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Deep Generative Models
Mijung Kim
Discriminative vs. Generative Learning
Discriminative Learning Generative Learning
Learn 𝑝(𝑦|𝑥) directly Model 𝑝 𝑦 , 𝑝 𝑥 𝑦 first,
Then derive the posterior distribution:
𝑝 𝑦 𝑥 =
𝑝 𝑥 𝑦 𝑝(𝑦)
𝑝(𝑥)
2
Undirected Graph vs. Directed Graph
Undirected Directed
• Boltzmann Machines
• Restricted Boltzmann Machines
• Deep Boltzmann Machines
• Sigmoid Belief Networks
• Variational Autoencoders (VAE)
• Generative Adversarial Networks
(GAN)
3
a b
c
a b
c
Deep Belief
Networks
Boltzmann Machines
• Stochastic Recurrent Neural Network and Markov Random
Field invented by Hinton and Sejnowski in 1985
• 𝑷 𝒙 =
𝐞𝐱𝐩(−𝑬 𝒙 )
𝒁
> E(x): Energy function
> Z: partition function where σ 𝑥 𝑃 𝑥 = 1
• Energy-based model: positive values all the time
• Single visible layer and single hidden layer
• Fully connected: not practical to implement
4
Restricted Boltzmann Machines
• Dimensionality reduction, classification, regression,
collaborative filtering, feature learning and topic modeling
• 𝑷 𝐯 = 𝒗, 𝐡 = 𝒉 =
𝟏
𝒁
𝐞𝐱𝐩(−𝑬 𝒗, 𝒉 )
• Two layers like BMs
• Building blocks of deep probabilistic models
• Gibbs sampling with Contrastive Divergence (CD) or Persistent
CD
5
Comparison btw BMs and RBMs
Boltzmann Machines Restricted Boltzmann Machines
v1 v2 v3
h1 h2 h3 h4
v1 v2 v3
h1 h2 h3 h4
6
𝑬 𝒗, 𝒉 = −𝒗 𝑻
𝑹𝒗 − 𝒉 𝑻
𝑺𝒉 − 𝒗 𝑻
𝑾𝒉 − 𝒃 𝑻
𝒗 − 𝒄 𝑻
𝒉 𝑬 𝒗, 𝒉 = −𝒃 𝑻
𝒗 − 𝒄 𝑻
𝒉 − 𝒗 𝑻
𝑾𝒉
Deep Belief
Networks
• Unsupervised
• Small dataset
• Stacked RBMs
• Pre-train each RBM
• Undirected + Directed
7
RBM
Sigmoid
Belief
Net
• 𝑃 v, h1, h2, h3 =
𝑃 v h1 𝑃 h1 h2 𝑃(h2, h3)
• 𝑃 v h1 = ς𝑖 𝑃(𝑣𝑖|h1)
• 𝑃 h1
h2
= ς 𝑗 𝑃(ℎ𝑗
1
|h2
)
• 𝑃 h2
, h3
=
1
𝑍(𝑊3)
exp(h2𝑇
𝑊3
h3
)
8
RBM
Sigmoid
Belief
Net
ℎ1
ℎ2
ℎ3
v
𝑊3
𝑊2
𝑊1
Sigmoid Belief Net RBM
Limitations of DBN (By Ruslan Salakhutdinov)
• Explaining away
• Greedy layer-wise pre-training
> no optimization over all layers
• Approximation inference is feed-forward
> no bottom-up and top-down
9
http://www.slideshare.net/zukun/p05-deep-boltzmann-machines-cvpr2012-deep-learning-methods-for-vision
Deep Boltzmann
Machines
• Unsupervised
• Small dataset
