REFINING DEEP GENERATIVE MODELS VIA DISCRIMINATOR GRADIENT FLOW

Abstract

Deep generative modeling has seen impressive advances in recent years, to the point where it is now commonplace to see simulated samples (e.g., images) that closely resemble real-world data. However, generation quality is generally inconsistent for any given model and can vary dramatically between samples. We introduce Discriminator Gradient f low (DGf low), a new technique that improves generated samples via the gradient flow of entropy-regularized f -divergences between the real and the generated data distributions. The gradient flow takes the form of a non-linear Fokker-Plank equation, which can be easily simulated by sampling from the equivalent McKean-Vlasov process. By refining inferior samples, our technique avoids wasteful sample rejection used by previous methods (DRS & MH-GAN). Compared to existing works that focus on specific GAN variants, we show our refinement approach can be applied to GANs with vector-valued critics and even other deep generative models such as VAEs and Normalizing Flows. Empirical results on multiple synthetic, image, and text datasets demonstrate that DGf low leads to significant improvement in the quality of generated samples for a variety of generative models, outperforming the state-of-the-art Discriminator Optimal Transport (DOT) and Discriminator Driven Latent Sampling (DDLS) methods.

1. INTRODUCTION

Deep generative models (DGMs) have excelled at numerous tasks, from generating realistic images (Brock et al., 2019) to learning policies in reinforcement learning (Ho & Ermon, 2016) . Among the variety of proposed DGMs, Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) have received widespread popularity for their ability to generate high quality samples that resemble real data. Unlike Variational Autoencoders (VAEs) (Kingma & Welling, 2014) and Normalizing Flows (Rezende & Mohamed, 2015; Kingma & Dhariwal, 2018) , GANs are likelihood-free methods; training is formulated as a minimax optimization problem involving a generator and a discriminator. The generator seeks to generate samples that are similar to the real data by minimizing a measure of discrepancy (between the generated samples and real samples) furnished by the discriminator. The discriminator is trained to distinguish the generated samples from the real samples. Once trained, the generator is used to simulate samples and the discriminator has traditionally been discarded. However, recent work has shown that discarding the discriminator is wasteful -it actually contains useful information about the underlying data distribution. This insight has led to sample improvement techniques that use this information to improve the quality of generated samples (Azadi et al., 2019; Turner et al., 