The L p convergence of finite Markov chains Guan-Yu Chen Department - - PowerPoint PPT Presentation

The Lp convergence of finite Markov chains

Guan-Yu Chen

Department of Applied Mathematics, NCTU

August 10, 2010

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 1 / 25

Outline

1

Ergodic Markovian semigroups

2

Lp mixing

3

The Lp cutoff

4

Recent works & references

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 2 / 25

Ergodic Markovian semigroups

Basic setting S: a set equipped with a σ-field B. T: the time, either {0, 1, 2, ...} or [0, ∞). p(t, x, ·): a family of probabilities (or transition functions) for all x ∈ S and t ∈ T. Pt: an operator defined by Ptf (x) =

• S

f (y)p(t, x, dy) for any bounded B-measurable function f . µPt: a probability on S given by µPt(A) =

• S

p(t, x, A)µ(dx).

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 3 / 25

Ergodic Markovian semigroups

Markov transition functions & Invariant measures p(t, x, ·) (or Pt) is called Markovian if for any A ∈ B and x ∈ S, p(s + t, x, A) =

• S

p(t, y, A)p(s, x, dy), p(0, x, {x}) = 1. A Markov process (Xt)t∈T with filtration Ft = σ(Xs : s ≤ t) ⊂ B has p(t, x, ·) as transition function provided E(f ◦ Xt|Fs) =

• S

f (y)p(t − s, Xs, dy) = Pt−sf (Xs) for all 0 < s < t and bounded measurable function f .

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 4 / 25

Ergodic Markovian semigroups

A probability π is called invariant if πPt = π for all t ∈ T. If µ ≪ π, then µPt ≪ π for all t > 0 and d(µPt)/dπ = P∗

t (dµ/dπ),

where P∗

t is the adjoint of Pt in L2(π).

Suppose |S| < ∞. In either of the following cases, (i) T = [0, ∞), Pt = e−t(I−K) where K is an irreducible stochastic matrix; (ii) T = {0, 1, 2, ...}, p(1, ·, ·) is irreducible and aperiodic; there is a unique invariant probability, say π, and lim

t→∞ p(t, x, y) = π(y)

∀x, y.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 5 / 25

Lp mixing

Lp distance & mixing time Let p(t, x, ·) be a Markov transition function with invariant probability π. For p ∈ [1, ∞], the Lp distance is defined by Dp(µ, t) = d(µPt)/dπ − 1p and the sup-Lp distance is defined by Dp(t) := sup

µ Dp(µ, t) = sup x Dp(δx, t).

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 6 / 25

Lp mixing

For t, s ∈ T such that t + s ∈ T, Dp(µ, t + s) ≤ Dp(µ, s)[Dp(t) ∧ 1]. This implies

(a) t → Dp(µ, t) is non-increasing. (b) t → Dp(t) is non-increasing and submultiplicative.

The Lp mixing time is defined by Tp(µ, ǫ) = inf{t ∈ T|Dp(µ, t) ≤ ǫ} and Tp(ǫ) = inf{t ∈ T|Dp(t) ≤ ǫ}.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 7 / 25

Lp mixing

Transitive group action Assume that S is a compact topological space and G is a compact group acting continuously and transitively on S. Then, S = G/Go where Go is the stabilizer of any fixed point o ∈ S. If p(t, x, ·) is invariant under the action of G, i.e., p(t, gx, gA) = p(t, x, A) ∀g ∈ G, x ∈ S, A ⊂ S, then Dp(δx, t) = Dp(δy, t) for all x, y ∈ G. π(A) = u(φ−1(A)), where φ : G → G/Go is the canonical projection map and u is the normalized Haar measure on G.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 8 / 25

Lp mixing

Random walks on finite groups Let Q be a probability on a finite group G. A random walk on G driven by Q is a Markov chain whose transition function p(t, x, y) satisfies:

(i) If T = {0, 1, ...}, then p(t, x, y) = Q∗t(x−1y) (ii) If T = [0, ∞), then p(t, x, y) = ∞

i=0 e−t ti i! Q∗i(x−1y)

where Q∗i is the convolution of Q for i times. The uniform distribution on G is invariant. Dp(t) = Dp(δx, t) for all x ∈ G. Remark: Dp(δx, t) = Dp(δy, t) for all x, y does not necessarily imply Dp(µ, t) = Dp(ν, t) for any probabilities µ, ν in general.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 9 / 25

Lp mixing

Spectral gap The spectral gap (of a Markov transition function) λ is defined to be the largest constant c such that Ptf − π(f )2 ≤ e−ctf 2 ∀f . If T = [0, ∞) and Pt is strongly continuous on L2(π), then λ = inf{−Af , f |f ∈ Dom(A), real-valued, π(f ) = 0, f 2 = 1} where A is the infinitesimal generator of Pt.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 10 / 25

Lp mixing

If T = {0, 1, 2, ...}, then e−λ is the second largest singular value of P1 on L2(π). That is, λ = − log(P1 − EπL2(π)→L2(π)). If S is finite, then for p ∈ [1, ∞], 1 ω ≤ Tp(1/e) ≤ 1 2λ(1 + log π−1

∗ )

where e−ωt is the spectral radius of Pt and π∗ = minx π(x).

