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Mixing Time – General Chains

Mixing Time – General Chains. Seminar on Random Walks on Graphs 2009/2010 Ilan Ben Bassat Omri Weinstein. Mixing Time - Lazy Chains. Given that: Then:. Proof Sketch. Bound distance to stationary distribution by some function h: Bound the value of h t (x) inductively.

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Mixing Time – General Chains

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  1. Mixing Time – General Chains Seminar on Random Walks on Graphs 2009/2010 Ilan Ben Bassat Omri Weinstein

  2. Mixing Time - Lazy Chains Given that: Then:

  3. Proof Sketch • Bound distance to stationary distribution by some function h: • Bound the value of ht(x) inductively. • Derive a bound on the mixing time.

  4. Proof Sketch • Bound distance to stationary distribution by some function h: • Bound the value of ht(x) inductively. • Derive a bound on the mixing time.

  5. Bounding Distributions Distance For every

  6. So, choosing for every S yields: And for every vertex w we can get:

  7. H(x) Values Order the graph vertices: And define . Find k such that: Then the value of ht(x) is obtained by: And we get:

  8. Function H - Analysis • Piece-wise linear function. • Concave • Breakpoints at • On the interval [0,1]: and

  9. Proof Sketch • Bound distance to stationary distribution by some function h: • Bound the value of ht(x) inductively. • Derive a bound on the mixing time.

  10. Proof Sketch and Intuition We will prove for every x and t: Intuition: Bound the value h(x) in time t by going one step backwards, and a bit towards the endpoints. Prove will be in three parts: • For a finite group of discrete x values . • For all x values satisfying (and symmetric case). • For all x values and

  11. Proof – Breakpoints Fix k and let

  12. Proof – cont’d

  13. Proof – second case

  14. Proof – Third Part So, for every x we get:

  15. Base Case

  16. Induction For t=0 trivial. For and

  17. Induction

  18. Proof Sketch • Bound distance to stationary distribution by some function h: • Bound the value of ht(x) inductively. • Derive a bound on the mixing time.

  19. Mixing Time Proof Given

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