Welcome back. Today. Welcome back. Today. Continue Sampling - PowerPoint PPT Presentation

Welcome back. Today.

Welcome back. Today. Continue Sampling combinatorial structures.

Welcome back. Today. Continue Sampling combinatorial structures. Random Walks.

Welcome back. Today. Continue Sampling combinatorial structures. Random Walks. Spectral Gap/Mixing Time.

Welcome back. Today. Continue Sampling combinatorial structures. Random Walks. Spectral Gap/Mixing Time. Expansion/Spectral Gap: Cheeger

Welcome back. Today. Continue Sampling combinatorial structures. Random Walks. Spectral Gap/Mixing Time. Expansion/Spectral Gap: Cheeger Example: partial orders.

Welcome back. Today. Continue Sampling combinatorial structures. Random Walks. Spectral Gap/Mixing Time. Expansion/Spectral Gap: Cheeger Example: partial orders. Cheeger and Tight Examples.

Welcome back. Today. Continue Sampling combinatorial structures. Random Walks. Spectral Gap/Mixing Time. Expansion/Spectral Gap: Cheeger Example: partial orders. Cheeger and Tight Examples.

Cycle Tight example for Other side of Cheeger?

Cycle Tight example for Other side of Cheeger? µ 2

Cycle Tight example for Other side of Cheeger? µ 2 = 1 − λ 2 2

Cycle Tight example for Other side of Cheeger? µ 2 = 1 − λ 2 ≤ h ( G ) 2

Cycle Tight example for Other side of Cheeger? µ 2 = 1 − λ 2 � ≤ h ( G ) ≤ 2 ( 1 − λ 2 ) 2

Cycle Tight example for Other side of Cheeger? µ 2 = 1 − λ 2 � � ≤ h ( G ) ≤ 2 ( 1 − λ 2 ) = 2 µ 2

Cycle Tight example for Other side of Cheeger? µ 2 = 1 − λ 2 � � ≤ h ( G ) ≤ 2 ( 1 − λ 2 ) = 2 µ 2 Will show other side of Cheeger is tight. Cycle on n nodes.

Cycle Tight example for Other side of Cheeger? µ 2 = 1 − λ 2 � � ≤ h ( G ) ≤ 2 ( 1 − λ 2 ) = 2 µ 2 Will show other side of Cheeger is tight. Cycle on n nodes. Edge expansion:Cut in half.

Cycle Tight example for Other side of Cheeger? µ 2 = 1 − λ 2 � � ≤ h ( G ) ≤ 2 ( 1 − λ 2 ) = 2 µ 2 Will show other side of Cheeger is tight. Cycle on n nodes. Edge expansion:Cut in half. | S | = n 2 , | E ( S , S ) | = 2

Cycle Tight example for Other side of Cheeger? µ 2 = 1 − λ 2 � � ≤ h ( G ) ≤ 2 ( 1 − λ 2 ) = 2 µ 2 Will show other side of Cheeger is tight. Cycle on n nodes. Edge expansion:Cut in half. | S | = n 2 , | E ( S , S ) | = 2 → h ( G ) = 4 n .

Cycle Tight example for Other side of Cheeger? µ 2 = 1 − λ 2 � � ≤ h ( G ) ≤ 2 ( 1 − λ 2 ) = 2 µ 2 Will show other side of Cheeger is tight. Cycle on n nodes. Edge expansion:Cut in half. | S | = n 2 , | E ( S , S ) | = 2 → h ( G ) = 4 n . Show eigenvalue gap µ ≤ 1 n 2 .

Cycle Tight example for Other side of Cheeger? µ 2 = 1 − λ 2 � � ≤ h ( G ) ≤ 2 ( 1 − λ 2 ) = 2 µ 2 Will show other side of Cheeger is tight. Cycle on n nodes. Edge expansion:Cut in half. | S | = n 2 , | E ( S , S ) | = 2 → h ( G ) = 4 n . Show eigenvalue gap µ ≤ 1 n 2 . Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1.

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1.

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2 Hit with M .