2019; Tanaka, 2019; Che et al., 2020) . Unfortunately, current methods either rely on wasteful rejection operations in the data space (Azadi et al., 2019; Turner et al., 2019) , or require a sensitive diffusion term to ensure sample diversity (Che et al., 2020) . Prior work has also focused on improving GANs with scalar-valued discriminators, which excludes a large family of GANs with vector-valued critics, e.g., MMDGAN (Li et al., 2017; Bińkowski et al., 2018) and OCFGAN (Ansari et al., 2020) , and likelihood-based generative models. < l a t e x i t s h a 1 _ b a s e 6 4 = " a 1 l V E J v n 2 F / + U J G y Z k z L Q X L Z b E U = " > A A A B 7 X i c b V D L S s N A F L 2 p r 1 p f V Z d u g k V w V Z L a 2 H Z l w Y 3 L C v Y B b S i T 6 a Q d O 8 m E m Y l Q Q v / B j Q t F 3 P o / 7 v w L P 8 F J W s T X g Q u H c + 7 l 3 n u 8 i F G p L O v d y K 2 s r q 1 v 5 D c L W 9 s 7 u 3 v F / Y O O 5 L H A p I 0 5 4 6 L n I U k Y D U l b U c V I L x I E B R 4 j X W 9 6 m f r d O y I k 5 e G N m k X E D d A 4 p D 7 F S G m p M x A T P r S G x Z J V t j K Y f 4 m 9 J K W L D 8 j Q G h b f B i O O 4 4 C E C j M k Z d + 2 I u U m S C i K G Z k X B r E k E c J T N C Z 9 T U M U E O k m 2 b V z 8 0 Q r I 9 P n Q l e o z E z 9 P p G g Q M p Z 4 O n O A K m J / O 2 l 4 n 9 e P 1 Z + 3 U 1 o G M W K h H i x y I + Z q b i Z v m 6 O q C B Y s Z k m C A u q b z X x B A m E l Q 6 o k I X Q q N q O 4 + j f G + d O 5 a y R k r p j V W t f I X Q q Z d s p W 9 f V U r O 1 S A P y c A T H c A o 2 1 K A J V 9 C C N m C 4 h X t 4 h C e D G w / G s / G y a M 0 Z y 5 l D + A H j 9 R M O Q J C I < / l a t e x i t > ⇢ 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " f h k M Z 6 C z 3 k + / G f K Z R l j b e i s 8 D P E = " > A A A B 7 X i c b V D L S s N A F L 2 p r 1 p f V Z d u g k V w V Z L a 2 H Z l w Y 3 L C v Y B b S i T 6 a Q d O 8 m E m Y l Q Q v / B j Q t F 3 P o / 7 v w L P 8 F J W s T X g Q u H c + 7 l 3 n u 8 i F G p L O v d y K 2 s r q 1 v 5 D c L W 9 s 7 u 3 v F / Y O O 5 L H A p I 0 5 4 6 L n I U k Y D U l b U c V I L x I E B R 4 j X W 9 6 m f r d O y I k 5 e G N m k X E D d A 4 p D 7 F S G m p M x A T P r S H x Z J V t j K Y f 4 m 9 J K W L D 8 j Q G h b f B i O O 4 4 C E C j M k Z d + 2 I u U m S C i K G Z k X B r E k E c J T N C Z 9 T U M U E O k m 2 b V z 8 0 Q r I 9 P n Q l e o z E z 9 P p G g Q M p Z 4 O n O A K m J / O 2 l 4 n 9 e P 1 Z + 3 U 1 o G M W K h H i x y I + Z q b i Z v m 6 O q C B Y s Z k m C A u q b z X x B A m E l Q 6 o k I X Q q N q O 4 + j f G + d O 5 a y R k r p j V W t f I X Q q Z d s p W 9 f V U r O 1 S A P y c A T H c A o 2 1 K A J V 9 C C N m C 4 h X t 4 h C e D G w / G s / G y a M 0 Z y 5 l D + A H j 9 R M P x J C J < / l a t e x i t > ⇢ 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " / z 5 Q x v a d + w 0 m r o R A G z 5 8 x L X 9 P X o = " > A