𝑊)
𝐹 𝑥 𝑦
𝑦∗ = argmax𝑦 𝐹(𝑥, 𝑦;𝑊)
→
𝐹(𝑥, 𝑦;𝑊)
Φ( )Φ( )
Θ( )
Θ( )
𝑎 𝑏 Φ 𝑎 Φ 𝑏 :
Φ(𝑥)𝑥 =
𝑎
𝑏
𝑐
Φ(𝑎)
Φ(𝑏)
Φ(𝑐)
𝑅𝑑
𝑢 𝑣 Θ 𝑢 Θ 𝑣 :
Θ(𝑦)
𝑢
𝑣
𝑧
Θ(𝑢)
Θ(𝑣)
Θ(𝑧)
𝑦 =
𝑅𝑒
•
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→
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→
→
𝑂(# 𝑐𝑙𝑎𝑠𝑠𝑒𝑠)
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Φ
→
•
X = 𝑥1, … , 𝑥𝑇
•
𝑥𝑡 𝜑 𝑥𝑡
•
1
𝑇 𝜑 𝑥𝑡𝑇𝑡=1
• →
• 𝜑𝑏𝑜𝑣 𝑥𝑡 = [0, … , 0, 1, 0, … , 0]
•
→
𝝋
𝝋
𝝋
𝝋
•
𝝋
•
•
{𝜇1, … , 𝜇𝑁}
X = 𝑥1, … , 𝑥𝑇
• 𝑁𝑁 𝑥𝑡 = argmin𝜇𝑖𝑥𝑡 − 𝜇𝑖
• 𝑣𝑖 = (𝑥𝑡 − 𝜇𝑖)𝑥𝑡:𝑁𝑁 𝑥𝑡 =𝜇𝑖
• 𝑣𝑖 ℓ2
3
x
v1 v2 v3 v4
v5
1
4
2
5
①
②
③
𝑣𝑖 = (𝑥𝑡 − 𝜇𝑖)𝑥𝑡:𝑁𝑁 𝑥𝑡 =𝜇𝑖
𝜑𝑣𝑙𝑎𝑑 𝑥𝑡 = 0,… , 0, (𝑥𝑡−𝜇𝑖), 0, … , 0
•
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→
𝐷
𝑫𝑵
𝑢𝜆 𝑥𝑡
𝑔𝜆 (𝑥𝑡) = 𝛻𝜆 log 𝑢𝜆(𝑥𝑡)
𝐹𝜆 = 𝐸𝑥~𝑢𝜆 𝑔𝜆(𝑥)𝑔𝜆(𝑥)𝑇
𝐾 𝑥, 𝑧 = 𝑔𝜆(𝑥)𝑇𝐹𝜆−1𝑔𝜆(𝑧)
𝐹𝜆 = 𝐸𝑥~𝑢𝜆 𝑔𝜆(𝑥)𝑔𝜆(𝑥)𝑇
𝐾 𝑥, 𝑧 = 𝑔𝜆(𝑥)𝑇𝐹𝜆−1𝑔𝜆(𝑧)
𝐹𝜆−1 = 𝐿𝜆
𝑇𝐿𝜆
𝜑𝜆𝑓𝑣(𝑥𝑡) = 𝐿𝜆 𝑔(𝑥𝑡)
𝑢𝜆
𝑢𝜆(𝑥) = 𝑤𝑖𝑢𝑖(𝑥)𝑁
𝑖=1
𝑢𝑖 𝑥 =1
(2𝜋)𝐷/2 Σ𝑖1/2 exp −
1
2(𝑥 − 𝜇𝑖)′Σ𝑖
−1(𝑥 − 𝜇𝑖) →
𝜆 = 𝑤𝑖 , 𝜇𝑖 , Σ𝑖 , 𝑖 = 1…𝑁
Σ𝑖 = 𝑑𝑖𝑎𝑔(𝜎𝑖2)