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise 

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 n 2 ))

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 → λ 2 ≥ 1 − O ( 1 n 2 )) n 2 )

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 → λ 2 ≥ 1 − O ( 1 n 2 )) n 2 ) µ = λ 1 − λ 2 = O ( 1 n 2 )

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 → λ 2 ≥ 1 − O ( 1 n 2 )) n 2 ) µ = λ 1 − λ 2 = O ( 1 n 2 ) n = Θ( √ µ ) h ( G ) = 2

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 → λ 2 ≥ 1 − O ( 1 n 2 )) n 2 ) µ = λ 1 − λ 2 = O ( 1 n 2 ) n = Θ( √ µ ) h ( G ) = 2 µ 2

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 → λ 2 ≥ 1 − O ( 1 n 2 )) n 2 ) µ = λ 1 − λ 2 = O ( 1 n 2 ) n = Θ( √ µ ) h ( G ) = 2 µ 2 = 1 − λ 2 2

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 → λ 2 ≥ 1 − O ( 1 n 2 )) n 2 ) µ = λ 1 − λ 2 = O ( 1 n 2 ) n = Θ( √ µ ) h ( G ) = 2 µ 2 = 1 − λ 2 ≤ h ( G ) 2

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 → λ 2 ≥ 1 − O ( 1 n 2 )) n 2 ) µ = λ 1 − λ 2 = O ( 1 n 2 ) n = Θ( √ µ ) h ( G ) = 2 µ 2 = 1 − λ 2 � ≤ h ( G ) ≤ 2 ( 1 − λ 2 ) 2

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 → λ 2 ≥ 1 − O ( 1 n 2 )) n 2 ) µ = λ 1 − λ 2 = O ( 1 n 2 ) n = Θ( √ µ ) h ( G ) = 2 µ 2 = 1 − λ 2 � � ≤ h ( G ) ≤ 2 ( 1 − λ 2 ) = 2 µ 2

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. � i − n / 4 if i ≤ n / 2 x i = 3 n / 4 − i if i > n / 2 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 → λ 2 ≥ 1 − O ( 1 n 2 )) n 2 ) µ = λ 1 − λ 2 = O ( 1 n 2 ) n = Θ( √ µ ) h ( G ) = 2 µ 2 = 1 − λ 2 � � ≤ h ( G ) ≤ 2 ( 1 − λ 2 ) = 2 µ 2 Tight example for upper bound for Cheeger.

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1.

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. x n / 2 ≈ n 4 � i − n / 4 if i ≤ n / 2 ··· ··· x i = 3 n / 4 − i if i > n / 2 x 1 ≈ − n x n ≈ − n 4 4

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. x n / 2 ≈ n 4 � i − n / 4 if i ≤ n / 2 ··· ··· x i = 3 n / 4 − i if i > n / 2 x 1 ≈ − n x n ≈ − n 4 4 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise 

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. x n / 2 ≈ n 4 � i − n / 4 if i ≤ n / 2 ··· ··· x i = 3 n / 4 − i if i > n / 2 x 1 ≈ − n x n ≈ − n 4 4 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 n 2 ))

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. x n / 2 ≈ n 4 � i − n / 4 if i ≤ n / 2 ··· ··· x i = 3 n / 4 − i if i > n / 2 x 1 ≈ − n x n ≈ − n 4 4 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 → λ 2 ≥ 1 − O ( 1 n 2 )) n 2 )

Find x ⊥ 1 with Rayleigh quotient, x T Mx x T x close to 1. x n / 2 ≈ n 4 � i − n / 4 if i ≤ n / 2 ··· ··· x i = 3 n / 4 − i if i > n / 2 x 1 ≈ − n x n ≈ − n 4 4 Hit with M .  − n / 4 + 1 / 2 if i = 1 , n   ( Mx ) i = n / 4 − 1 if i = n / 2  x i otherwise  → x T Mx = x T x ( 1 − O ( 1 → λ 2 ≥ 1 − O ( 1 n 2 )) n 2 ) µ = λ 1 − λ 2 = O ( 1 n 2 )