Robust reconstruction on trees is determined by the second eigenvalue



The Annals of Probability

Robust reconstruction on trees is determined by the second eigenvalue

Svante Janson and Elchanan Mossel

Source: Ann. Probab. Volume 32, Number 3B (2004), 2630-2649.

Abstract

Consider a Markov chain on an infinite tree T=(V,E) rooted at ρ. In such a chain, once the initial root state σ({ρ}) is chosen, each vertex iteratively chooses its state from the one of its parent by an application of a Markov transition rule (and all such applications are independent). Let μj denote the resulting measure for σ({ρ})=j. The resulting measure μj is defined on configurations $\sigma=(\sigma(x))_{x\in V}\in \mathcal {A}^{V}$, where $\mathcal {A}$ is some finite set. Let μjn denote the restriction of μ to the sigma-algebra generated by the variables σ(x), where x is at distance exactly n from ρ. Letting $\alpha_{n}=\max_{i,j\in \mathcal {A}}d_{\mathrm{TV}}(\mu_{i}^{n},\mu_{j}^{n})$, where dTV denotes total variation distance, we say that the reconstruction problem is solvable if lim inf n→∞αn>0. Reconstruction solvability roughly means that the nth level of the tree contains a nonvanishing amount of information on the root of the tree as n→∞.

In this paper we study the problem of robust reconstruction. Let ν be a nondegenerate distribution on $\mathcal {A}$ and ɛ>0. Let σ be chosen according to μjn and σ' be obtained from σ by letting for each node independently, σ(v)=σ'(v) with probability 1−ɛ and σ'(v) be an independent sample from ν otherwise. We denote by μjn[ν,ɛ] the resulting measure on σ'. The measure μjn[ν,ɛ] is a perturbation of the measure μjn. Letting $\alpha_{n}(\nu,\varepsilon )=\max_{i,j\in \mathcal {A}}d_{\mathrm{TV}}(\mu_{i}^{n}[\nu,\varepsilon ],\mu_{j}^{n}[\nu,\varepsilon ])$, we say that the reconstruction problem is ν-robust-solvable if lim inf n→∞αn(ν,ɛ)>0 for all 0<ɛ<1. Roughly speaking, the reconstruction problem is robust-solvable if for any noise-rate and for all n, the nth level of the tree contains a nonvanishing amount of information on the root of the tree.

Standard techniques imply that if T is the rooted B-ary tree (where each node has B children) and if B2(M)|2>1, where λ2(M) is the second largest eigenvalue of M (in absolute value), then for all nondegenerate ν, the reconstruction problem is ν-robust-solvable. We prove a converse and show that the reconstruction problem is not ν-robust-solvable if B2(M)|2<1. This proves a conjecture by the second author and Y. Peres. We also consider other models of noise and general trees.

Primary Subjects: 60K35
Secondary Subjects: 60J80, 82B26
Keywords: Robust phase transition; reconstruction on trees; branching number

Full-text: Open access

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1091813626
Digital Object Identifier: doi:10.1214/009117904000000153
Mathematical Reviews number (MathSciNet): MR2078553
Zentralblatt MATH identifier: 02121709