→ 𝑤𝑖 , 𝜇𝑖 𝜎𝑖
𝛾𝑡(𝑖) 𝑥𝑡 𝑖
𝜑𝑤 𝑥𝑡 =𝛾𝑡(1)
𝑤1, … ,𝛾𝑡(𝑁)
𝑤𝑁𝜑𝑏𝑜𝑣 𝑥𝑡 = [0,… , 0, 1, 0, … , 0]
→
𝜑𝜇 𝑥𝑡 = 𝛾𝑡 1
𝜎1 𝑤1𝑥𝑡 − 𝜇1 , … ,
𝛾𝑡 1
𝜎1 𝑤𝑁𝑥𝑡 − 𝜇𝑁 𝜑𝑣𝑙𝑎𝑑 𝑥𝑡 = 0,… , (𝑥𝑡−𝜇𝑖), … , 0
→
𝜑𝜎 𝑥𝑡 = 𝛾𝑡 1
2𝑤1
𝑥𝑡−𝜇12
𝜎12− 1 ,… ,
𝛾𝑡 𝑁
2𝑤𝑁
𝑥𝑡−𝜇𝑁2
𝜎𝑁2− 1
→
→ 𝜑𝜇 𝜑𝜎 → 𝟐𝑫𝑵
𝜑𝑏𝑜𝑣 𝑥 = [0,… , 0, 1, 0,… , 0]
𝑤𝑇𝜑𝑏𝑜𝑣(𝑥)
𝜑𝑏𝑜𝑣 𝑥 = [0,… , 0, 1, 0,… , 0]
𝑤𝑇𝜑𝑏𝑜𝑣(𝑥)
→
𝜑𝑣𝑙𝑎𝑑 𝑥 = 0,… , (𝑥 − 𝜇𝑖), … , 0
𝑤𝑇𝜑𝑣𝑙𝑎𝑑(𝑥)
𝜑𝑣𝑙𝑎𝑑 𝑥 = 0,… , (𝑥 − 𝜇𝑖), … , 0
𝑤𝑇𝜑𝑣𝑙𝑎𝑑(𝑥)
→
𝜑𝑓𝑣 𝑥𝑡 = … ,𝛾𝑡 𝑖
𝜎𝑖 𝑤𝑖𝑥𝑡 − 𝜇𝑖 ,
𝛾𝑡 𝑖
2𝑤𝑖
𝑥𝑡 − 𝜇𝑖2
𝜎𝑖2− 1 ,…
𝑤𝑇𝜑𝑓𝑣(𝑥)
𝜑𝑓𝑣 𝑥𝑡 = … ,𝛾𝑡 𝑖
𝜎𝑖 𝑤𝑖𝑥𝑡 − 𝜇𝑖 ,
𝛾𝑡 𝑖
2𝑤𝑖
𝑥𝑡 − 𝜇𝑖2
𝜎𝑖2− 1 ,…
𝑤𝑇𝜑𝑓𝑣(𝑥)
→
→
•
ℓ2•
𝑧 𝑠𝑖𝑔𝑛 𝑧 𝑧 𝛼 0 ≤ 𝛼 ≤•
→ 𝛼 = 1/2
𝜆 = 𝑤𝑖 , 𝜇𝑖 , Σ𝑖 , 𝑖 = 1…𝑁
X = 𝑥1, … , 𝑥𝑇• 𝑥𝑡:
• 𝛾𝑡 𝑖 =𝑤𝑖𝑢𝑖 𝑥𝑡
𝑤𝑘𝑢𝑘 𝑥𝑡𝑁𝑘=1
• 𝜑𝜇 += … ,𝛾𝑡 𝑖
𝜎𝑖 𝑤𝑖𝑥𝑡 − 𝜇𝑖 , …
𝜑𝜎 += … ,𝛾𝑡 𝑖
2𝑤𝑖
𝑥𝑡−𝜇𝑖2
𝜎𝑖2 − 1 ,…
• ℓ2
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𝑢
𝑣
𝑧
Θ(𝑢)
Θ(𝑏)
Θ(𝑐)
𝑅𝑒
𝒴 = 1,… , 𝑘
Θ 𝑦 = [0,… , 0, 1, 0, … , 0]
• {−1,+1} 𝐵(1/2)
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→
• {−1,+1} 𝐵(1/2)