A A B 7 X i c b V D L S s N A F L 2 p r 1 p f V Z d u g k V w V Z L a 2 H Z l w Y 3 L C v Y B b S i T 6 a Q d O 8 m E m Y l Q Q v / B j Q t F 3 P o / 7 v w L P 8 F J W s T X g Q u H c + 7 l 3 n u 8 i F G p L O v d y K 2 s r q 1 v 5 D c L W 9 s 7 u 3 v F / Y O O 5 L H A p I 0 5 4 6 L n I U k Y D U l b U c V I L x I E B R 4 j X W 9 6 m f r d O y I k 5 e G N m k X E D d A 4 p D 7 F S G m p M x A T P q w M i y W r b G U w / x J 7 S U o X H 5 C h N S y + D U Y c x w E J F W Z I y r 5 t R c p N k F A U M z I v D G J J I o S n a E z 6 m o Y o I N J N s m v n 5 o l W R q b P h a 5 Q m Z n 6 f S J B g Z S z w N O d A V I T + d t L x f + 8 f q z 8 u p v Q M I o V C f F i k R 8 z U 3 E z f d 0 c U U G w Y j N N E B Z U 3 2 r i C R I I K x 1 Q I Q u h U b U d x 9 G / N 8 6 d y l k j J X X H q t a + Q u h U y r Z T t q 6 r p W Z r k Q b k 4 Q i O 4 R R s q E E T r q A F b c B w C / f w C E 8 G N x 6 M Z + N l 0 Z o z l j O H 8 A P G 6 y c R S J C K < / l a t e x i t > ⇢ 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " L M 0 2 l Y j 2 P a x / x m l O v D p 8 k 9 y n i l U H u 5 9 x 4 / 5 E w b 2 3 6 3 V l b X 1 j c 2 c 1 v 5 7 Z 3 F 1 4 M L h n H u 5 9 x 4 / 5 E w b 2 3 6 3 V l b X 1 j c 2 c 1 v 5 7 Z 3 F 1 4 M L h n H u 5 9 x 4 / 5 E w b 2 3 6 3 V l b X 1 j c 2 c 1 v 5 7 Z 3  = " > A A A B 7 X i c b V D L S s N A F L 2 p r 1 p f V Z d u g k V w V Z L a 2 H Z l w Y 0 r q W B b o Q 1 l M p 2 0 Y y e Z M D M R S u g / u H G h i F v / x 5 1 / 4 S c 4 S Y v 4 O n D h c M 6 9 3 H u P F z E q l W W 9 G 7 m l 5 Z X V t f x 6 Y W N z a 3 u n u L v X k T w W m L Q x Z 1 z c e E g S R k P S V l Q x c h M J g g K P k a 4 3 O U / 9 7 h 0 R k v L w W k 0 j 4 g Z o F F K f Y q S 0 1 O m L M R 9 c D o o l q 2 x l M P 8 S e 0 F K Z x + Q o T U o v v W H H M c B C R V m S M q e b U X K T Z B Q F D M y K / R j S S K E J 2 h E e p q G K C D S T b J r Z + a R V o a m z 4 W u U J m Z + n 0 i Q Y G U 0 8 D T n Q F S Y / n b S 8 X / v F 6 s / L q b 0 D C K F Q n x f J E f M 1 N x M 3 3 d H F J B s G J T T R A W V N 9 q 4 j E S C C s d U C E L o V G 1 H c f R v z d O n c p J I y V 1 x 6 r W v k L o V M q 2 U 7 a u q q V m a 5 4 G 5 O E A D u E Y b K h B E y 6 g B W 3 A c A v 3 8 A h P B j c e j G f j Z d 6 a M x Y z + / A D x u s n O 7 i Q p g = = < / l a t e x i t > ⇢ N < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 W Q 0 o v C F 7 X Q M T m x O c a I q J k p w B 3 U = " > A A A B 6 3 i c b V D L S s N A F L 3 x W e u r 6 t L N Y B F c l a Q 2 t l 1 Z c O O y g n 1 A G 8 p k O m 2 H z k z C z E Q o o b / g x o U i b v 0 h d / 6 F n 2 C S F v F 1 4 M L h n d v f 3 C w W F b B 5 E i t E U C H q i u j z X l T N K W Y Y b T b q g o F j 6 n H X 9 6 l f q d O 6 o 0 C + S t m Y X U E 3 g s 2 Y g R b F K p r 5 k Y F I p 2 y c 6 A / h J n S Y q X H 5 C h O S i 8 9 Y c B i Q S V h n C s d c + x Q + P F W B l G O J 3 n + 5 G m I S Z T P K a 9 h E o s q P b i 7 N Y 5 O k 2 U I R o F K i l p U K Z + n 4 i x 0 H o m / K R T Y D P R v 7 1 U / M / r R W Z U 8 2 I m w 8 h Q S R a L R h F H J k D p 4 2 j I F C W G z x K C i W L J r Y h M s M L E J P H k s x D q F c d 1 3 e T 3 + o V b P q + n p O b a l e p X C O 1 y y X F L 9 k 2 l 2 G g u 0 o A c H M M J n I E D V W j A N T S h B Q Q m c A + P 8 G Q J 6 8 F 6 t l 4 W r S v W c u Y I f s B 6 / Q T n 6 o / l < / l a t e x i t > ⇠ < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 W Q 0 o v C F 7 X Q M T m x O c a I q J k p w B 3 U = " > A A A B 6 3 i c b V D L S s N A F L 3 x W e u r 6 t L N Y B F c l a Q 2 t l 1 Z c O O y g n 1 A G 8 p k O m 2 H z k z C z E Q o o b / g x o U i b v 0 h d / 6 F n 2 C S F v d v f 3 C w W F b B 5 E i t E U C H q i u j z X l T N K W Y Y b T b q g o F j 6 n H X 9 6 l f q d O 6 o 0 C + S t m Y X U E 3 g s 2 Y g R b F K p r 5 k Y F I p 2 y c 6 A / h J n S Y q X H 5 C h O S i 8 9 Y c B i Q S V h n C s d c + x Q + P F W B l G O J 3 n + 5 G m I S Z T P K a 9 h E o s q P b i 7 N Y 5 O k 2 U I R o F K i l p U K Z + n 4 i x 0 H o m / K R T Y D P R v 7 1 U / M / r R W Z U 8 2 I m w 8 h Q S R a L R h F H J k D p 4 2 j I F C W G z x K C i W L J r Y h M s M L E J P H k s x D q F c d 1 3 e T 3 + o V b P q + n p O b a l e p X C O 1 y y X F L 9 k 2 l 2 G g u 0 o A c H M M J n I E D V W j A N T S h B Q Q m c A + P 8 G Q J 6 8 F 6 t l 4 W r S v W c u Y I f s B 6 / Q T n 6 o / l < / l a t e x i t > ⇠ < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 W Q 0 o v C F 7 X Q M T m x O c a I q J k p w B 3 U = " > A A A B 6 3 i c b V D L S s N A F L 3 x W e u r 6 t L N Y B F c l a Q 2 t l 1 Z c O O y g n 1 A G 8 p k O m 2 H z k z C z E Q o o b / g x o U i b v 0 h d / 6 F n 2 C S F v d v f 3 C w W F b B 5 E i t E U C H q i u j z X l T N K W Y Y b T b q g o F j 6 n H X 9 6 l f q d O 6 o 0 C + S t m Y X U E 3 g s 2 Y g R b F K p r 5 k Y F I p 2 y c 6 A / h J n S Y q X H 5 C h O S i 8 9 Y c B i Q S V h n C s d c + x Q + P F W B l G O J 3 n + 5 G m I S Z T P K a 9 h E o s q P b i 7 N Y 5 O k 2 U I R o F K i l p U K Z + n 4 i x 0 H o m / K R T Y D P R v 7 1 U / M / r R W Z U 8 2 I m w 8 h Q S R a L R h F H J k D p 4 2 j I F C W G z x K C i W L J r Y h M s M L E J P H k s x D q F c d 1 3 e T 3 + o V b P q + n p O b a l e p X C O 1 y y X F L 9 k 2 l 2 G g u 0 o A c H M M J n I E D V W j A N T S h B Q Q m c A + P 8 G Q J 6 8 F 6 t l 4 W r S v W c u Y I f s B 6 / Q T n 6 o / l < / l a t e x i t > ⇠ < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 W Q 0 o v C F 7 X Q M T m x O c a I q J k p w B 3 U = " > A A A B 6 3 i c b V D L S s N A F L 3 x W e u r 6 t L N Y B F c l a Q 2 t l 1 Z c O O y g n 1 A G 8 p k O m 2 H z k z C z E Q o o b / g x o U i b v 0 h d / 6 F n 2 C S F v F 1 4 M L h n H u 5 9 x 4 / 5 E w b 2 