Sandwich Theorem Lecture 1 3 Sandwich Theorem Modular

Sandwich Theorem Lecture 1 -3

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