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−1…+1
+1…+1
+1…−1
• {−1,+1} 𝐵(1/2)
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→
Θ 6 =
→
→
→
→
7 × 5
1 = 2 = 3 = 𝐴 = 𝐵 =
7 × 5
• →
•
1 = 2 = 3 = 𝐵 =
Θ 𝑦 = Φ(𝑠𝑦𝑛𝑡ℎ𝑒𝑠𝑖𝑠 𝑦 )
𝐴 =
𝑛 × 𝑘 𝑌
→
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𝐹
𝑦∗ = argmax𝑦 𝐹(𝑥, 𝑦;𝑊)
Φ(𝑥) ∈ 𝑅𝑑 Θ(𝑦) ∈ 𝑅𝑒
𝒅 ≠ 𝒆
𝐹(𝑥, 𝑦;𝑊)
Φ( )Φ( )
Θ( )
Θ( )
𝑤𝑦 𝑑 = 𝑒
𝐹 𝑥, 𝑦 = 𝑤𝑦𝑇Φ 𝑥
• 𝑊 = 𝑤1 , … , 𝑤𝑘 𝑑 × 𝑘 𝑘
• Θ(𝑦) Θ 𝑦 = [0, … , 0, 1, 0, … , 0]𝑇
𝐹 𝑥, 𝑦;𝑊 = [ Φ 𝑥 𝑇 ] 𝑊 Θ(𝑦)
𝑑 ≠ 𝑒:
𝐹 𝑥, 𝑦;𝑊 = [ Φ 𝑥 𝑇 ] 𝑊 Θ(𝑦)
𝑊 𝑑 × 𝑒
→
→ 𝑊
𝑑 ≠ 𝑒:
𝐹 𝑥, 𝑦;𝑊 = [ Φ 𝑥 𝑇 ] 𝑊 Θ(𝑦)
𝑊 𝑑 × 𝑒
• 𝐹 𝑥, 𝑦;𝑊 = −| 𝑊𝑇Φ 𝑥 − Θ 𝑦 |2
• 𝐹 𝑥, 𝑦;𝑊 = −| Φ 𝑥 −𝑊Θ 𝑦 |2
→
U 𝑊 𝑊 = 𝑈𝑇𝑉
• 𝑈 𝑟 × 𝑑
• 𝑉 𝑟 × 𝑒
𝐹 𝑥, 𝑦;𝑊 = Φ 𝑥 𝑇𝑊Θ(𝑦)
𝐹 𝑥, 𝑦; 𝑈, 𝑉 = 𝑈Φ 𝑥𝑇𝑉Θ 𝑦 = Φ′ 𝑥 𝑇Θ′ 𝑦
Φ′ 𝑥 = 𝑈Φ 𝑥 Θ′ 𝑦 = 𝑉Θ 𝑦
→ 𝑟
→
𝑟 ≪ 𝑑, 𝑒
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→
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• Ψ 𝑥, 𝑦 = Φ 𝑥 ⊗ Θ 𝑦 𝑑𝑒
• 𝑤 𝑑𝑒 𝑊
𝐹 𝑥, 𝑦;𝑊 = Φ 𝑥 𝑇𝑊Θ 𝑦 = 𝑤𝑇Ψ 𝑥, 𝑦
→
→
Θ = [Θ 1 ,… , Θ 𝑘 𝑒 × 𝑘
𝐹 𝑥, . ;𝑊 = Θ𝑇(Φ 𝑥 𝑇𝑊)