• Stacked RBMs
• Pre-train each RBM
• Undirected
10
• 𝑃 𝜃 v =
1
𝑍(𝜃)
σh1,h2,h3 exp( v 𝑇 𝑊1h1 +
h1𝑇 𝑊2h2 + h2𝑇 𝑊3h3)
• 𝜃 = {𝑊1, 𝑊2, 𝑊3}
• Bottom-up and Top-down:
• 𝑃 ℎ𝑗
2
= 1|h1, h3 =
𝜎(σ 𝒌 𝑾 𝒌𝒋
𝟑
𝒉 𝒌
𝟑
+ σ 𝒎 𝑾 𝒎𝒋
𝟐
𝒉 𝒎
𝟏 )
11
ℎ1
ℎ2
ℎ3
v
𝑊3
𝑊2
𝑊1
Variational Autoencoders (VAE)
12
Encoder Decoderx x’
μ
σ
Z
𝑞(𝑧|𝑥) 𝑝(𝑥|𝑧)
Sampling Reconstruct
ℒ 𝑞 = −𝑫 𝑲𝑳 𝒒 𝒛 𝒙 ∥ 𝒑 𝒎𝒐𝒅𝒆𝒍 𝒛 + 𝔼 𝒛~𝒒 𝒛 𝒙 𝐥𝐨𝐠 𝒑 𝒎𝒐𝒅𝒆𝒍(𝒙|𝒛)
𝔃 ~ 𝓝(𝝁, 𝝈)
http://www.slideshare.net/KazukiNitta/variational-autoencoder-68705109
Stochastic Gradient Variational Bayes
(SGVB) Estimator
13
Encoder Decoderx x’
μ
σ
Z
Back Prop.
Feed Forward
𝝐 ~ 𝓝 𝟎, 𝑰
𝔃 = 𝝁 + 𝝐𝝈
14http://dpkingma.com/wordpress/wp-content/uploads/2014/05/2014-03_talk_iclr.pdf
Generative Adversarial Networks(GAN)
15
Data Sample Discriminator
Generator
Sample
Noise
Yes or No
Generator
https://ishmaelbelghazi.github.io/ALI
Convolutional Neural Networks
16
Image Classification
Deep Convolutional Generative
Adversarial Networks (DCGAN)
17
https://openai.com/blog/generative-models/
Real Images vs. Generated images
18
http://kenkihara.com/projects/GAN.html

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Deep Generative Models

  • 2. Discriminative vs. Generative Learning Discriminative Learning Generative Learning Learn 𝑝(𝑦|𝑥) directly Model 𝑝 𝑦 , 𝑝 𝑥 𝑦 first, Then derive the posterior distribution: 𝑝 𝑦 𝑥 = 𝑝 𝑥 𝑦 𝑝(𝑦) 𝑝(𝑥) 2
  • 3. Undirected Graph vs. Directed Graph Undirected Directed • Boltzmann Machines • Restricted Boltzmann Machines • Deep Boltzmann Machines • Sigmoid Belief Networks • Variational Autoencoders (VAE) • Generative Adversarial Networks (GAN) 3 a b c a b c Deep Belief Networks
  • 4. Boltzmann Machines • Stochastic Recurrent Neural Network and Markov Random Field invented by Hinton and Sejnowski in 1985 • 𝑷 𝒙 = 𝐞𝐱𝐩(−𝑬 𝒙 ) 𝒁 > E(x): Energy function > Z: partition function where σ 𝑥 𝑃 𝑥 = 1 • Energy-based model: positive values all the time • Single visible layer and single hidden layer • Fully connected: not practical to implement 4
