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DS Decomposition

DS Decomposition. Lecture 2-1. Fundamental Theorem. Mukund Narasimhan , Jeff A. Bilmes : A Submodular- supermodular Procedure with Applications to Discriminative Structure Learning. UAI 2005 : 404-412

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DS Decomposition

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  1. DS Decomposition Lecture 2-1

  2. Fundamental Theorem MukundNarasimhan, Jeff A. Bilmes: A Submodular-supermodular Procedure with Applications to Discriminative Structure Learning. UAI 2005: 404-412 RishabhIyer and Jeff Bilmes: Algorithm for approximate Minimization of the difference between submodular functions, with application, arXiv:1207.0560v4 24 August 2013

  3. Fundamental Theorem

  4. Weak Version Proof

  5. Any Submodular Function Proof

  6. Fundamental Theorem Proof

  7. Open Problem • Given a set function, there are many DS decompositions for it. • Given a set function, can we find a DS function for it efficiently?

  8. Theorem Proof

  9. Proof

  10. Thank You, end

  11. Examples Lecture 1-2

  12. 1st Example Viral Marketing of Games

  13. Problem

  14. Examples

  15. Examples

  16. Upper Bound Lemma 1

  17. Lemma 2

  18. Wang, Zhefeng, et al. "Activity Maximization by Effective Information Diffusion in Social Networks." IEEE Transactions on Knowledge and Data Engineering 29.11 (2017): 2374-2387.

  19. 2nd Example Find Effector

  20. “Rumor” Source • Finding sources of activation is an important issue. • Given social network and diffusion model as well as all active nodes, find k effectors which best fit for the role of sources..

  21. Problem Formulation Theorem

  22. DS Decomposition

  23. References

  24. 3rd Example Composed Influence

  25. Problem • Two or more active persons together may give stronger influence than individual. • Composed influence can be formulated into hyper-edge. • With composed influence, the influence spread is neither submodular nor supermodular.

  26. Submodular Upper Bound • Create “super nodes” representing the start node of super-edges. • Add edges from nodes contained in super-node to the super nodes. • A super node is active if it contains at least one active node.

  27. Difference • What is the difference between upper bound and influence spread? • At least one super node is activated by an active node together with an inactive node.

  28. DS Decomposition of the difference • By the inclusive-exclusive formula, the difference between the upper bound and the influence spread can be expressed as MC IC

  29. Mutually-exclusive Cascade (MC)

  30. Mutually-exclusive Cascade

  31. Independent Cascade (IC)

  32. Independent Cascade

  33. Kempe-Kleinberg-Tardos Conjecture This conjecture is proved by Mossel and Roch in 2007 (STOC’07)

  34. The 4th Example Active Friending

  35. LinkedIn • Do you receive invitations from LinkedIn everyday? • Does LinkedIn have the following function: LinkedIn may suggest a list of invitations when you want to include a target person into friend list.

  36. Problem Formulation Theorem

  37. References

  38. Thank You, end

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