→
Φ(𝑥) 𝑧 = 𝑊𝑇Φ 𝑥 Θ𝑇𝑧
𝑊 Θ
Θ = [Θ 1 ,… , Θ 𝑘 𝑒 × 𝑘
𝐹 𝑥, . ;𝑊 = Θ𝑇(Φ 𝑥 𝑇𝑊)
→
•
Φ(𝑥) 𝑧 = 𝝈(𝑊𝑇Φ 𝑥 ) Θ𝑇𝑧
𝑊 Θ
Θ = [Θ 1 ,… , Θ 𝑘 𝑒 × 𝑘
𝐹 𝑥, . ;𝑊 = Θ𝑇(Φ 𝑥 𝑇𝑊)
→
•
•
→
Φ(𝑥) 𝑧 = 𝝈(𝑊𝑇Φ 𝑥 ) Θ𝑇𝑧
𝑊 Θ
•
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→ 𝑊
•
→ 𝑊 Θ
𝚯 𝑊
𝑊
→
𝚯 𝑊
𝑊
→
𝚯 𝑊
argmax𝑊1
𝑛 𝐹(𝑥𝑖 , 𝑦𝑖𝑛
𝑖=1;𝑊)
𝑊
𝐹 𝑥, 𝑦;𝑊 = −| 𝑊𝑇Φ 𝑥 − Θ 𝑦 |2 𝐹 𝑥, 𝑦;𝑊 = −| Φ 𝑥 −𝑊Θ 𝑦 |2
→
𝐹 𝑥, 𝑦; 𝑈, 𝑉 = −| 𝑈Φ 𝑥 − 𝑉Θ 𝑦 |2
→
𝚯 𝑊
𝑥 = , 𝑦+ = , 𝑦− =
𝐹 𝑥, 𝑦+;𝑊 > 𝐹 𝑥, 𝑦−;𝑊
→
𝚯 𝑊
ℓ 𝑥, 𝑦;𝑊 = max𝑗 Δ 𝑦, 𝑦𝑗 − 𝐹 𝑥, 𝑦;𝑊 + 𝐹 𝑥, 𝑦𝑗;𝑊
Δ 𝑦, 𝑦𝑗 𝑦 𝑦𝑗• 𝑦 = 𝑦𝑗 𝑦 ≠ 𝑦𝑗
•
→ arg max𝑊1
𝑛 ℓ(𝑥𝑖 , 𝑦𝑖𝑛𝑖=1 ;𝑊)
𝑊
𝚯 𝑊
ℓ 𝑥, 𝑦;𝑊 = max 0, Δ 𝑦, 𝑦𝑗 − 𝐹 𝑥, 𝑦;𝑊 + 𝐹 𝑥, 𝑦𝑗;𝑊𝑘𝑗=1
𝚯 𝑊
ℓ 𝑥, 𝑦;𝑊 = max 0, Δ 𝑦, 𝑦𝑗 − 𝐹 𝑥, 𝑦;𝑊 + 𝐹 𝑥, 𝑦𝑗;𝑊𝑘𝑗=1
𝑦 = , 𝑥+ = , 𝑥− =
𝐹 𝑥+, 𝑦;𝑊 > 𝐹 𝑥−, 𝑦;𝑊
𝚯 𝑊
Δ 𝑦+, 𝑦− − 𝐹 𝑥, 𝑦+;𝑊 + 𝐹 𝑥, 𝑦−;𝑊 = Δ 𝑦+, 𝑦− − 𝑥𝑇𝑊 𝑦+ − 𝑦−
→
𝑊 𝚯
•
•
𝑊 𝚯
arg max𝑊,Θ1
𝑛 ℓ(𝑥𝑖 , 𝑦𝑖𝑛
𝑖=1;𝑊, Θ) +
𝜆
2Θ − Θ𝑝𝑟𝑖𝑜𝑟
2
→
• Θ = Θ𝑝𝑟𝑖𝑜𝑟 𝑊
• Θ 𝑊
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∞
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