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Author: Mossel, Elchanan ×

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Now showing 1 - 10 of 73
  • Publication
    Co-evolution Is Incompatible With the Markov Assumption in Phylogenetics
    (2011-11-01) Tuller, Tamir; Mossel, Elchanan
    Markov models are extensively used in the analysis of molecular evolution. A recent line of research suggests that pairs of proteins with functional and physical interactions co-evolve with each other. Here, by analyzing hundreds of orthologous sets of three fungi and their co-evolutionary relations, we demonstrate that co-evolutionary assumption may violate the Markov assumption. Our results encourage developing alternative probabilistic models for the cases of extreme co-evolution.
  • Publication
    Learning DNF From Random Walks
    (2003-10-11) Bshouty, Nader; Mossel, Elchanan; O'Donnell, Ryan; Servedio, Rocco A
    We consider a model of learning Boolean functions from examples generated by a uniform random walk on {0, 1}n. We give a polynomial time algorithm for learning decision trees and DNF formulas in this model. This is the first efficient algorithm for learning these classes in a natural passive learning model where the learner has no influence over the choice of examples used for learning.
  • Publication
    Exact Thresholds for Ising–Gibbs Samplers on General Graphs
    (2013-01-01) Mossel, Elchanan; Sly, Allan
    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 are bounded by Cn log n. Moreover, the spectral gap is uniformly bounded away from 0 for all such graphs, as well as for infinite graphs of maximal degree d. We further show that when d tanh β < 1, with high probability over the Erdős–Rényi random graph G(n,d/n), it holds that the mixing time of Gibbs samplers is n1 + Θ(1/loglogn). Both results are tight, as it is known that the mixing time for random regular and Erdős–Rényi random graphs is, with high probability, exponential in n when (d − 1) tanh β > 1, and d tanh β > 1, respectively. To our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs. Moreover, our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees.
  • Publication
    Reconstruction on Trees: Beating the Second Eigenvalue
    (2001-01-01) Mossel, Elchanan
    We consider a process in which information is transmitted from a given root node on a noisy d-ary tree network T. We start with a uniform symbol taken from an alphabet A. Each edge of the tree is an independent copy of some channel (Markov chain) M, where M is irreducible and aperiodic on A. The goal is to reconstruct the symbol at the root from the symbols at the nth level of the tree. This model has been studied in information theory, genetics and statistical physics. The basic question is: is it possible to reconstruct (some information on)the root? In other words, does the probability of correct reconstruction tend to 1/A as n →∞? It is known that reconstruction is possible if dλ22(M) > 1, where λ2(M) is the second eigenvalue of M. Moreover, in this case it is possible to reconstruct using a majority algorithm which ignores the location of the data at the boundary of the tree. When M is a symmetric binary channel, this threshold is sharp. In this paper we show that, both for the binary asymmetric channel and for the symmetric channel on many symbols, it is sometimes possible to reconstruct even when dλ22(M) < 1. This result indicates that, for many (maybe most) tree-indexed Markov chains, the location of the data on the boundary plays a crucial role in reconstruction problems.
  • Publication
    Reaching Consensus on Social Networks
    (2009-01-01) Mossel, Elchanan; Schoenebeck, Grant
    Research in sociology studies the effectiveness of social networks in achieving computational tasks. Typically the agents who are supposed to achieve a task are unaware of the underlying social network except their immediate friends. They have limited memory, communication, and coordination. These limitations result in computational obstacles in achieving otherwise trivial computational problems. One of the simplest problems studied in the social sciences involves reaching a consensus among players between two alternatives which are otherwise indistinguishable. In this paper we formalize the computational model of social networks. We then analyze the consensus problem as well as the problem of reaching a consensus which is identical to the majority of the original signals. In both models we seek to minimize the time it takes players to reach a consensus.
  • Publication
    Complete Characterization of Functions Satisfying the Conditions of Arrow's Theorem
    (2012-06-01) Mossel, Elchanan; Tamuz, Omer
    Arrow’s theorem implies that a social welfare function satisfying Transitivity, the Weak Pareto Principle (Unanimity), and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When non-strict preferences are also allowed, a dictatorial social welfare function is defined as a function for which there exists a single voter whose strict preferences are followed. This definition allows for many different dictatorial functions, since non-strict preferences of the dictator are not necessarily followed. In particular, we construct examples of dictatorial functions which do not satisfy Transitivity and IIA. Thus Arrow’s theorem, in the case of non-strict preferences, does not provide a complete characterization of all social welfare functions satisfying Transitivity, the Weak Pareto Principle, and IIA. The main results of this article provide such a characterization for Arrow’s theorem, as well as for follow up results by Wilson. In particular, we strengthen Arrow’s and Wilson’s result by giving an exact if and only if condition for a function to satisfy Transitivity and IIA (and the Weak Pareto Principle). Additionally, we derive formulae for the number of functions satisfying these conditions.
