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30
Optimal phylogenetic reconstruction
 In STOC ’06: Proceedings of the thirtyeighth annual ACM symposium on Theory of computing
, 2006
"... One of the major tasks of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. This problem is of critical importance in almost all areas of biology and has a very clear mathematical formulation. The evolutionary model is given by a Markov chain on the true evolution ..."
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Cited by 43 (9 self)
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One of the major tasks of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. This problem is of critical importance in almost all areas of biology and has a very clear mathematical formulation. The evolutionary model is given by a Markov chain on the true evolutionary tree. Given samples from this Markov chain at the leaves of the tree, the goal is to reconstruct the evolutionary tree. It is crucial to minimize the number of samples, i.e., the length of genetic sequences, as it is constrained by the underlying biology, the price of sequencing etc. It is well known that in order to reconstruct a tree on n leaves, sequences of length Ω(log n) are needed. It was conjectured by M. Steel that for the CFN evolutionary model, if the mutation probability on all edges of the tree is less than p ∗ = ( √ 2 −1)/2 3/2 than the tree can be recovered from sequences of length O(log n). This was proven by the second author in the special case where the tree is “balanced”. The second author also proved that if all edges have mutation probability larger than p ∗ then the length needed is n Ω(1). This “phasetransition ” in the number of samples needed is closely related to the phase transition for the reconstruction problem (or extremality of free measure) studied extensively in statistical physics and probability.
Combinatorial Criteria for Uniqueness of Gibbs Measures, Random Structures and Algorithms
, 2005
"... We generalize previously known conditions for uniqueness of the Gibbs measure in statistical physics models by presenting conditions of any finite size for models on any underlying graph. We give two dual conditions, one requiring that the total influence on a site is small, and the other that the t ..."
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Cited by 26 (2 self)
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We generalize previously known conditions for uniqueness of the Gibbs measure in statistical physics models by presenting conditions of any finite size for models on any underlying graph. We give two dual conditions, one requiring that the total influence on a site is small, and the other that the total influence of a site is small. Our proofs are combinatorial in nature and use tools from the analysis of discrete Markov chains, in particular the path coupling method. The implications of our conditions for the mixing time of natural Markov chains associated with the models are discussed as well. We also present some examples of models for which the conditions hold.
The KestenStigum reconstruction bound is tight for roughly symmetric binary channels
 In Proceedings of IEEE FOCS 2006
, 2006
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Exact Thresholds for IsingGibbs Samplers on General Graphs
"... We establish tight results for rapid mixing of Gibbs Samplers for the Ferromagnetic Ising model on general graphs. We show that if (d − 1)tanhβ < 1, then there exists a constant C such that the discrete time mixing time of Gibbs Samplers for the Ferromagnetic Ising model on any graph of n vertice ..."
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Cited by 14 (3 self)
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We establish tight results for rapid mixing of Gibbs Samplers for the Ferromagnetic Ising model on general graphs. We show that if (d − 1)tanhβ < 1, then there exists a constant C such that the discrete time mixing time of Gibbs Samplers for the Ferromagnetic Ising model on any graph of n vertices and maximal degree d, where all interactions are bounded by β, and arbitrary external fields is bounded by Cn log n. We further show the when d tanhβ < 1, with high probability over the ErdősRényi random graph on n vertices with average degree d, it holds that the mixing time of Gibbs Samplers is n 1+Θ ( 1 log log n). Both result are tight as it is known that the mixing time for random regular and ErdősRényi random graphs is, with high probability, exponential in n when if (d − 1)tanhβ> 1 and d tanhβ> 1 respectively.
Rapid mixing of gibbs sampling on graphs that are sparse on average
, 2007
"... Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the ErdősRényi random graph G(n, d/n), where each edge is chosen independently with probability d/n and d is f ..."
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Cited by 13 (3 self)
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Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the ErdősRényi random graph G(n, d/n), where each edge is chosen independently with probability d/n and d is fixed. While the average degree in G(n, d/n) is d(1 − o(1)), it contains many nodes of degree of order log n / loglog n. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n, p) such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature β, the mixing time of Gibbs sampling is at least n 1+Ω(1 / log log n). Recall that the Ising model with inverse temperature β defined on a graph G = (V, E) is the distribution over {±} V given by P(σ) = 1
Evolutionary Trees and the Ising Model on the Bethe Lattice: a Proof of Steel’s Conjecture
, 2008
"... One of the major tasks of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on the true evolutionary tree. Given samples from this Markov chain at the leaves of the tree, the goal is to reconstruct the evolutionary ..."
