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Geometric Interpretation to Generative Models and Optimal Transportation

Samadi Bahrami, Shirin | 2022

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 55000 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Bahraini, Alireza
  7. Abstract:
  8. As the core of this thesis, we express the relation between optimal transportation and convex geometry especially the variational approach to solve Alexandrov problem, which leads to a geometric interpretation to adversarial models and propose a novel framework for these models. Along the way, we peruse generative adversarial networks from optimal transportation view and show that generator calculates the transportation map and the discriminator computes Wasserestein distance, which is equivalent to Kantorovich potential. By using optimal mass transportation theory and choosing an especial cost function c, we see that the generator and discriminator are equivalent. Therefore, once the potential Kantorovich reaches the optimum, the generator map can be obtained directly without training. We introduce Minkowski and Alexandrov problems in convex geometry, which can be described by Monge-Ampere equation as well and this intrinsic connection gives a geometric interpretation to optimal transportation map with L^2 transportation cost. We also peruse the semi−discrete transportation problem from geometric point of view, which is useful in practice. The result that obtained is proposing a novel generative framework that is combination of generator and discriminator and decouples the two encoding/decoding and probability measure transformation processes
  9. Keywords:
  10. Generative Adversarial Networks ; Optimal Transport ; Extended Kantorovich Method ; Convex Geometry ; Alexandrov Theorem ; Kantorovich Potential

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