Lecture 04 florent perronnin - large-scale visual recognition with ecplicit embedding

𝐹 𝑥 𝑦

𝑦∗ = argmax𝑦 𝐹(𝑥, 𝑦;𝑊)

𝐹(𝑥, 𝑦;𝑊)

𝐹 𝑥 𝑦

Φ( )Φ( )

𝑎 𝑏 Φ 𝑎 Φ 𝑏 :

Φ(𝑥)𝑥 =

Φ(𝑎)

Φ(𝑏)

Φ(𝑐)

𝑅𝑑

𝑢 𝑣 Θ 𝑢 Θ 𝑣 :

Θ(𝑦)

Θ(𝑢)

Θ(𝑣)

Θ(𝑧)

𝑦 =

𝑅𝑒

𝑂(# 𝑐𝑙𝑎𝑠𝑠𝑒𝑠)

X = 𝑥1, … , 𝑥𝑇

𝑥𝑡 𝜑 𝑥𝑡

𝑇 𝜑 𝑥𝑡𝑇𝑡=1

• →

• 𝜑𝑏𝑜𝑣 𝑥𝑡 = [0, … , 0, 1, 0, … , 0]

{𝜇1, … , 𝜇𝑁}

X = 𝑥1, … , 𝑥𝑇

• 𝑁𝑁 𝑥𝑡 = argmin𝜇𝑖𝑥𝑡 − 𝜇𝑖

• 𝑣𝑖 = (𝑥𝑡 − 𝜇𝑖)𝑥𝑡:𝑁𝑁 𝑥𝑡 =𝜇𝑖

• 𝑣𝑖 ℓ2

v1 v2 v3 v4

𝑣𝑖 = (𝑥𝑡 − 𝜇𝑖)𝑥𝑡:𝑁𝑁 𝑥𝑡 =𝜇𝑖

𝜑𝑣𝑙𝑎𝑑 𝑥𝑡 = 0,… , 0, (𝑥𝑡−𝜇𝑖), 0, … , 0

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 −

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

Θ(𝑢)

Θ(𝑏)

Θ(𝑐)

𝑅𝑒

𝒴 = 1,… , 𝑘

Θ 𝑦 = [0,… , 0, 1, 0, … , 0]

• {−1,+1} 𝐵(1/2)

−1…+1

+1…+1

+1…−1

• {−1,+1} 𝐵(1/2)

Θ 6 =

7 × 5

1 = 2 = 3 = 𝐴 = 𝐵 =

7 × 5

• →

1 = 2 = 3 = 𝐵 =

Θ 𝑦 = Φ(𝑠𝑦𝑛𝑡ℎ𝑒𝑠𝑖𝑠 𝑦 )

𝐴 =

𝑛 × 𝑘 𝑌

Φ(𝑥) ∈ 𝑅𝑑 Θ(𝑦) ∈ 𝑅𝑒

𝒅 ≠ 𝒆

Φ( )Φ( )

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

• 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

Φ(𝑥) 𝑧 = 𝝈(𝑊𝑇Φ 𝑥 ) Θ𝑇𝑧

𝑊 Θ

Θ = [Θ 1 ,… , Θ 𝑘 𝑒 × 𝑘

Φ(𝑥) 𝑧 = 𝝈(𝑊𝑇Φ 𝑥 ) Θ𝑇𝑧

𝑊 Θ

→ 𝑊

→ 𝑊 Θ

𝚯 𝑊

argmax𝑊1

𝑛 𝐹(𝑥𝑖 , 𝑦𝑖𝑛

𝑖=1;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)

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

arg max𝑊,Θ1

𝑛 ℓ(𝑥𝑖 , 𝑦𝑖𝑛

𝑖=1;𝑊, Θ) +

2Θ − Θ𝑝𝑟𝑖𝑜𝑟

• Θ = Θ𝑝𝑟𝑖𝑜𝑟 𝑊

• Θ 𝑊

Lecture 04 florent perronnin - large-scale visual recognition with ecplicit embedding

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