Generative Adversarial Networks, and Applications
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1 Generative Adversarial Networks, and Applications Ali Mirzaei Nimish Srivastava Kwonjoon Lee Songting Xu CSE 252C 4/12/17 2/44 Outline: Generative Models vs Discriminative Models (Background) Generative Adversarial Networks (GANs) Mathematical Proofs GAN Results GAN Failures Deep Convolutional GAN Architecture (Related Work) GAN-based Applications (Related Work) 1
2 3/44 Generative Models vs Discriminative Models: Data: Pixels Features: X Labels: Y (Zebra, No zebra) Discriminative model: model p(y. Generative model: model p(x,y). Learn p(y indirectly using Bayes rule. (Bayes rule: p(y p(x y)p(y)) 4/44 Discriminative Model: Map each sample to feature space. Learns decision boundary. For each point (x ) in feature space: P(Zebra x )+P(No Zebra x )=1 2
3 5/44 Generative Model: Learns probability distribution of each class of : P(x Zebra)dx=1 P(x No Zebra)dx=1 6/44 Generative Adversarial Networks (GANs): There is a game between two networks: Generative network. Discriminative network. Example: Counterfeit Money. G is a crook and is trying to generate fake money. D is a teller at the Bank. G s goal is to generate samples that D classifies as real. G should learn the underlying distribution of real money. 3
4 7/44 Generative Network Input: Random noise (z). Output: A fake image (x =G(z)). x Discriminator Network Input: A real image (, or a fake image (x =G(z)). Output: Probability that input is real. Input noise (z) G D P (input is real) x G s goal: D(G(z))=1 D s goal: D(=1 D(G(z))=0 Objective Function: 8/44 Remember that: x: Real image G(z): Fake image D s goals: D(=1 D(G(z))=0 G s goal: D(G(z))=1 4
5 9/44 Desired Convergence: We will prove that the models will approach equilibrium: D(=0.5 D(G(z))=0.5 D cannot discriminate. G will learn underlying distribution of real. (P ) Training Algorithm & Mathematical Proofs 5
6 11/44 Training Algorithm: From: Generative Adversarial Nets, Goodfellow et al, /44 Training Algorithm: The Algorithm takes k steps to optimize for the Discriminative net, D, and then one step of optimizing G, the generative net based on the outcome from D after above k steps. While optimizing for D, we work on the entire value function. For G, we only use the second term of the value function, as the gradient w.r.t. for the first term is zero. 6
7 13/44 Training Algorithm: It is important to note that we ascend the gradient when optimizing for D, as it is a maximization problem, and descend the gradient when optimizing for G as it is a minimization problem. Initially when G is far from optimal, the gradients in second optimization might be very small. Instead, we can ascend the gradient for log(d(g(z))) initially. 14/44 Training Algorithm: From: Generative Adversarial Nets, Goodfellow et al,
8 15/44 Global Optimality of p g =p This is done using A proposition for finding an optimal discriminator D for a fixed G (Proposition 1) A theorem for finding the global minimum for G (Theorem 1) Proposition 1: For G fixed, the optimal discriminator D is: D * G p p p g 16/44 Proof for Proposition 1: Expected Value: E [ g ( x )] p ( x ) g ( x ) p ( x ) g ( x ) dx Change of Variable: If Y = u(x) and u is invertible, then X=v(Y) where v is the inverse of u. By using the change of variable rule on the density functions for the distribution of Y to the distribution of X, we get: x Now: V ( G, D) Ex ~ p( [log( D( )] Ez ~ pz( z) [log(1 D( G( z)))] x x p p log( D( ) dx z pz( z)log(1 D( G( z))) dz log( D( ) dx pg( log(1 D( ) dx 8