References

Berger, N., Kenyon, C., Mossel, E. and Peres, Y. (2004). Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Related Fields. To appear.
Mathematical Reviews (MathSciNet): MR2123248
Digital Object Identifier: doi:10.1007/s00440-004-0369-4
Bleher, P. M., Ruiz, J. and Zagrebnov, V. A. (1995). On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Statist. Phys. 79 473--482.
Mathematical Reviews (MathSciNet): MR1325591
Digital Object Identifier: doi:10.1007/BF02179399
Evans, W. S., Kenyon, C., Peres, Y. and Schulman, L. J. (2000). Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 410--433.
Mathematical Reviews (MathSciNet): MR1768240
Digital Object Identifier: doi:10.1214/aoap/1019487349
Project Euclid: euclid.aoap/1019487349
Evans, W. S. and Schulman, L. J. (1999). Signal propagation and noisy circuits. IEEE Trans. Inform. Theory 45 2367--2373.
Mathematical Reviews (MathSciNet): MR1725124
Digital Object Identifier: doi:10.1109/18.796377
Felsenstein, J. (2003). Inferring Phylogenies. Sinauer, New York.
Furstenberg, H. (1970). Intersections of Cantor sets and transversality of semigroups. In Problems in Analysis (R. C. Gunning, ed.) 41--59. Princeton Univ. Press.
Mathematical Reviews (MathSciNet): MR354562
Georgii, H. O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
Mathematical Reviews (MathSciNet): MR956646
Zentralblatt MATH: 0657.60122
Hajek, B. and Weller, T. (1991). On the maximum tolerable noise for reliable computation by formulas. IEEE Trans. Inform. Theory 37 388--291.
Higuchi, Y. (1977). Remarks on the limiting Gibbs states on a $(d+1)$-tree. Publ. Res. Inst. Math. Sci. 13 335--348.
Mathematical Reviews (MathSciNet): MR676482
Ioffe, D. (1996). On the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phys. 37 137--143.
Mathematical Reviews (MathSciNet): MR1391195
Digital Object Identifier: doi:10.1007/BF00416016
Kenyon, C., Mossel, E. and Peres, Y. (2001). Glauber dynamics on trees and hyperbolic graphs. In 42nd IEEE Symposium on on Foundations of Computer Science 568--578. IEEE, Los Alamitos, CA.
Mathematical Reviews (MathSciNet): MR1948746
Kesten, H. and Stigum, B. P. (1966). Additional limit theorems for indecomposable multidimensional Galton--Watson processes. Ann. Math. Statist. 37 1463--1481.
Mathematical Reviews (MathSciNet): MR200979
Lyons, R. (1989). The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 337--353.
Mathematical Reviews (MathSciNet): MR1016874
Digital Object Identifier: doi:10.1007/BF01217911
Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931--958.
Mathematical Reviews (MathSciNet): MR1062053
Martin, J. (2003). Reconstruction thresholds on regular trees. In Discrete Random Walks (C. Banderier and C. Krattenthaler, eds.) 191--204. Available at http://dmtcs. loria.fr/proceedings/dmACind.html.
Mathematical Reviews (MathSciNet): MR2042387
Zentralblatt MATH: 1073.68702
Martinelli, F., Sinclair, A. and Weitz, D. (2003). The Ising model on trees: Boundary conditions and mixing time. In Proceedings of the Forty Fourth Annual Symposium on Foundations of Computer Science 628--639.
Martinelli, F., Sinclair, A. and Weitz, D. (2004). The Ising model on trees: Boundary conditions and mixing time. Comm. Math. Phys. To appear.
Mathematical Reviews (MathSciNet): MR2094519
Digital Object Identifier: doi:10.1007/s00220-004-1147-y
Mossel, E. (2001). Reconstruction on trees: Beating the second eigenvalue. Ann. Appl. Probab. 11 285--300.
Mathematical Reviews (MathSciNet): MR1825467
Project Euclid: euclid.aoap/998926994
Mossel, E. (2003). On the impossibility of reconstructing ancestral data and phylogenies. J. Comput. Biol. 10 669--678.
Mossel, E. (2004). Phase transitions in phylogeny. Trans. Amer. Math. Soc. 356 2379--2404.
Mathematical Reviews (MathSciNet): MR2048522
Digital Object Identifier: doi:10.1090/S0002-9947-03-03382-8
Mossel, E. (2004). Survey: Information flow on trees. In Graphs, Morphisms and Statistical Physiscs (J. Nesetril and P. Winkler, eds.) 155--170. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR2056226
Zentralblatt MATH: 1066.94006
Mossel, E. and Peres, Y. (2003). Information flow on trees. Ann. Appl. Probab. 13 817--844.
Mathematical Reviews (MathSciNet): MR1994038
Digital Object Identifier: doi:10.1214/aoap/1060202828
Project Euclid: euclid.aoap/1060202828
Mossel, E. and Steel, M. (2004). A phase transition for a random cluster model on phylogenetic trees. Mathematical Biosciences 187 189--203.
Mathematical Reviews (MathSciNet): MR2043825
Digital Object Identifier: doi:10.1016/j.mbs.2003.10.004
Pemantale, R. and Peres, Y. (1995). Recursions on trees and the Ising model at critical temperatures. Unpublished manuscript.
Pemantle, R. and Steif, J. E. (1999). Robust phase transitions for Heisenberg and other models on general trees. Ann. Probab. 27 876--912.
Mathematical Reviews (MathSciNet): MR1698979
Digital Object Identifier: doi:10.1214/aop/1022677452
Project Euclid: euclid.aop/1022677389
Semple, C. and Steel, M. (2003). Phylogenetics. Oxford Univ. Press.
Mathematical Reviews (MathSciNet): MR2060009
Zentralblatt MATH: 1043.92026
Sober, E. and Steel, M. A. (2002). Testing the hypothesis of common ancestry. J. Theor. Biol. 218 395--408.
Mathematical Reviews (MathSciNet): MR2027378
Spitzer, F. (1975). Markov random fields on an infinite tree. Ann. Probab. 3 387--398.
Mathematical Reviews (MathSciNet): MR378152
von Neumann, J. (1956). Probabilistic logics and the synthesis of reliable organisms from unreliable components. In Automata Studies (C. E. Shannon and J. McCarthy, eds.) 43--98. Princeton Univ. Press.
Mathematical Reviews (MathSciNet): MR77479

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