 # Sandwich Theorem Lecture 2 2 Sandwich Theorem Modular

• Slides: 28 Sandwich Theorem Lecture 2 -2 Sandwich Theorem Modular function  Sandwich Theorem gives a surprising property of the set function. Proof • DS decomposition • Sandwich theorem for submodular function Modular upper bound for submodular function Modular lower bound for submodular function • Proof of Sandwich theorem Fundamental Theorem Modular Upper Bound for submodular function   Modular Lower Bound for submodular function Proof of Sandwich Theorem Thank You, end Submodular-Supermodular Procedure Lecture 1 -4 Minimize DS function Difference of Submodular functions! Rishabh Iyer and Jeff Bilmes: Algorithm for approximate Minimization of the difference between submodular functions, with application, ar. Xiv: 1207. 0560 v 4 24 August 2013 Submodular-Supermodular (Sub. Sup) Procedure ? ? ? ? Subdifferential Sub-differential Extreme Point Minimum unconstrained submodular function can be computed in polynomial-time Theorem Proof What is the local minimum? • Two ways to understand: • Subdifferential: what is the subdifferential of a DS function? • Operation-based: what operations? Variations Supermodular-Submodular (Sup. Sub) Procedure ? Modular-Modular (Mod. Mod) Procedure Thank You, end Quality of Bounds Difference of Submodular functions! DS maximization! Discrete DC maximization! Difference of Convex functions! Bound of Any Submodular Function