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Lecture11 - Lecture 11 A Local Cheeger Inequality Using...

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Lecture 11: A Local Cheeger Inequality Using Lazy Walks Transcriber: Franklin Kenter The goal of this lecture is to prove an inequality involving h S , the Cheeger constant using the Dirchelet eigenvalues. Goal Theorem: h S λ S α 2 4 - log | S | t Definition of α : α = min u S { h f : u Z k , k t, f : V [0 , ) , k f k 1 = 1 } That is, we can bound the Cheeger constant for a given subset, h S without actually necessarily computing the Dirchelet eigenvalues. But instead, we can find a good enough cut using the lazy walk to bound λ S , and hence, h S . Secton 1: The Extensions of f In order to prove the theorem, we will, first, provide from extensions of f . Below we will use a subscript for each of the extensions. However, afterwards, since all of the extensions have different domains, we plan to abuse notation and remove the subscripts. First Extension: f on the edges of G f E : V × V [0 , ) defined as f ( u, v ) = f ( u ) d u 1 u v Note that f E is not symmetric. Also, note f E is defined on non-edges to be 0.
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