3 6 3 V l b X 1 j c 2 c 1 v 5 7 Z 3 d v f 3 C w W F b B 5 E i t E U C H q i u j z X l T N K W Y Y b T b q g o F j 6 n H X 9 6 l f q d O 6 o 0 C + S t m Y X U E 3 g s 2 Y g R b F K p r 5 k Y F I p 2 y c 6 A / h J n S Y q X H 5 C h O S i 8 9 Y c B i Q S V h n C s d c + x Q + P F W B l G O J 3 n + 5 G m I S Z T P K a 9 h E o s q P b i 7 N Y 5 O k 2 U I R o F K i l p U K Z + n 4 i x 0 H o m / K R T Y D P R v 7 1 U / M / r R W Z U 8 2 I m w 8 h Q S R a L R h F H J k D p 4 2 j I F C W G z x K C i W L J r Y h M s M L E J P H k s x D q F c d 1 3 e T 3 + o V b P q + n p O b a l e p X C O 1 y y X F L 9 k 2 l 2 G g u 0 o A c H M M J n I E D V W j A N T S h B Q Q m c A + P 8 G Q J 6 8 F 6 t l 4 W r S v W c u Y I f s B 6 / Q T n 6 o / l < / l a t e x i t > ⇠ < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 W Q 0 o v C F 7 X Q M T m x O c a I q J k p w B 3 U = " > A A A B 6 3 i c b V D L S s N A F L 3 x W e u r 6 t L N Y B F c l a Q 2 t l 1 Z c O O y g n 1 A G 8 p k O m 2 H z k z C z E Q o o b / g x o U i b v 0 h d / 6 F n 2 C S F v F 1 4 M L h n H u 5 9 x 4 / 5 E w b 2 3 6 3 V l b X 1 j c 2 c 1 v 5 7 Z 3 d v f 3 C w W F b B 5 E i t E U C H q i u j z X l T N K W Y Y b T b q g o F j 6 n H X 9 6 l f q d O 6 o 0 C + S t m Y X U E 3 g s 2 Y g R b F K p r 5 k Y F I p 2 y c 6 A / h J n S Y q X H 5 C h O S i 8 9 Y c B i Q S V h n C s d c + x Q + P F W B l G O J 3 n + 5 G m I S Z T P K a 9 h E o s q P b i 7 N Y 5 O k 2 U I R o F K i l p U K Z + n 4 i x 0 H o m / K R T Y D P R v 7 1 U / M / r R W Z U 8 2 I m w 8 h Q S R a L R h F H J k D p 4 2 j I F C W G z x K C i W L J r Y h M s M L E J P H k s x D q F c d 1 3 e T 3 + o V b P q + n p O b a l e p X C O 1 y y X F L 9 k 2 l 2 G g u 0 o A c H M M J n I E D V W j A N T S h B Q Q m c A + P 8 G Q J 6 8 F 6 t l 4 W r S v v Z O f n e v J Y N I Y N L E A Q t E x 0 O S M M p J U 1 H F S C c U B P k e I 2 1 v c p H 4 7 T s i J A 3 4 j Z q G x P X R i N M h x U h p 6 b Y n x k E / v j q x Z / 1 8 w S p a K c y / x F 6 Q w v k H p G j 0 8 2 + 9 Q Y A j n 3 C F G Z K y a 1 u h c m M k F M W M z H K 9 S J I Q 4 Q k a k a 6 m H P l E u n F 6 8 c w 8 0 s r A H A Z C F 1 d m q n 6 f i J E v 5 d T 3 d K e P 1 F j + 9 h L x P 6 8 b q W H V j S k P I 0 U 4 n i 8 a R s x U g Z m 8 b w 6 o I F i x q S Y I C 6 p v N f E Y C Y S V D i m X h l A r 2 4 7 j 6 N 9 r Z 0 7 p t J a Q q m O V K 1 8 h t E p F 2 y l a 1 + V C v T F P A 7 J w A I d w D D Z U o A 6 X 0 I A m Y O B w D 4 / w Z E j j w X g 2 X u a t G W M x s w 8 / Y L x + A u F j k i Q = < / l a t e x i t > ⇢ N 1 < l a t e x i t A W h L R L y U H R d L C l n A W 0 p p j j t R o J i 3 + W 0 4 0 4 u U r 9 z S 4 V k Y X C j p h F 1 f D w K m M c I V l q 6 v h u g Q a G I S i i D + Z d Y C 1 I 8 / 4 A M z U H h r T 8 M S e z T Q B G O p e x Z K F J O g o V i h N N Z v h 9 L G m E y w S P a 0 z T A P p V O