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 11 / 25

Lp mixing

Logarithmic Sobolev inequality & Poincar´ e inequality Gross (1975) introduced the concept of logarithmic Sobolev inequality

• n continuous spaces.

Diaconis & Saloff-Coste (1996) introduced a discrete version as

• follows. Write Pt = e−t(I−K) and let π be the invariant probability.

Define the Dirichlet form associated with K by E(f , g) = ℜf , g and set L(f ) = π(|f |2 log |f |2) − π(|f |2) log π(|f |2). The logarithmic Sobolev inequality is what follows. cL(f ) ≤ E(f , f ) ∀f .

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 12 / 25

Lp mixing

The logarithmic Sobolev constant denoted by α is the largest constant such that the above inequality holds. That is, α = inf E(f , f ) L(f )

• L(f ) = 0
• .

The Poincar´ e inequality is f − π(f )2 ≤ CE(f , f ) ∀f . The spectral gap λ is the inverse of the smallest C such that the Poincar´ e inequality holds. In general, 2α ≤ λ ≤ ω.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 13 / 25

Lp mixing

If K is reversible, i.e. Pt is self-adjoint in L2(π), then For 2 ≤ p ≤ ∞, 1 2α ≤ Tp(1/e) ≤ 1 2α(3 + log+ log π−1

∗ ).

For 1 < p < 2, 1 2mpα ≤ Tp(1/e) ≤ 1 4α(4 + log+ log π−1

∗ ),

where log+ t = 0 ∨ log t and mp = 1 + ⌈(2 − p)/(2p − 2)⌉. Remark: For non-reversible cases, the upper bound is similar. (multiply the right side with 2) but, however, the lower is unknown.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 14 / 25

The Lp cutoff

Lp cutoff Consider a family of ergodic Markov transition functions F = {pn(t, ·, ·)|n = 1, 2, ...}. For p ∈ [1, ∞], F is said to present a Lp cutoff if there is a sequence tn such that lim

n→∞ Dn,p(ctn) =

• ∞

if 0 < c < 1 if c > 1 . tn is called the Lp cutoff sequence (or cutoff time). Equivalently, F presents a Lp cutoff if and only if lim

n→∞

Tn,p(ǫ) Tn,p(δ) = 1 ∀0 < ǫ < δ < ∞. In particular, if tn is a Lp cutoff sequence, then tn ∼ Tn,p(ǫ) for all ǫ > 0.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 15 / 25

The Lp cutoff

Theorem Fix p ∈ (1, ∞). Consider a family F of Markov transition functions pn(t, ·, ·) with t ∈ T = [0, ∞) and invariant probability πn. Let λn be the spectral gap and Tn,p(ǫ) be the Lp mixing time of the nth transition function. (i) If there is ǫ > 0 such that lim

n→∞ λnTn,p(ǫ) = ∞,

then F has a Lp cutoff. (ii) If pn(t, ·, ·) is normal (that is, Pn,tP∗

n,t = P∗ n,tPn,t in L2(πn)) and F

has a Lp cutoff, then lim

n→∞ λnTn,p(ǫ) = ∞

∀ǫ > 0. Remark: Similar result holds true for the discrete time case.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 16 / 25

The Lp cutoff

Random transposition Random transposition is a random walk on the symmetric group Sn of degree n driven by Q, a probability satisfying Q(id) = 1/n and Q(i, j) = 2/n2. Diaconis & Shahshahani (1981) showed the existence of Lp cutoff for 1 ≤ p ≤ 2 at time n log n. Representation theory yields all eigenvalues so that λn ∼ 2/n. Random transpositions present the Lp cutoff with 2 < p < ∞. (In fact, L∞ cutoff exists.) But, however, the cutoff sequence is unknown.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 17 / 25

The Lp cutoff

Regular expander graphs A lazy simple random walk on a graph is a Markov chain whose transit evolves in the way that

(i) Remain at current state with probability 1/2. (ii) Move to neighbors uniformly at random with probability 1/2.

Fix k > 0 and consider a family of k-regular graphs Gn = (Vn, En) satisfying for all n, min |{edges between A and Ac}| |A|

• A = ∅, |A| ≤ |Vn|/2
• ≥ ǫ > 0.

The simple random walk on Gn has Tn,p(ǫ) ≥ ck(ǫ) log |Vn|, inf

n λn > 0.

