Markov random fields are used to model high dimensional distributions in a number of applied areas. Much recent interest has been devoted to the reconstruction of the dependency structure from independent samples from the Markov random fields. We analyze a simple algorithm for reconstructing the underlying graph defining a Markov random field on @math nodes and maximum degree @math given observations. We show that under mild non-degeneracy conditions it reconstructs the generating graph with high probability using @math samples where @math depend on the local interactions. For most local interaction @math are of order @math . Our results are optimal as a function of @math up to a multiplicative constant depending on @math and the strength of the local interactions. Our results seem to be the first results for general models that guarantee that the generating model is reconstructed. Furthermore, we provide explicit @math running time bound. In cases where the measure on the graph has correlation decay, the running time is @math for all fixed @math . We also discuss the effect of observing noisy samples and show that as long as the noise level is low, our algorithm is effective. On the other hand, we construct an example where large noise implies non-identifiability even for generic noise and interactions. Finally, we briefly show that in some simple cases, models with hidden nodes can also be recovered.
Subsequent to our work being posted on the Arxiv, Santhanam and Wainwright @cite_7 again considered essentially the problem for the Ising model, producing nearly matching lower and upper bounds on the asymptotic sampling complexity. Again their conditions do not apply to the low temperature regime. Another key difference from our work is that they restrict attention to the Ising model, i.e. Markov random fields with pairwise potentials and where each variable takes two values. Our results are not limited to pairwise interactions and apply to the more general setting of MRFs with potentials on larger cliques.
@cite_7
2030344865 2065448588 2029903428 1993711495
We consider the two-dimensional stochastic Ising model in finite square Λ with free boundary conditions, at inverse temperature β>β0 and zero external field. Using duality and recent results of Ioffe on the Wulff construction close to the critical temperature, we extend some of the results obtained by Martinelli in the low-temperature regime to any temperature below the critical one. In particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of Λ, with a rate constant determined by the surface tension along one of the coordinate axes. We also extend to the same range of temperatures the result due to Shlosman on the equilibrium large deviations of the magnetization with free boundary conditions. We study the asymptotic behavior as timet → + ∞ of certain nonstationary Markov chains, and prove the convergence of the annealing algorithm in Monte Carlo simulations. We find that in the limitt → + ∞, a nonstationary Markov chain may exhibit “phase transitions.” Nonstationary Markov chains in general, and the annealing algorithm in particular, lead to biased estimators for the expectation values of the process. We compute the leading terms in the bias and the variance of the sample-means estimator. We find that the annealing algorithm converges if the temperatureT(t) goes to zero no faster thanC log(t t0) ast→+∞, with a computable constantC andt0 the initial time. The bias and the variance of the sample-means estimator in the annealing algorithm go to zero likeO(t−1+e) for some 0⩽e<1, with e=0 only in very special circumstances. Our results concerning the convergence of the annealing algorithm, and the rate of convergence to zero of the bias and the variance of the sample-means estimator, provide a rigorous procedure for choosing the optimal “annealing schedule.” This optimal choice reflects the competition between two physical effects: (a) The “adiabatic” effect, whereby if the temperature is loweredtoo abruptly the system may end up not in a ground state but in a nearby metastable state, and (b) the “super-cooling” effect, whereby if the temperature is loweredtoo slowly the system will indeed approach the ground state(s) but may do so extremely slowly. This work extends to dimension d≥3 the main result of Dehghanpour and Schonmann. We consider the stochastic Ising model on Zd evolving with the Metropolis dynamics under a fixed small positive magnetic field h starting from the minus phase. When the inverse temperature β goes to ∞, the relaxation time of the system, defined as the time when the plus phase has invaded the origin, behaves like exp(βκd). The value κd is equal to κd=1d+1(Γ1+⋯+Γd), where Γi is the energy of the i-dimensional critical droplet of the Ising model at zero temperature and magnetic field h. We study the phase separation phenomenon in the Ising model in dimensions d ≥ 3. To this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization m*. We confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase. Moreover, the rescaled droplet is asymptotically close to a definite deterministic shape, the Wulff crystal, which minimizes the surface free energy. In the course of the proof we establish a surface order large deviation principle for the magnetization. Our results are valid for temperatures T below a limit of slab-thresholds T c conjectured to agree with the critical point T c . Moreover, T should be such that there exist only two extremal translation invariant Gibbs states at that temperature, a property which can fail for at most countably many values and which is conjectured to be true for every T. The proofs are based on the Fortuin-Kasteleyn representation of the Ising model along with coarse-graining techniques. To handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis. Finally, we propose a heuristic picture that for subcritical temperatures close enough to T c , the dominant minus spin cluster of the Wulff droplet permeates the entire box and has a strictly positive local density everywhere.