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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−λ2 2

Cycle

Tight example for Other side of Cheeger?

µ 2 = 1−λ2 2

≤ h(G)

Cycle

Tight example for Other side of Cheeger?

µ 2 = 1−λ2 2

≤ h(G) ≤

• 2(1−λ2)

Cycle

Tight example for Other side of Cheeger?

µ 2 = 1−λ2 2

≤ h(G) ≤

• 2(1−λ2) =
• 2µ

Cycle

Tight example for Other side of Cheeger?

µ 2 = 1−λ2 2

≤ h(G) ≤

• 2(1−λ2) =
• 2µ

Will show other side of Cheeger is tight. Cycle on n nodes.

Cycle

Tight example for Other side of Cheeger?

µ 2 = 1−λ2 2

≤ h(G) ≤

• 2(1−λ2) =
• 2µ

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−λ2 2

≤ h(G) ≤

• 2(1−λ2) =
• 2µ

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−λ2 2

≤ h(G) ≤

• 2(1−λ2) =
• 2µ

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−λ2 2

≤ h(G) ≤

• 2(1−λ2) =
• 2µ

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

n2 .

Cycle

Tight example for Other side of Cheeger?

µ 2 = 1−λ2 2

≤ h(G) ≤

• 2(1−λ2) =
• 2µ

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

n2 .

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 Hit with M.

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ)

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ) µ 2

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ) µ 2 = 1−λ2 2

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ) µ 2 = 1−λ2 2

≤ h(G)

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ) µ 2 = 1−λ2 2

≤ h(G) ≤

• 2(1−λ2)

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ) µ 2 = 1−λ2 2

≤ h(G) ≤

• 2(1−λ2) =
• 2µ

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ) µ 2 = 1−λ2 2

≤ h(G) ≤

• 2(1−λ2) =
• 2µ

Tight example for upper bound for Cheeger.

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 ··· ···

x1 ≈ − n

4

xn ≈ − n

4

xn/2 ≈ n

4

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 ··· ···

x1 ≈ − n

4

xn ≈ − n

4

xn/2 ≈ n

4

Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 ··· ···

x1 ≈ − n

4

xn ≈ − n

4

xn/2 ≈ n

4

Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 ··· ···

x1 ≈ − n

4

xn ≈ − n

4

xn/2 ≈ n

4

Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 ··· ···

x1 ≈ − n

4

xn ≈ − n

4

xn/2 ≈ n

4

Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 ··· ···

x1 ≈ − n

4

xn ≈ − n

4

xn/2 ≈ n

4

Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ)

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 ··· ···

x1 ≈ − n

4

xn ≈ − n

4

xn/2 ≈ n

4

Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ) µ 2

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 ··· ···

x1 ≈ − n

4

xn ≈ − n

4

xn/2 ≈ n

4

Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ) µ 2 = 1−λ2 2

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 ··· ···

x1 ≈ − n

4

xn ≈ − n

4

xn/2 ≈ n

4

Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ) µ 2 = 1−λ2 2

≤ h(G)

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 ··· ···

x1 ≈ − n

4

xn ≈ − n

4

xn/2 ≈ n

4

Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ) µ 2 = 1−λ2 2

≤ h(G) ≤

• 2(1−λ2)

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 ··· ···

x1 ≈ − n

4

xn ≈ − n

4

xn/2 ≈ n

4

Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ) µ 2 = 1−λ2 2

≤ h(G) ≤

• 2(1−λ2) =
• 2µ

Find x ⊥ 1 with Rayleigh quotient, xT Mx

xT x close to 1.

xi =

• i −n/4

if i ≤ n/2 3n/4−i if i > n/2 ··· ···

x1 ≈ − n

4

xn ≈ − n

4

xn/2 ≈ n

4

Hit with M. (Mx)i =      −n/4+1/2 if i = 1,n n/4−1 if i = n/2 xi

• therwise

→ xT Mx = xT x(1−O( 1

n2 ))

→ λ2 ≥ 1−O( 1

n2 )

µ = λ1 −λ2 = O( 1

n2 )

h(G) = 2

n = Θ(√µ) µ 2 = 1−λ2 2

≤ h(G) ≤

• 2(1−λ2) =
• 2µ

Tight example for upper bound for Cheeger.

Eigenvalues of cycle?

Eigenvalues: cos 2πk

n .

Eigenvalues of cycle?

Eigenvalues: cos 2πk

n .

xi = cos 2πki

n

Eigenvalues of cycle?

Eigenvalues: cos 2πk

n .

xi = cos 2πki

n

(Mx)i

Eigenvalues of cycle?

Eigenvalues: cos 2πk

n .

xi = cos 2πki

n

(Mx)i = cos

• 2πk(i+1)

n

• +cos
• 2πk(i−1)

n

Eigenvalues of cycle?

Eigenvalues: cos 2πk

n .

xi = cos 2πki

n

(Mx)i = cos

• 2πk(i+1)

n

• +cos
• 2πk(i−1)

n

• = 2cos
• 2πk

n

• cos
• 2πki

n

Eigenvalues of cycle?

Eigenvalues: cos 2πk

n .

xi = cos 2πki

n

(Mx)i = cos

• 2πk(i+1)

n

• +cos
• 2πk(i−1)

n

• = 2cos
• 2πk

n

• cos
• 2πki

n

• Eigenvalue: cos 2πk

n .

Eigenvalues of cycle?

Eigenvalues: cos 2πk

n .

xi = cos 2πki

n

(Mx)i = cos

• 2πk(i+1)

n

• +cos
• 2πk(i−1)

n

• = 2cos
• 2πk

n

• cos
• 2πki

n

• Eigenvalue: cos 2πk

n .

Eigenvalues:

Eigenvalues of cycle?

Eigenvalues: cos 2πk

n .

xi = cos 2πki

n

(Mx)i = cos

• 2πk(i+1)

n

• +cos
• 2πk(i−1)

n

• = 2cos
• 2πk

n

• cos
• 2πki

n

• Eigenvalue: cos 2πk

n .

Eigenvalues: vibration modes of system.

Eigenvalues of cycle?

Eigenvalues: cos 2πk

n .

xi = cos 2πki

n

(Mx)i = cos

• 2πk(i+1)

n

• +cos
• 2πk(i−1)

n

• = 2cos
• 2πk

n

• cos
• 2πki

n

• Eigenvalue: cos 2πk

n .

Eigenvalues: vibration modes of system. Fourier basis.

Eigenvalues of cycle?

Eigenvalues: cos 2πk

n .

xi = cos 2πki

n

(Mx)i = cos

• 2πk(i+1)

n

• +cos
• 2πk(i−1)

n

• = 2cos
• 2πk

n

• cos
• 2πki

n

• Eigenvalue: cos 2πk

n .

Eigenvalues: vibration modes of system. Fourier basis.

Sum up.

Sampling by random walks.

Sum up.

Sampling by random walks. Random Walks mix if µ is “large”.

Sum up.

Sampling by random walks. Random Walks mix if µ is “large”. If expanding µ is large. “Cheeger.

Sum up.

Sampling by random walks. Random Walks mix if µ is “large”. If expanding µ is large. “Cheeger. Example: partial orders. More songs about People and Food

Sum up.

Sampling by random walks. Random Walks mix if µ is “large”. If expanding µ is large. “Cheeger. Example: partial orders. More songs about People and Food ...or Cheeger.

See you on Tuesday.

See you on Tuesday. Not!