If |Vn| → ∞, then the family has a Lp cutoff for 1 < p ≤ ∞. But, the L1 cutoff is open.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 18 / 25

The Lp cutoff

Example Sn = {0, 1}n, Kn(x, y) = 1/2 if yi+1 = xi for 1 ≤ i < n, where y = (y1, y2, ..., yn) and x = (x1, x2, ..., xn). If T = {0, 1, 2, ...}, then λn = 0 and Tn,p(ǫ) ∼ n for all ǫ > 0. If T = [0, ∞), then λ ≤ 1/n and, for all ǫ ∈ (0, 1), Tn,1(ǫ) ∼ n, and for p ∈ (1, ∞] and ǫ > 0, Tn,p(ǫ) ∼ (1−1/p)(log 2)n

1−21/p−1

if p ∈ (1, ∞) (2 log 2)n if p = ∞ . There is a Lp cutoff for p ∈ [1, ∞], but supn λnTn,p(ǫ) < ∞ for all ǫ.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 19 / 25

The Lp cutoff

Comparison of mixing times For any Markov transition function p(t, ·, ·) with invariant probability π, the adjoint operator of Pt in L2(π) is given formally by P∗

t f (x) =

• S

f (y)p∗(t, x, dy), p∗(t, x, A) =

• A p(t, z, dx)π(dz)

π(dx) . For 1 ≤ p ≤ q ≤ ∞, Tp(ǫ) ≤ Tq(ǫ). For 1 ≤ p ≤ ∞ and 1/p + 1/p′ = 1, T∞(ǫ2) ≤ Tp(ǫ) + T ∗

p′(ǫ).

For 1 < p < q ≤ ∞ and 1/p + 1/p′ = 1/q + 1/q′ = 1, Tq(ǫmp,q) ≤ mp,q max{Tp(ǫ), T ∗

p (ǫ)}

where mp,q = ⌈p′/q′⌉.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 20 / 25

The Lp cutoff

Theorem Let T = [0, ∞) and F be a family of Markov transition functions pn(t, ·, ·) with invariant probability πn, spectral gap λn and Lp mixing time Tn,p(ǫ). If pn(t, ·, ·) is reversible (that is, Pn,t is self-adjoint on L2(πn)), then the following areequivalent. (i) F has a Lp cutoff for some p ∈ (1, ∞]. (ii) F has a Lp cutoff for all p ∈ (1, ∞]. (iii) λnTn,p(ǫ) → ∞ for some p ∈ (1, ∞] and ǫ > 0. (iv) λnTn,p(ǫ) → ∞ for all p ∈ (1, ∞] and ǫ > 0.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 21 / 25

The Lp cutoff

Brownian motions Let d be a positive integer. For n ≥ 1, (Mn, gn) is a compact Riemannian manifolds of dimension d without boundary and πn is a normalized Riemannian measure πn. The heat semigroup Pn,t on Mn is the Markov semigroup with the infinitesimal generator the Laplace-Beltrami operator on (Mn, gn). This corresponds to Brownian motion and has πn as invariant

• measure. In fact, Pn,t is self-adjoint in L2(πn).

λn is the spectral gap and Tn,p(ǫ) is the Lp mixing time.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 22 / 25

The Lp cutoff

If Mn has non-negative Ricci curvature, then there exist universal constants c(d), C(d) and a(d), A(d) such that c(d)δ−2

n

≤ λn ≤ C(d)δ−2

n ,

a(d)δ−2

n

≤ Tn,p(1) ≤ A(d)δ−2

n

where δn is the diameter of Mn. This implies that the family has no Lp cutoff for 1 < p ≤ ∞. In fact, no L1 cutoff exists, either.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 23 / 25

Recent works & references

Recent results in Lp cutoffs Consider the case T = [0, ∞) and |S| < ∞. Let F be a family of reversible Markov transition functions with spectral gaps λn and logarithmic Sobolev constants αn. If λn/αn → ∞, then F presents a Lp cutoff for 1 < p ≤ ∞. If λn/αn is bounded and F has an Lp cutoff for some p ∈ (1, ∞], then for all q ∈ (1, ∞) and ǫ > 0, Tn,q(ǫ) ∼ Tn,2(ǫ) and Tn,∞(ǫ) ∼ 2Tn,2(ǫ). Chen & Saloff-Coste (2010) give a criterion on the L2 cutoff (with specified initial distributions) and display a formula on the L2 cutoff sequence using spectral decomposition.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 24 / 25

Recent works & references

Reference

Chen, G.-Y. and Saloff-Coste, L. The cutoff phenomenon for ergodic Markov processes. Electronic Journal of Probability, 13, p. 26–78, 2008. Chen, G.-Y. and Saloff-Coste, L. The L2-cutoff for reversible Markov

• processes. J. Funct. Anal. 258, p. 2246-2315, 2010.

Diaconis, P. and Saloff-Coste, L. Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab., 6, p. 695–750, 1996. Gross, L. Logarithmic Sobolev inequalities. Amer. J. Math., 97, p. 346–374, 1976. Saloff-Coste, L. Lectures on finite Markov chains. Lectures on probability theory and statistics (Saint-Flour Summer School, 1996), 301–413, Lecture Notes in Math., 1665, Springer, Berlin, 1997.

G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 25 / 25