  • 5. Restricted Boltzmann Machines • Dimensionality reduction, classification, regression, collaborative filtering, feature learning and topic modeling • 𝑷 𝐯 = 𝒗, 𝐡 = 𝒉 = 𝟏 𝒁 𝐞𝐱𝐩(−𝑬 𝒗, 𝒉 ) • Two layers like BMs • Building blocks of deep probabilistic models • Gibbs sampling with Contrastive Divergence (CD) or Persistent CD 5
  • 6. Comparison btw BMs and RBMs Boltzmann Machines Restricted Boltzmann Machines v1 v2 v3 h1 h2 h3 h4 v1 v2 v3 h1 h2 h3 h4 6 𝑬 𝒗, 𝒉 = −𝒗 𝑻 𝑹𝒗 − 𝒉 𝑻 𝑺𝒉 − 𝒗 𝑻 𝑾𝒉 − 𝒃 𝑻 𝒗 − 𝒄 𝑻 𝒉 𝑬 𝒗, 𝒉 = −𝒃 𝑻 𝒗 − 𝒄 𝑻 𝒉 − 𝒗 𝑻 𝑾𝒉
  • 7. Deep Belief Networks • Unsupervised • Small dataset • Stacked RBMs • Pre-train each RBM • Undirected + Directed 7 RBM Sigmoid Belief Net
  • 8. • 𝑃 v, h1, h2, h3 = 𝑃 v h1 𝑃 h1 h2 𝑃(h2, h3) • 𝑃 v h1 = ς𝑖 𝑃(𝑣𝑖|h1) • 𝑃 h1 h2 = ς 𝑗 𝑃(ℎ𝑗 1 |h2 ) • 𝑃 h2 , h3 = 1 𝑍(𝑊3) exp(h2𝑇 𝑊3 h3 ) 8 RBM Sigmoid Belief Net ℎ1 ℎ2 ℎ3 v 𝑊3 𝑊2 𝑊1 Sigmoid Belief Net RBM
  • 9. Limitations of DBN (By Ruslan Salakhutdinov) • Explaining away • Greedy layer-wise pre-training > no optimization over all layers • Approximation inference is feed-forward > no bottom-up and top-down 9 http://www.slideshare.net/zukun/p05-deep-boltzmann-machines-cvpr2012-deep-learning-methods-for-vision
  • 10. Deep Boltzmann Machines • Unsupervised • Small dataset • Stacked RBMs • Pre-train each RBM • Undirected 10
  • 11. • 𝑃 𝜃 v = 1 𝑍(𝜃) σh1,h2,h3 exp( v 𝑇 𝑊1h1 + h1𝑇 𝑊2h2 + h2𝑇 𝑊3h3) • 𝜃 = {𝑊1, 𝑊2, 𝑊3} • Bottom-up and Top-down: • 𝑃 ℎ𝑗 2 = 1|h1, h3 = 𝜎(σ 𝒌 𝑾 𝒌𝒋 𝟑 𝒉 𝒌 𝟑 + σ 𝒎 𝑾 𝒎𝒋 𝟐 𝒉 𝒎 𝟏 ) 11 ℎ1 ℎ2 ℎ3 v 𝑊3 𝑊2 𝑊1
  • 12. Variational Autoencoders (VAE) 12 Encoder Decoderx x’ μ σ Z 𝑞(𝑧|𝑥) 𝑝(𝑥|𝑧) Sampling Reconstruct ℒ 𝑞 = −𝑫 𝑲𝑳 𝒒 𝒛 𝒙 ∥ 𝒑 𝒎𝒐𝒅𝒆𝒍 𝒛 + 𝔼 𝒛~𝒒 𝒛 𝒙 𝐥𝐨𝐠 𝒑 𝒎𝒐𝒅𝒆𝒍(𝒙|𝒛) 𝔃 ~ 𝓝(𝝁, 𝝈) http://www.slideshare.net/KazukiNitta/variational-autoencoder-68705109
  • 13. Stochastic Gradient Variational Bayes (SGVB) Estimator 13 Encoder Decoderx x’ μ σ Z Back Prop. Feed Forward 𝝐 ~ 𝓝 𝟎, 𝑰 𝔃 = 𝝁 + 𝝐𝝈
  • 15. Generative Adversarial Networks(GAN) 15 Data Sample Discriminator Generator Sample Noise Yes or No Generator https://ishmaelbelghazi.github.io/ALI
  • 17. Deep Convolutional Generative Adversarial Networks (DCGAN) 17 https://openai.com/blog/generative-models/
  • 18. Real Images vs. Generated images 18 http://kenkihara.com/projects/GAN.html