Markov Chains on Finite Groups Maarit Hietalahti Postgraduate - - PowerPoint PPT Presentation

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Markov Chains on Finite Groups

Maarit Hietalahti Postgraduate Seminar in Theoretical Computer Science 17.11.2003

Based on Sections 15 and 16 of E. Behrends. Introduction to Markov Chains, with Special Emphasis on Rapid Mixing. Vieweg & Sohn, Braunschweig Wiesbaden, 2000. and

• P. Diaconis. Group Representations in Probability and Statistics.

Institute of Mathematical Statistics, Hayward CA, 1988.

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Contents

• 1. Preliminaries: Algebraic terms
• 2. Markov chains on groups: definition
• 3. Goal and the path
• 4. k-step transitions
• 5. Convolutions
• 6. Characters
• 7. Lemma 15.3
• 8. Fourier transforms
• 9. Variation distance
• 10. Conclusion: Rapid mixing in Markov chains on finite commutative groups
• 11. Remark on the non-commutative case

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Refresher on Algebra

group (G, ◦): G set, ◦ associative multiplication between elements of G: if g, h ∈ G then g ◦ h ∈ G. Identity: g ◦ id = id ◦ g = g. Inverse g−1g = id. subgroup H ∈ G: id ∈ H and h1 ◦ h2 ∈ H when h1, h2 ∈ H. (H is closed with respect to ◦) group generator g ∈ G is said to generate the group G, if for all elements of h ∈ G there is a k s.t. h = gk. conjugacy class H subgroup, left (right) conjugacy classes are sets of the form H ◦ g (g ◦ H) with g ∈ G. group homomorphism is a map between two groups G, H such that 1) f(g1g2) = f(g1)f(g2) and 2) f(idG) = idH.

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Markov chains on finite commutative groups

(G, ◦) is a finite group, g, h, . . . ∈ G are the states of a Markov chain. P0 is a probability measure on G. Transition probabilities: pg,h◦g := P0({h}) lemma 15.1

• pg,h◦g are the entries of a (doubly) stochastic matrix. Thus, the uniform

distribution of this matrix is the equilibrium distribution.

• H subgroup generated by supp := {h|P0(h) > 0}. The irreducible subsets of

the chain are precisely the sets of the form H ◦ g with g ∈ G, that is, the left conjugacy classes. In particular, the chain is irreducible iff suppP0 generates G.

• The chain is aperiodic and irreducible iff there is a k s. t. every element of G

can be written as the product of k elements, each lying in suppP0.

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Outline: the train of thought

Problem: How fast does the chain converge to its equilibrium? –> What is the distribution after k steps of a walk which starts at 0? Answer: P(k∗) . Notion Matrix doubly stochastic: the uniform distribution is the equilibrium distribution! –> How fast does the P(k∗) tend to the uniform distribution? –> How close is a distribution P0 to the uniform distribution? Notion Variation distance can be calculated with the help of the Fourier transformation –> How small are the ˆ P0(χ) for the nontrivial characters χ?

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k-step transitions

Probability P0 on G for the one-step transitions.

• Start: g0 arbitrary.
• g0 + h0 with probability P0({h0}) for h0.
• (g0 + h0) + h1 with probability P0({h1}) for h1.
• and so on.

Note that h0 and h1 are independent. 2-step transitions: g0 → (g0 + h0) + h1 = g0 + h for which the probability is Σh0+h1=hP0({h0})P0({h1}) = Σh0P0({h0})P2({h − h0})).

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Convolutions of probability measures

Definition 15.9 Let P1, P2 be probability measures on G. (i) We define the convolution P1 ∗ P2 of P1, P2 by (P1 ∗ P2)({h}) := Σh0P1({h0})P2({h − h0}) (ii) In the special case P1 = P2 = P0 we put P(k∗) := P0 ∗ P0. This is extended to a definition for arbitrary integer exponents P((k+1)∗) := P(k∗) ∗ P0.

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Characters

Relating abstract groups to complex numbers: Definition 15.2 Denote by (Γ, ·) the multiplicative group of all complex numbers

• f modulus one. Then a character on G is a group homomorphism χ from G to Γ:

χ(g + h) = χ(g)χ(h) for all g, h ∈ G. Properties of characters:

• (χ(g) = χ(g)) is a character. (Also, χ is the inverse 1/χ of χ.)
• χ1χ2 is a character when χ1, χ2 are.
• The trivial character: χtriv : g → 1.
• ˆ

G, the collection of all characters, forms a commutative group with resp. to pointwise multiplication.

• If G has N elements, the range of any character on G is contained in the set of

the N’th roots of unity (exp(2πij/N), j = 0, . . . , N − 1, i = √−1)

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Lemma 15.3 and corollary

Let (G, +) be a commutative group with N elements. The N-dimensional vector space of all mappings from G to C will be denoted by XG, and this space will be provided with the scalar product < f1, f2 >G:= Σgf1(g)f2(g)/N. (i) Let χ be a character which is not the trivial character χtriv. Then Σgχ(g) = 0. (ii) In the Hilbert space (Xg, < ·, · >G) the family of characters forms an

• rthonormal system.

(iii) Any collection of characters is linearly independent (iv) ˆ G has at most N elements. (v) In fact there exists N different characters so that ˆ G is an orthonormal basis of

• XG. Also (G, +) is isomorphic with ( ˆ

G, ·).

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✬ ✫ ✩ ✪ Corollary 15.4 (i) Let f be any element of XG. Then f can be written as a linear combination of the χ ∈ ˆ G as follows: f = Σχ < f, χ >G χ. (ii) For different g, h ∈ G there is a character χ s.t. χ(g) = χ(h).

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Fourier transform

Fourier transform of measure P0: ˆ P0 : ˆ G → C, χ → Σgχ(g)P0({g}) Fourier transform of convolutions: For probability measures P1, P2 on (G, +) the Fourier transform of P2 ∗ P1 is just the (pointwise) product of the functions ˆ P1 and ˆ

• P2. In particular it follows that, for

any probability P0, the Fourier transform of P(k∗) is the k’th power of the Fourier transform of P0.

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Calculating the variation distance

Lemma 15.8 Let P0, P1, P2 be probability measures on the finite commutative group G. By U we denote the uniform distribution. (i) P0 = U iff ˆ P0(χ) is one for the trivial character and zero for the other χ. (ii) The variation distance P1 − P2 can be estimated by (Σχ| ˆ P1(χ) − ˆ P2(χ)|2)1/2/2; in particular P1 − U is less than or equal to (Σχ=χtriv| ˆ P1(χ)|2)1/2/2, where the summation runs over all nontrivial characters χ. (iii) Conversely, the distance of ˆ P1 and ˆ P2 with respect to the maximum norm is bounded by 2P1 − P2.

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Rapid mixing: Conclusion

Combining previous results gives us P(k∗) − U2 ≤ 1 4Σχ=χtriv| ˆ P0(χ)|2k

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Remark: Generalization to arbitrary finite groups

Relating the abstract group to something more concrete is done by using

• representations. Characters will no longer do, as they are homomorphisms with

commutative ranges, which cannot distinguish between different elements of a non-commutative groups. The use of representations leads to more demanding technicalities. In other respects, the construction follows the same principles.

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