9 17/44 Proof for Proposition 1 (Contd. ): From the last equation on previous slide: V ( G, D) p x D x pg ( ( )log( ( )) log(1 D( G( x x )))) For a function f: y a log(y) + b log(1-y), the maximum occurs at y = Therefore: * DG p p( p g a a+b. dx 18/44 Global Optimality of p g =p For a given G, the minmax value function can be defined as: C( G) max E E x ~ p x ~ p D [log( D [log( p V ( G, D) * G )] E p( p x ~ p g [log(1 D g )] E x ~ p )] [log( p pg( p )] Theorem 1: The global minimum of the virtual training criterion for G, i.e. C(G) is achieved if and only if p g =p. At that point, C(G) achieves the value of -log(4). g * G g 9
10 19/44 Proof of Theorem 1: For p g =p, D * G = 1/2; So, Also, C( G) E x ~ p log(4) x~ p [ log 2 log 2 log( p p log( ( p ( ~ GC ) E x~ p [log( 2)] E xpg [log( 2)]log( )4 p( )] E pg( p( ) pg( ) / 2 x~ pg( x ~ p [ log 2 log 2 log( p g pg( log( ( p pg( ) pg( ) / 2 pg( )] pg( Proof of Theorem 1 (Contd. ): 20/44 KL-Divergence between two probability distributions p and q: From the last equation on previous slide: CG ( ) log( 4) p pi p( KL ( p q) pi log( ) p( log( ) dx qi q( log( ( p i p ) pg )/2 pg log( ( p KL(p q) 0 for all p, q; equality holds when p=q. Therefore the minimum of C(G)=-log(4) obtained when p g =p. x pg ) pg ( ))/2 x x~ p x~ pg KL(p p + p g ) KL(p 2 g p + p g 2 ) 10
11 21/44 Convergence of Training Algorithm: For this, we use the following proposition. Proposition 2: If G and D have enough capacity, and at each step of Algorithm 1, the discriminator is allowed to reach its optimum given G, and p g is updated to improve the criterion: * * Ex ~ p[log D ] Ex ~ pg[log(1 D )] then p g converges to p. G G GAN Failures Deep Convolutional GAN Architecture (related work) 11
12 23/44 GAN Results 24/44 Failure 1: Mode Collapse Successful Unsuccessful (mode collapse) Class 0 Class 7 Class 6 Class 5 Class 4 Class 0 Class 7 Class 6 Class 5 Class 4 Class 1 Class 2 Class 3 Class 1 Class 2 Class 3 Problem Goal of GAN: To generate fake examples imitating real samples Easy way of achieving goal: Just generate easy modes (classes). Image Source: Luke Metz, Ben Poole, David Pfau, and Jascha Sohl-Dickstein. Unrolled generative adversarial networks (2016). 12
13 25/44 Failure 1: Mode Collapse (demo) Image Source: CSE 253 (Winter 2017) Project of Kwonjoon Lee, Andrew Leverentz, Ali Mirzaei, and Andrew Durnford. 26/44 Failure 2: Vanishing Gradient Problem The representational power (or capacity) between discriminator and generator is not balanced 13
14 27/44 DCGAN: The First Stabilized GAN (01/07/2016) Convolution (4, 4) w/ stride (2, 2) LeakyReLU (0.2) Subsampling (2, 2) Convolution (4, 4) LeakyReLU (0.2) Subsampling (2, 2) Convolution (4, 4) LeakyReLU (0.2) Fully Connected (4*4*512, 1) sigmoid P (input is real) Discriminator 64 x 64 x 3 32 x 32 x x 16 x x 8 x x 4 X Latent Random Noise Fully Connected (100, 4*4*512) Deconvolution (4, 4) w/ stride (2, 2) ReLU Upsampling (2, 2) Convolution (4, 4) ReLU Upsampling (2, 2) Convolution (4, 4) tanh Generator x 4 X x 8 x x 16 x x 32 x x 64 x 3 28/44 DCGAN: Results Bedroom images Face images Image Source: Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks (2016). 14
15 GAN-Based Applications (related work) 30/44 DCGAN: Vector Arithmetic on Face Samples (01/07/2016) Inspiration Vector( King ) Vector( Man ) + Vector( Woman ) Vector( Queen ) Simple arithmetic operations reveal rich linear structure in representation space (Mikolov et al., 2013) Question & Method Whether similar structure exists in Z representation(input) of our Generator e.g. Smiling Woman Neutral Woman + Neutral Man Smiling Man Difficulties Results are unstable if working on only single sample, but averaging the Z vector for three exemplars yields consistent and stable generations. 15
16 31/44 DCGAN: Vector Arithmetic on Face Samples Result Vector arithmetic for visual concepts, from Radford, Alec, Luke Metz, and Soumith Chintala. "Unsupervised representation learning with deep convolutional generative adversarial networks." arxiv preprint arxiv: (2015). 32/44 DCGAN: Vector Arithmetic on Face Samples Result Vector arithmetic for visual concepts, from Radford, Alec, Luke Metz, and Soumith Chintala. "Unsupervised representation learning with deep convolutional generative adversarial networks." arxiv preprint arxiv: (2015). 16