k p 0 6 M 4 + 1 M j S 9 U O g K l J m p 3 y c S 7 E s 5 9 V 3 d 6 W M 1 l r + 9 V 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L M B a E t k j I Q 9 F 1 s a S c B b S l m O K 0 G w m K f Z f T j j u 5 S P 3 O L R W S h c G N m k b U 8 f E o Y B 4 j W G n p + m 5 Q H h S K Z s n M g P 4 S a 0 G K 5 x + Q o T k o v P W H I Y l 9 G i j C s Z Q 9 y 4 y U k 2 C h G O F 0 l u / H k k a Y T P C I 9 j Q N s E + l k 2 S n z t C x V o b I C 4 W u Q K F M / T 6 R Y F / K q e / q T h + r s f z t p e J / X i 9 W X s 1 J W B D F i g Z k v s i L O V I h S v 9 G Q y Y o U X y q C S a C 6 V s R G W O B i d L p 5 L M Q 6 h X L t m 3 9 e / 3 M L p / W U 1 K z z U r 1 K 4 R 2 u W T Z J f O q U m w 0 5 2 l A D g 7 h C E 7 A g i o 0 4 B K a 0 A I C I 7 i H R 3 g y u P F g P B s v 8 9 Y l Y z F z A D 9 g v H 4 C 2 H u P Q Q = = < / l a t e x i t > z 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " / K f K s l h V 4 0 K 5 p m a I v J N q c Y D g h I Y = " > A A A B 7 n i c b V D L S s N A F L 2 p r 1 p f V Z d u g k V w Y 0 l q Y 9 u V B T e u p I J 9 Q B v K Z D p t h 0 4 m Y W Y i 1 N C P c O N C E b d + j z v / w k 9 w k h b x d e D C 4 Z x 7 u f c e L 2 R U K s t 6 N z J L y y u r a 9 n 1 3 M b m 1 v Z O f n e v J Y N I Y N L E A Q t E x 0 O S M M p J U 1 H F S C c U B P k e I 2 1 v c p H 4 7 V s i J A 3 4 j Z q G x P X R i N M h x U h p q X 3 X j 6 9 O 7 F k / X 7 C K V g r z L 7 E X p H D + A S k a / f x b b x D g y C d c Y Y a k 7 N p W q N w Y C U U x I 7 N c L 5 I k R H i C R q S r K U c + k W 6 c n j s z j 7 Q y M I e B 0 M W V m a r f J 2 L k S z n 1 P d 3 p I z W W v 7 1 E / M / r R m p Y d W P K w 0 g R j u e L h h E z V W A m v 5 s D K g h W b K o J w o L q W 0 0 8 R g J h p R P K p S H U y r b j O P r 3 2 p l T O q 0 l p O p Y 5 c p X C K 1 S 0 X a K 1 n W 5 U G / M 0 4 A s H M A h H I M N F a j D J T S g C R g m c A + P 8 G S E x o P x b L z M W z P G Y m Y f f s B 4 / Q S j 4 Z D b < / l a t e x i t > z N 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " L b P h c H T 5 F B P D j A b 3 M n f p C K v T g 5 s = " > A A A B 7 H i c b V D L S s N A F L 3 x W e u r 6 t J N s A i u S l I b 2 6 4 s u H E l F U x b a E O Z T K f t 0 M k k z E y E G v o N b l w o 4 t Y P c u d f + A l O 0 i K + D l w 4 n H M v 9 9 7 j R 4 x K Z V n v x t L y y u r a e m 4 j v 7 m 1 v b N b 2 N t v y T A W m L g 4 Z K H o + E g S R j l x F V W M d C J B U O A z 0 v Y n F 6 n f v i V C 0 p D f q G l E v A C N O B 1 S j J S W 3 L t + c j X r F 4 p W y c p g / i X 2 g h T P P y B D s 1 9 4 6 w 1 C H A e E K 8 y Q l F 3 b i p S X I K E o Z m S W 7 8 W S R A h P 0 I h 0 N e U o I N J L s m N n 5 r F W B u Y w F L q 4 M j P 1 + 0 S C A i m n g a 8 7 A 6 T G 8 r e X i v 9 5 3 V g N a 1 5 C e R Q r w v F 8 0 T B m p g r N 9 H N z Q A X B i k 0 1 Q V h Q f a u J x 0 g g r H Q + + S y E e s