Geometric View of Optimal Transportation & Generative Models

1. GAN & Optimal Transportation Zhizhong LI Nov 23, 2017

2. A geometric view of Optimal transportation and generative model • Na Lei, Kehua Sun, Li Cui, Shing-Tung Yau, David Xianfeng Gu • References • Santambrogio, Filippo. "Optimal transport for applied mathematicians." Birkäuser, NY (2015). • Villani, Cédric. Optimal transport: old and new. Vol. 338. Springer Science & Business Media, 2008. • Goodfellow, Ian, et al. "Generative adversarial nets." Advances in neural information processing systems. 2014. • Arjovsky, Martin, Soumith Chintala, and Léon Bottou. "Wasserstein generative adversarial networks." International Conference on Machine Learning. 2017.

3. Glossary ConceptsDefinitions

4. Ian Goodfellow • Research scientist at Google Brain

5. GAN

6. Monge • (9 May 1746 – 28 July 1818) • Inventor of descriptive geometry (mathematical basis of technical drawing) • Father of differential geometry • Minister of the Marine

7. Monge’s Problem • Given two densities of mass on , with , • Find a map , pushing the first to the other, • And minimizing the cost • (note: two densities are for earth and embankment)

8. Push-forward operator • If is a Borel measure, is a Borel map. • Then the push-forward of is a measure on , defined by

9. Measure-Preserving Map • : metric spaces with probability measures , respectively. • is measure preserving if measurable set , • can be written as

10. Monge’s Optimal Mass Transport • Given a transportation cost function , find the measure preserving map that minimizes the total transportation cost

11. Kantorovich • (19 January 1912 – 7 April 1986) • Founder of linear programming • Stalin Prize in 1949 • Nobel Memorial Prize in Economic Sciences in 1975

12. Transportation Plan • Any strategy for sending to can be represented by a joint measure on , • is the transport plan of the amount of moving from to . • Equivalent written in terms of projections and ,

13. Kantorovich’s Problem • Finding the transportation plan that minimizing the transportation cost • It is a relaxation of Monge problem in that it allows more general movement. • is not a distance in general. E.g. when .

14. Roland Dobrushin • (July 20, 1929 – November 12, 1995) • A mathematician who made important contributions to probability theory, mathematical physics, and information theory. • The name "Wasserstein distance" was coined by R. L. Dobrushin in 1970.

15. Leonid Vaseršteĭn • Professor of Mathematics at Penn State University • He is well known for providing a simple proof of the Quillen–Suslin theorem, a result in commutative algebra.

16. Wasserstein distance • When the cost is defined in terms of a distance, is a distance between and . Called the Wasserstein distance. In general, for ,

17. Dual Kantorovich Problem • The Kantorovich problem is a linear optimization under linear constraints. • The dual problem is • Where , . • is called the Kantorovich potential.

18. Sun Wukong • 72 Di Sha transformation • Jin Dou Cloud • Eye of Truth

19. Example Explained • A company managing both bakeries (located at ) and cafés (located at ). • The transportation cost of sending bread from bakery to café is . • The primal Kantorovich problem is to minimize transportation cost.

20. Example Explained – Competitive pricing • Consider Sun Wukong want to build a business like this: • 1) buy breads from bakeries (say, bakery ) at price , and • 2) sell them to cafés (say, café ) at price • A competitive pricing is

21. Example Explained – inequality • With the help of Wukong, the transportation cost for the company is lower than the previous optimal transport plan

22. Example Explained – tight price • To get maximum profit, Wukong need the pair of prices and be tight • It would be great if he can achieve upper bound in previous slide’s equation. • To do that, Wukong only need to consider tight pairs. • Tight pairs only exist if is -convex.

23. convex • A function is convex only if it is a supreme of a family of linear functions • Which are parametrized by gradients , and determined by intercept .

24. -convex • Given function • A function is -convex only if exists , s.t. • The graph of a -convex function can entirely caress from below with a tool whose shape is negative of cost function.

25. Lipschitz • (14 May 1832 – 7 October 1903) • Known for • Lipschitz continuity • Lipschitz integral condition • Lipschitz quaternion

26. Example of -Convexity • , convex functions • , -Lipschitz functions

27. -transform • For -convex function , the -transform is • , Legendre transform • , identity functor • for tight pairs

28. Kantorovich Duality • Under some condition, there is duality

29. Kantorovich potential • is called the Kantorovich potential. • Once we know the optimal , we know the optimal cost.

30. G. S. Rubinstein • Professor

31. Kantorovich-Rubinstein formula • When , , and -convexity is -Lipschitz, • Note that is the Kantorovich potential. • This equation is the loss for discriminator of WGAN.

32. Martin Arjovsky • PhD student at the Courant Institute of Mathematical Sciences.

33. WGAN

34. WGAN Discriminator Loss • : data distribution • : generated distribution • : discriminator network with weight clipping

35. Brenier • (born January 1st 1957) • Centre de mathématiques Laurent Schwartz, • Ecole Polytechnique, FR-91128 Palaiseau, France

36. Brenier potential • If , and , where strictly convex, then the optimal transport plan • send each to a unique destination , • is the gradient of a convex function . This is called the Brenier potential. • Note is not ok.

37. Relation between potentials and • Note for Kantorovich potential • When is optimal in the dual Kantorovich problem, infimum is achieved. • For , the Brenier potential is ,

38. Minkovski • (22 June 1864 – 12 January 1909) • Created and developed the geometry of numbers. • He showed that his former student's, Albert Einstein, special theory of relativity could be understood geometrically as a theory of four-dimensional space–time, since known as the "Minkowski spacetime".

39. Minkovski problem • Suppose are unit vectors which span , • s.t. . Then • Exists convex polytope with codim-faces, s.t., • are their outward normal, and • are their volume.

40. Alexandrov • (August 4, 1912 – July 27, 1999) • Stalin Prize (1942) • Lobachevsky International Prize (1951) • Euler Gold Medal of the Russian Academy of Sciences (1992)

41. Alexandrov problem • compact convex polytope • , -th coordinates are negative • , s.t. , then • Exists convex polytope with codim-faces, s.t., • are normal of faces, • are volume of intersection of and projected faces.

42. Xianfeng Gu • Associate Professor • Department of Computer Science • Department of Applied Mathematics • State University of New York at Stony Brook

43. Feng Luo • Professor of Mathematics, Rutgers University

44. Jian SUn • Associate Professor at Tsinghua University

45. Shing-Tung Yau • (born April 4, 1949) • Director, The Institute of Mathematical Sciences, The Chinese University of Hong Kong • Department of Mathematics, Harvard University • Fields Medal in 1982 • Wolf Prize in Mathematics in 2010

46. Convex Polytope by Linear function • Given , and , the graph of • is a convex polytope in . • The projection induces a cell decomposition of . • Given probability measure on , volume of is defined as

47. Gu-Luo-Sun-Yau • Let be compact convex domain in , a probability measure on ,be distinct points, then • For any , , Exists , s.t. . • minimizes the quadratic cost • among transport maps , where the Dirac measure .

48. Semi-discrete optimal mass transport Geometry Transportation Given target locations with probability weight . Find transportation map . The transportation map is determined by the Brenier potential . Each cell is mapped to . • Given normal , and each associate with a volume . • Find the convex polytope satisfy normal condition and volume condition. • The convex polytope is determined by the graph of .

49. Constructive result • Alexandrov’s proof is non-variational and non-constructive. • Gu-Luo-Sun-Yau’s theorem has a variational proof and produce an algorithm for finding , and thus the Brenier potential . • In fact, is the maximum of concave function • On a open convex set.

50. Claims What Does It Want To Say