Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques

When faced with a complex task, is it better to be systematic or to proceed by making random adjustments? We study aspects of this problem in the context of generating random elements of a finite group. For example, suppose we want to fill n empty spaces with zeros and ones such that the probability of configuration x = (x1, . . . , xn) is θ n−|x|(1− θ)|x|, with |x| the number of ones in x. A systematic scan approach works left to right, filling each successive place with a θ coin toss. A random scan approach picks places at random, and a given site may be hit many times before all sites are hit. The systematic approach takes order n steps and the random approach takes order 1 4n log n steps. Realistic versions of this toy problem arise in image analysis and Ising-like simulations, where one must generate a random array by a Monte Carlo Markov chain. Systematic updating and random updating are competing algorithms that are discussed in detail in Section 2. There are some successful analyses for random scan algorithms, but the intuitively appealing systematic scan algorithms have resisted analysis. Our main results show that the binary problem just described is exceptional; for the examples analyzed in this paper, systematic and random scans converge in about the same number of steps. Let W be a finite Coxeter group generated by simple reflections s1, s2, . . . , sn, where s2 i = id. For example, W may be the permutation group Sn+1 with si = (i, i + 1). The length function `(w) is the smallest k such that w = si1si2 · · · sik . Fix 0 < θ ≤ 1 and define a probability distribution on W by π(w) = θ −`(w) PW (θ−1) , where PW(θ −1) = ∑

[1]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[2]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[3]  C. L. Mallows.  NON-NULL RANKING MODELS. I , 1957 .

[4]  John G. Kemeny,et al.  Finite Markov Chains. , 1960 .

[5]  Nicolas Bourbaki,et al.  Eléments de Mathématique , 1964 .

[6]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[7]  P. Prescott,et al.  Monte Carlo Methods , 1964, Computational Statistical Physics.

[8]  P. Deligne,et al.  Les immeubles des groupes de tresses généralisés , 1972 .

[9]  Nicolas Bourbaki,et al.  Eléments de mathématique : groupes et algèbres de Lie , 1972 .

[10]  Egbert Brieskorn,et al.  Artin-Gruppen und Coxeter-Gruppen , 1972 .

[11]  Louis Solomon,et al.  On the Theorem of Feit-Higman , 1973, J. Comb. Theory, Ser. A.

[12]  Peter Norbert Hoefsmit.  Representations of Hecke Algebras of finite groups with BN-Pairs of classical type , 1974 .

[13]  John G. Kemeny,et al.  Finite Markov chains , 1960 .

[14]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[15]  G. Letac.  Problemes classiques de probabilite sur un couple de Gelfand , 1981 .

[16]  G. Letac.  Les fonctions sphériques d'un couple de Gelfand symétrique et les chaînes de Markov , 1982, Advances in Applied Probability.

[17]  Roger W. Carter,et al.  Finite groups of Lie type: Conjugacy classes and complex characters , 1985 .

[18]  Meinolf Geck,et al.  Finite groups of Lie type , 1985 .

[19]  D. Critchlow.  Metric Methods for Analyzing Partially Ranked Data , 1986 .

[20]  David J. Benson,et al.  METHODS OF REPRESENTATION THEORY With Applications to Finite Groups and Orders, Volume II , 1988 .

[21]  E. Loh.  Multigrid Monte Carlo methods , 1988 .

[22]  P. Diaconis.  Group representations in probability and statistics , 1988 .

[23]  Goodman,et al.  Multigrid Monte Carlo method. Conceptual foundations. , 1989, Physical review. D, Particles and fields.

[24]  P. Barone,et al.  Improving Stochastic Relaxation for Gussian Random Fields , 1990, Probability in the Engineering and Informational Sciences.

[25]  Y. Amit.  On rates of convergence of stochastic relaxation for Gaussian and non-Gaussian distributions , 1991 .

[26]  U. Grenander,et al.  Comparing sweep strategies for stochastic relaxation , 1991 .

[27]  P. Diaconis,et al.  Eigen Analysis for Some Examples of the Metropolis Algorithm , 1992 .

[28]  Joseph S. Verducci,et al.  Probability Models and Statistical Analyses for Ranking Data , 1992 .

[29]  K. Ross,et al.  Hypergroup deformations and Markov chains , 1994 .

[30]  S. Ariki,et al.  A Hecke Algebra of (Z/rZ)Sn and Construction of Its Irreducible Representations , 1994 .

[31]  Persi Diaconis,et al.  What do we know about the Metropolis algorithm? , 1995, STOC '95.

[32]  Arun Ram.  Seminormal Representations of Weyl Groups and Iwahori‐Hecke Algebras , 1995, math/9511223.

[33]  Bruna Tanaka Cremonini,et al.  Buildings , 1995, Data, Statistics, and Useful Numbers for Environmental Sustainability.

[34]  Coordinate Selection Rules for Gibbs Sampling , 1996 .

[35]  W. Fulton.  Young Tableaux: With Applications to Representation Theory and Geometry , 1996 .

[36]  J. Marden.  Analyzing and Modeling Rank Data , 1996 .

[37]  Y. Amit.  Convergence properties of the Gibbs sampler for perturbations of Gaussians , 1996 .

[38]  Svante Janson,et al.  Metrics on Compositions and Coincidences among Renewal Sequences , 1996 .

[39]  G. Roberts,et al.  Updating Schemes, Correlation Structure, Blocking and Parameterization for the Gibbs Sampler , 1997 .

[40]  Igor Pak,et al.  WHEN AND HOW n CHOOSE k , 1997 .

[41]  Eric David Belsley.  Rates of convergence of random walk on distance regular graphs , 1998 .

[42]  W. Shannon,et al.  Combining classification trees using MLE. , 1999, Statistics in medicine.

[43]  Random Walks on Wreath Products of Groups , 2000, math/0006118.

[44]  M. Talagrand,et al.  Lectures on Probability Theory and Statistics , 2000 .

[45]  Louis J. Billera,et al.  Geometry of the Space of Phylogenetic Trees , 2001, Adv. Appl. Math..

[46]  A. V. D. Vaart,et al.  Lectures on probability theory and statistics , 2002 .