V 2 H E f / X j 9 z y q f 1 l N Q c q 1 L 9 C q F V L t l O y b q u F B v N e R q Q g 0 M 4 g h O w o Q o N u I Q m u I C B w j 0 8 w p P B j Q f j 2 X i Z t y 4 Z i 5 k D + A H j 9 R P H s Z B p < / l a t e x i t > z N < l a t e x i t s h a 1 _ b a s e 6 4 = " m F F G x i 1 f T Y 2 G X k w g c O N w 6 Q g B + f I = " > A A A B 8 n i c b V D L S s N A F L 3 x W e u r 6 t J N s A i u S l I b 2 6 4 s u H F Z w T 4 0 D W U y n b Z D J 5 k w M x F K 6 G e 4 c a G I W 7 / G n X / h J z h J i / g 6 M H A 4 5 1 7 m 3 O N H j E p l W e / G 0 v L K 6 t p 6 b i O / u b W 9 s 1 v Y 2 2 9 L H g t M W p g z L r o + k o T R k L Q U V Y x 0 I 0 F Q 4 D P S 8 S c X q d + 5 I 0 J S H l 6 r a U S 8 A I 1 C O q Q Y K S 2 5 v Q C p M U Y s u Z 3 1 C 0 W r Z G U w / x J 7 Q Y r n H 5 C h 2 S + 8 9 Q Y c x w E J F W Z I S t e 2 I u U l S C i K G Z n l e 7 E k E c I T N C K u p i E K i P S S L P L M P N b K w B x y o V + o z E z 9 v p G g Q M p p 4 O v J N K L 8 7 a X i f 5 4 b q 2 H N S 2 g Y x Y q E e P 7 R M G a m 4 m Z 6 v z m g g m D F p p o g L K j O a u I x E g g r 3 V I + K 6 F e s R 3 H 0 b f X z 5 z y a T 0 l N c e q V L 9 K a J d L t l O y r i r F x s 2 8 D c j B I R z B C d h Q h Q Z c Q h N a g I H D P T z C k 6 G M B + P Z e J m P L h m L n Q P 4 A e P 1 E 2 F K k x c = < / l a t e x i t > Z < l a t e x i t s h a 1 _ b a s e 6 4 = " R S 0 k 4 p r 8 / S K b e w o E c p 6 u s 1 7 a H P 0 We further present a generalized framework that employs existing pre-trained discriminators to refine samples from a variety of deep generative models: we demonstrate our method can be applied to GANs with vector-valued critics, and even likelihood-based models such as VAEs and Normalizing Flows. Empirical results on synthetic datasets, and benchmark image (CIFAR10, STL10) and text (Billion Words) datasets demonstrate that our gradient flow-based approach outperforms DOT and DDLS on multiple quantitative evaluation metrics. = " > A A A B 9 H i c b V D L T g I x F L 2 D L 8 Q X 6 t J N I z F x R W a Q E V h J 4 s Y l J v I w M C G d U q C h 8 7 D t k J A J 3 + H G h c a 4 9 W P c + R d + g p 2 B G F 8 n a X J y z r 2 5 p 8 c N O Z P K N N + N z M r q 2 v p G d j O 3 t b 2 z u 5 f f P 2 j J I B K E N k n A A 9 F x s a S c + b S p m O K 0 E w q K P Z f T t j u 5 T P z 2 l A r J A v 9 G z U L q e H j k s y E j W G n J 6 X l Y j Q n m c W P e L / X In summary, this paper's key contributions are: • DGf low, a method to refine deep generative models using the gradient flow of f -divergences; • a framework that extends DGf low to GANs with vector-valued critics, VAEs, and Normalizing Flows; • experiments on a variety of generative models trained on synthetic, image (CIFAR10 & STL10), and text (Billion Words) datasets demonstrating that DGf low is effective in improving samples from generative models.

