Bounding the mixing time via Spectral Gap

Ilan Ben-Bassat Omri Weinstein

Conductance For any 𝑆⊆ Ω denote 𝛷𝑆 = 𝜋(𝑆)−1 𝑄(𝑆,𝑆ҧ)

where 𝑄(𝑆,𝑆ҧ) = π(ω) P(ω,σ) Intuition: 𝛷𝑆 is the steady state probability of moving from 𝑆 to 𝑆ҧ, Conditioned on being in 𝑆.

𝛷= min𝛷𝑆 ∶ 𝑆⊆ Ω, 0 < 𝜋(𝑆) ≤ 1/2

If ℳ is lazy , 𝛷𝑆 ≤ 1/2. 𝛷𝑆𝜋ሺ𝑆ሻ= Qሺ𝑆,𝑆ҧሻ= Qሺ𝑆ҧ,𝑆ሻ= 𝛷𝑆ҧ𝜋ሺ𝑆ҧሻ

Qሺ𝑆,𝑆ҧሻ= QሺΩ,𝑆ҧሻ− Qሺ𝑆ҧ,𝑆ҧሻ= 𝜋ሺ𝑆ҧሻ− Qሺ𝑆ҧ,𝑆ҧሻ= Qሺ𝑆ҧ,𝑆ሻ

Lemma: If ℳ is lazy, irreducible and aperiodic, then all eigenvalues of 𝑃 are positive.

Reversible Chains

Properties

Denote for any 𝑦∈ℝ𝑁: 𝜀ሺ𝑦,𝑦ሻ= 𝜋𝑖𝑃𝑖,𝑗(𝑦𝑖 − 𝑦𝑗)2𝑖<𝑗

Lemma: If ℳ is reversible, then

1− 𝜆1 = 𝑚𝑖𝑛 𝜋𝑇𝑦=0 𝜀ሺ𝑦,𝑦ሻσ 𝜋𝑖𝑖 𝑦𝑖2 Proof. Recall ,𝑆= 𝐷1/2𝑃𝐷−1/2, 𝑒(0). Note that 𝜆1 = 𝑚𝑎𝑥<𝑒(0),𝑥>=0 <𝑆𝑥,𝑥><𝑥,𝑥>

Now consider the matrix 𝐷1/2(𝐼− 𝑃)𝐷−1/2. What are it’s eigenvalues?…

If ℳ is reversible then

1− 𝜆1 ≥ 𝛷22

(*)

( ) 1NS

Corollary 𝜏ሺ𝜀ሻ= 2|log (𝜀𝜋𝑚𝑖𝑛 )|𝛷2 ඈ

Proof. 1log (𝜆𝑚𝑎𝑥−1 ) ≤ 1log ((1−𝛷22 )−1) ≤ 2𝛷2

Consider a random walk on a Graph G=(V,E). By definition, we have

If G is a r-regular graph

Example: A walk on the discrete cube 𝑄𝑛. For 𝑆⊆ 𝑄𝑛 Let 𝑖𝑛(𝑆) denote the number of edges which are wholly contained in 𝑆. Lemma: if 𝑆 ≠ ∅, 𝑖𝑛(𝑆) ≤ 12 |𝑆|𝑙𝑜𝑔2|𝑆| Proof. By induction on n, using the inequality: 𝑥𝑙𝑜𝑔2𝑥 + 𝑦𝑙𝑜𝑔2𝑦 + 2y ≤ (𝑥+ 𝑦)𝑙𝑜𝑔2(x+ y) By summing the degrees at each vertex of 𝑆 we have 𝑒ሺ𝑆,𝑆ҧሻ+ 2𝑖𝑛ሺ𝑆ሻ= 𝑛|𝑆| So by the Lemma we get

𝑒ሺ𝑆,𝑆ҧሻ≥ 𝑛ȁA𝑆ȁA− 12ȁA𝑆ȁA𝑙𝑜𝑔2ȁA𝑆ȁA= ȁA𝑆ȁA൬𝑛 − 12𝑙𝑜𝑔2ȁA𝑆ȁA൰≥ |𝑆|

x>=y

It follows that Φ ≥ 1𝑛 . Adding self-loops will halve the conductance -

the denominator σ 𝑑𝑣𝑣∈𝑆 doubles without changing the numerator in 𝛷𝑠. By (*), this gives us an estimate of 1− 18𝑛2 for the spectral gap (so we’re off by a factor of n)

Upper Bound of the Spectral Gap

Path Congestion