17 DCGAN: Vector Arithmetic on Face Samples Result Turn Vector By adding interpolations to random samples(z) we were able to transform their pose. 33/44 Turn Vector, from Radford, Alec, Luke Metz, and Soumith Chintala. "Unsupervised representation learning with deep convolutional generative adversarial networks." arxiv preprint arxiv: (2015). 34/44 DCGAN: Vector Arithmetic on Face Samples Future Conditional generative models can learn to convincingly model object attributes like scale, rotation, and position (Dosovitskiy et al., 2014) Further exploring the mentioned vector arithmetic could dramatically reduce the amount of needed for conditional generative modeling of complex image distributions. e.g. Gender + Expression + Race + Age + Pose = Individual with many details. 17
18 35/44 Video GAN: (Conditional) Video Generation (09/2016) Video GAN Basic Generator tries to generate a synthetic video, while Discriminator tries to discriminate synthetic from real videos Structure(Two-stream): From Vondrick, Carl, Hamed Pirsiavash, and Antonio Torralba. "Generating videos with scene dynamics." Advances In Neural Information Processing Systems /44 Video GAN: (Conditional) Video Generation Video Generation: 5000-hour videos from Flickr for training Experiment with four scene categories: golf course, hospital rooms, beaches, and train stations. Result: 18
19 37/44 Video GAN: (Conditional) Video Generation Conditional Video GAN Structure Given a static image x0, extrapolate a video of possible consequent frames. From Vondrick, Carl, Hamed Pirsiavash, and Antonio Torralba. "Generating videos with scene dynamics."advances In Neural Information Processing Systems Result 38/44 CycleGAN: Collection Style Transfer (03/30/2017) Image to Image Translation (11/21/2016) Paired Data Conditional GAN From Isola, Phillip, et al. "Image-to-image translation with conditional adversarial networks." arxiv preprint arxiv: (2016). 19
20 39/44 CycleGAN: Collection Style Transfer (03/30/2017) CycleGAN Basics Discovering special characteristics of each image collection and how these characteristics could be translated into the other image collection, all in the absence of any paired training examples. Generator1: Input x -> Output G( Discriminator: Tell G( from x Generator2: Input y -> Output F(y) Discriminator2: Tell F(y) from y Original Two GANs Loss Add a Cycle Consistency Loss that encourages F(G() = x and G(F(y)) = y L1 norm in this case Figures from Zhu, Jun-Yan, et al. "Unpaired Image-to-Image Translation using Cycle- Consistent Adversarial Networks." arxiv preprint arxiv: (2017). CycleGAN: Collection Style Transfer Objtective Minic the style of a whole set of art work, instead of a single piece of art work. e.g. Transfer photos(x) to Van Gogh style(y), rather than just in the style of Starry Night. Method Train the model on landscape photographs downloaded from Flickr(X) and art work from WikiArt(Y) 40/44 Some art works of Vincent Van Gogh, from WikiArt Starry Night, 1889, from WikiArt 20
21 41/44 Result Figures from Zhu, Jun-Yan, et al. "Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks." arxiv preprint arxiv: (2017). CycleGAN: Other Applications Photo generation from paintings Also include an identity loss that encourages identity mapping(photo and painting should not vary much) 42/44 Figures from Zhu, Jun-Yan, et al. "Unpaired Image-to-Image Translation using Cycle- Consistent Adversarial Networks." arxiv preprint arxiv: (2017). 21
22 43/44 CycleGAN: Other Applications Season Transfer Figures from Zhu, Jun-Yan, et al. "Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks." arxiv preprint arxiv: (2017). Thanks! 22
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