2. BACKGROUND: GRADIENT FLOWS

The following gives a brief introduction to gradient flows; we refer readers to the excellent overview by Santambrogio (2017) for a more thorough introduction. Let (X , • 2 ) be a Euclidean space and F : X → R be a smooth energy function. The gradient flow of F is the smooth curve {x t } t∈R+ that follows the direction of steepest descent, i.e., x (t) = -∇F (x(t)). (1) The value of the energy F is minimized along this curve. This idea of steepest descent curves can be characterized in arbitrary metric spaces via the minimizing movement scheme (Jordan et al., 1998) . Of particular interest is the metric space of probability measures that is endowed with the Wasserstein distance (W p ); the Wasserstein distance is a metric and the W p topology satisfies weak convergence of probability measures (Villani, 2008, Theorem 6.9) . Gradient flows in the 2-Wasserstein space (P 2 (Ω), W 2 ) -i.e., the space of probability measures with finite second moments and the 2-Wasserstein metric -have been studied extensively. Let {ρ t } t∈R+ be the gradient flow of a functional F in the 2-Wasserstein space, where ρ t is absolutely continuous with respect to the Lebesgue measure. The curve {ρ t } t∈R+ satisfies the continuity equation (Ambrosio et al., 2008, Theorem 8.3.1) , ∂ t ρ t + ∇ • (ρ t v t ) = 0. (2) The velocity field v t in Eq. ( 2) is given by v t (x) = -∇ x δF δρ (ρ),



W c u Y I f s B 6 / Q T n 6 o / l < / l a t e x i t > ⇠ < l a t e x i t s h a 1 _ b a s e 6 4 = " T x O L w t r q x Y a 9 6 2 C D Z i y F p S a 5 S D o = " > A A A B 8 X i c b V D L S s N A F L 2 p r 1 p f V Z d u g k V w Y 0 l q Y 9 u V B T e u p I J 9 Y B v K Z D p t h 0 4 m Y W Y i l N C / c O N C E b f + j T v / w k 9 w k h b x d e D C 4 Z x 7 u f c e L 2 R U K s t 6 N z J L y y u r a 9 n 1 3 M b m 1

s h a 1 _ b a s e 6 4 = " x y 2 s b a o 8 B B Z L Y l 4 t 2 P 7 Y 1 i A m 3 g c = " > A A A B 6 n i c b V D J S g N B E K 1 x j X G L e v Q y G A R P o S d m T H I y 4 M V j R L N A M o S e T k / S p G e h u 0 e I Q z 7 B i w d F v P p F 3 v w L P 8 G e S R C 3 B w W P 9 6 q o q u d G n E m F 0 L u x t L y y u r a e 2 8 h v b m 3 v 7 B b 2 9 t s y j

d z D I z w Z 3 H g w n o 2 X e e u S s Z g 5 g B 8 w X j 8 B 1 X O P P w = = < / l a t e x i t > z 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " u V D / z N v 0 B V x S P w W p T 8 f E o / / e J 0 Q = " > A A A B 6 n i c b V D J S g N B E K 1 x j X G L e v T S G A R P Y S Z m T H I y 4 M V j R L N A M o S e T k / S p G e h u 0 e I Q z 7 B i w d F v P p F 3 v w L P 8 G e S R C 3 B w W P 9 6 q o

d z D I z w Z 3 H g w n o 2 X e e u S s Z g 5 g B 8 w X j 8 B 1 v e P Q A = = < / l a t e x i t > z 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " L I 6 F y 3 d R 2 X w S 5 h u O b H x t s A f z F 2 c = " > A A A B 6 n i c b V D J S g N B E K 1 x j X G L e v T S G A R P Y S Z m T H I y 4 M V j R L N A M o S e T k / S p G e h u 0 e I Q z 7 B i w d F v P p F 3 v w L P 8 G e S R C 3 B w W P 9 6 q o

Figure 1: An illustration of refinement using DGf low, with the gradient flow in the 2-Wasserstein space P2 (top) and the corresponding discretized SDE in the latent space Z (bottom). The image samples from the densities along the gradient flow are shown in the middle.