  • Publication
    On the Hardness of Sampling Independent Sets Beyond the Tree Threshold
    (2009-03-01) Mossel, Elchanan; Weitz, Dror; Wormald, Nicholas
    We consider local Markov chain Monte–Carlo algorithms for sampling from the weighted distribution of independent sets with activity λ, where the weight of an independent set I is λ|I|. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree d and λ < λ c (d), where λ c (d) is the critical activity for uniqueness of the Gibbs measure (i.e., for decay of correlations with distance in the weighted distribution over independent sets) on the d-regular infinite tree. We show that for d ≥ 3, λ just above λ c (d) with high probability over d-regular bipartite graphs, any local Markov chain Monte–Carlo algorithm takes exponential time before getting close to the stationary distribution. Our results provide a rigorous justification for “replica” method heuristics. These heuristics were invented in theoretical physics and are used in order to derive predictions on Gibbs measures on random graphs in terms of Gibbs measures on trees. A major theoretical challenge in recent years is to provide rigorous proofs for the correctness of such predictions. Our results establish such rigorous proofs for the case of hard-core model on bipartite graphs. We conjecture that λ c is in fact the exact threshold for this computational problem, i.e., that for λ > λ c it is NP-hard to approximate the above weighted sum over independent sets to within a factor polynomial in the size of the graph.
  • Publication
    Non-Interactive Correlation Distillation, Inhomogeneous Markov Chains, and the Reverse Bonami-Beckner Inequality
    (2006-12-01) Mossel, Elchanan; O'Donnell, Ryan; Regev, Oded; Steif, Jeffrey E; Sudakov, Benny
    In this paper we study non-interactive correlation distillation (NICD), a generalization of noise sensitivity previously considered in [5, 31, 39]. We extend the model toNICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly random string and this string is distributed throughout the network, with the edges of the tree acting as independent binary symmetric channels. The goal of the players is to agree on a shared random bit without communicating. Our new contributions include the following: • In the case of a k-leaf star graph (the model considered in [31]), we resolve the open question of whether the success probability must go to zero as k → ∞. We show that this is indeed the case and provide matching upper and lower bounds on the asymptotically optimal rate (a slowly-decaying polynomial). • In the case of the k-vertex path graph, we show that it is always optimal for all players to use the same 1-bit function. • In the general case we show that all players should use monotone functions. We also show, somewhat surprisingly, that for certain trees it is better if not all players use the same function. Our techniques include the use of thereverse Bonami-Beckner inequality. Although the usual Bonami-Beckner has been frequently used before, its reverse counterpart seems not to be well known. To demonstrate its strength, we use it to prove a new isoperimetric inequality for the discrete cube and a new result on the mixing of short random walks on the cube. Another tool that we need is a tight bound on the probability that a Markov chain stays inside certain sets; we prove a new theorem generalizing and strengthening previous such bounds [2, 3, 6]. On the probabilistic side, we use the “reflection principle” and the FKG and related inequalities in order to study the problem on general trees.
  • Publication
    A Spectral Approach to Analysing Belief Propagation for 3-Colouring
    (2009-11-01) Coja-Oghlan, Amin; Mossel, Elchanan; Vilenchik, Dan
    Belief propagation (BP) is a message-passing algorithm that computes the exact marginal distributions at every vertex of a graphical model without cycles. While BP is designed to work correctly on trees, it is routinely applied to general graphical models that may contain cycles, in which case neither convergence, nor correctness in the case of convergence is guaranteed. Nonetheless, BP has gained popularity as it seems to remain effective in many cases of interest, even when the underlying graph is ‘far’ from being a tree. However, the theoretical understanding of BP (and its new relative survey propagation) when applied to CSPs is poor. Contributing to the rigorous understanding of BP, in this paper we relate the convergence of BP to spectral properties of the graph. This encompasses a result for random graphs with a ‘planted’ solution; thus, we obtain the first rigorous result on BP for graph colouring in the case of a complex graphical structure (as opposed to trees). In particular, the analysis shows how belief propagation breaks the symmetry between the 3! possible permutations of the colour classes.
  • Publication
    Iterative Maximum Likelihood on Networks
    (2010-07-01) Mossel, Elchanan; Tamuz, Omer
    We consider n agents located on the vertices of a connected graph. Each agent v receives a signal Xv(0)∼N(μ,1) where μ is an unknown quantity. A natural iterative way of estimating μ is to perform the following procedure. At iteration t+1 let Xv(t+1) be the average of Xv(t) and of Xw(t) among all the neighbors w of v. It is well known that this procedure converges to X (∞) = ½ |E|-1 ∑dvXv where dv is the degree of v. In this paper we consider a variant of simple iterative averaging, which models “greedy” behavior of the agents. At iteration t, each agent v declares the value of its estimator Xv(t) to all of its neighbors. Then, it updates Xv(t+1) by taking the maximum likelihood (or minimum variance) estimator of μ, given Xv(t) and Xw(t) for all neighbors w of v, and the structure of the graph. We give an explicit efficient procedure for calculating Xv(t), study the convergence of the process as t→∞ and show that if the limit exists then Xv(∞)=Xw(∞) for all v and w. For graphs that are symmetric under actions of transitive groups, we show that the process is efficient. Finally, we show that the greedy process is in some cases more efficient than simple averaging, while in other cases the converse is true, so that, in this model, “greed” of the individual agents may or may not have an adverse affect on the outcome. The model discussed here may be viewed as the maximum likelihood version of models studied in Bayesian Economics. The ML variant is more accessible and allows in particular to show the significance of symmetry in the efficiency of estimators using networks of agents.