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Cited by 13 (6 self)
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One of the major tasks of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on the true evolutionary tree. Given samples from this Markov chain at the leaves of the tree, the goal is to reconstruct the evolutionary tree. It is well known that in order to reconstruct a tree on n leaves, sequences of length Ω(log n) are needed. It was conjectured by M. Steel that for the CFN/Ising evolutionary model, if the mutation probability on all edges of the tree is less than p ∗ = ( √ 2 − 1)/2 3/2, then the tree can be recovered from sequences of length O(log n). The value p ∗ is exactly the transition point for the extremality of the free Gibbs measure for the Ising model on the binary tree. Steel’s conjecture was proven by the second author in the special case where the tree is “balanced”. The second author also proved that if all edges have mutation probability larger than p ∗ then the length needed is n Ω(1). Here we complete the proof of Steel’s conjecture and give a reconstruction algorithm that requires optimal (up to a multiplicative constant) sequence length. Our results further extend to obtain an optimal reconstruction algorithm for the JukesCantor model with short edges. All reconstruction algorithms run in polynomial time. Our proof and results demonstrate that extremality of the free Gibbs measure on the infinite binary tree, which has been studied before in probability, statistical physics and computer science, determines how distinguishable are Gibbs measures on finite binary trees.
Reconstruction for colorings on trees
, 2008
"... Consider kcolorings of the complete tree of depth ℓ and branching factor ∆. If we fix the coloring of the leaves, for what range of k is the root uniformly distributed over all k colors (in the limit ℓ → ∞)? This corresponds to the threshold for uniqueness of the infinitevolume Gibbs measure. It i ..."
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Cited by 11 (4 self)
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Consider kcolorings of the complete tree of depth ℓ and branching factor ∆. If we fix the coloring of the leaves, for what range of k is the root uniformly distributed over all k colors (in the limit ℓ → ∞)? This corresponds to the threshold for uniqueness of the infinitevolume Gibbs measure. It is straightforward to show the existence of colorings of the leaves which “freeze ” the entire tree when k ≤ ∆ + 1. For k ≥ ∆ + 2, Jonasson proved the root is “unbiased ” for any fixed coloring of the leaves and thus the Gibbs measure is unique. What happens for a typical coloring of the leaves? When the leaves have a nonvanishing influence on the root in expectation, over random colorings of the leaves, reconstruction is said to hold. Nonreconstruction is equivalent to extremality of the Gibbs measure. When k < ∆ / ln ∆, it is straightforward to show that reconstruction is possible (and hence the measure is not extremal). We prove that for C> 2 and k = C∆ / ln ∆, nonreconstruction holds, i.e., the Gibbs measure is extremal. We prove a strong form of extremality: with high probability over the colorings of the leaves the influence at the root decays exponentially fast with the depth of the tree. These are the first results coming close to the threshold for extremality for colorings. Extremality on trees and random graphs has received considerable attention recently since it may have connections to the efficiency of local algorithms.
Mixing time of critical Ising model on trees is polynomial in the height
 Comm. Math. Phys
"... Abstract. In the heatbath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuoustime chain exhibits the following behavior. For some critical inversetemperature βc, the inversegap is O(1) for β < βc, polynomial in the surface area for β = ..."
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Cited by 9 (4 self)
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Abstract. In the heatbath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuoustime chain exhibits the following behavior. For some critical inversetemperature βc, the inversegap is O(1) for β < βc, polynomial in the surface area for β = βc and exponential in it for β> βc. This has been proved for Z 2 except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inversegap is bounded for β < βc and exponential for β> βc were established, where βc is the critical spinglass parameter, and the treeheight h plays the role of the surface area. In this work, we complete the picture for the inversegap of the Ising model on the bary tree, by showing that it is indeed polynomial in h at criticality. The degree of our polynomial bound does not depend on b, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixingtime of the chain. In addition, we study the near critical behavior, and show that for β> βc, the inversegap and mixingtime are both exp[Θ((β − βc)h)]. 1.