exploratory analysis of spatiotemporal patterns of cellular automata by clustering compressibility in this paper we study the classification of spatiotemporal pattern of one dimensional cellular automata ca whereas the classification comprises ca rules including their initial conditions we propose an exploratory analysis method based on the normalized compression distance ncd of spatiotemporal patterns which is used as dissimilarity measure for a hierarchical clustering our approach is different with respect to the following points first the classification of spatiotemporal pattern is comparative because the ncd evaluates explicitly the difference of compressibility among two objects e g strings corresponding to spatiotemporal patterns this is in contrast to all other measures applied so far in a similar context because they are essentially univariate second kolmogorov complexity which underlies the ncd was used in the classification of ca with respect to their spatiotemporal pattern third our method is semiautomatic allowing us to investigate hundreds or thousands of ca rules or initial conditions simultaneously to gain insights into their organizational structure our numerical results are not only plausible confirming previous classification attempts but also shed light on the intricate influence of random initial conditions on the classification results

approximation of the two part mdl code approximation of the optimal two part minimum description length mdl code for given data through successive monotonically length decreasing two part mdl codes has the following properties i computation of each step may take arbitrarily long ii we may not know when we reach the optimum or whether we will reach the optimum at all iii the sequence of models generated may not monotonically improve the goodness of fit but iv the model associated with the optimum has almost the best goodness of fit to express the practically interesting goodness of fit of individual models for individual data sets we have to rely on kolmogorov complexity

finite group extensions and the atiyah conjecture the atiyah conjecture for a discrete group g states that the l 2 betti numbers of a finite cw complex with fundamental group g are integers if g is torsion free and in general that they are rational numbers with denominators determined by the finite subgroups of g here we establish conditions under which the atiyah conjecture for a torsion free group g implies the atiyah conjecture for every finite extension of g the most important requirement is that h g z p is isomorphic to the cohomology of the p adic completion of g for every prime p an additional assumption is necessary e g that the quotients of the lower central series or of the derived series are torsion free we prove that these conditions are fulfilled for a certain class of groups which contains in particular artin s pure braid groups and more generally fundamental groups of fiber type arrangements free groups fundamental groups of orientable compact surfaces certain knot and link groups primitive positive one relator groups and products of these therefore every finite in fact every elementary amenable extension of these groups satisfies the atiyah conjecture as a consequence if such an extension h is torsion free then the group ring ch contains no non trivial zero divisors i e h fulfills the zero divisor conjecture in the course of the proof we prove that if these extensions are torsionfree then they have plenty of non trivial torsion free quotients which are virtually nilpotent all of this applies in particular to artin s full braid group therefore answering question b6 on www grouptheory org our methods also apply to the baum connes conjecture this is discussed in 47 where for example the baum connes conjecture is proved for the full braid groups msc 55n25 homology with local coecients 16s34 group rings laurent rings 57m25 knots and links

distance spectra of graphs a survey abstract in 1971 graham and pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems they also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph computing the distance characteristic polynomial and its coefficients was the first research subject of interest thereafter the eigenvalues attracted much more attention in the present paper we report on the results related to the distance matrix of a graph and its spectral properties

on the similarity metric and the distance metric similarity and dissimilarity measures are widely used in many research areas and applications when a dissimilarity measure is used it is normally required to be a distance metric however when a similarity measure is used there is no formal requirement in this article we have three contributions first we give a formal definition of similarity metric second we show the relationship between similarity metric and distance metric third we present general solutions to normalize a given similarity metric or distance metric

a metric between probability distributions on finite sets of different cardinalities and applications to order reduction in this paper we define a metric distance between probability distributions on two distinct finite sets of possibly different cardinalities the metric is defined in terms of a joint distribution on the product of the two sets which has the two given distributions as its marginals and has minimum entropy computing the metric exactly turns out to be np hard therefore an efficient greedy algorithm is presented for finding an upper bound on the distance we then study the problem of optimal order reduction in the metric defined here it is shown that every optimal reduced order approximation must be an aggregation of the original distribution and that optimal reduced order approximation is equivalent to finding an aggregation with maximum entropy this problem also turns out to be np hard so again a greedy algorithm is constructed for finding a suboptimal reduced order approximation taken together all the results presented here permit the approximation of an independent and identically distributed i i d process over a set of large cardinality by another i i d process over a set of smaller cardinality in future work attempts will be made to extend this work to markov processes over finite sets

a really simple approximation of smallest grammar in this paper we present a really simple linear time algorithm constructing a context free grammar of size 4 g log 3 2 n g for the input string where n is the size of the input string and g the size of the optimal grammar generating this string the algorithm works for arbitrary size alphabets but the running time is linear assuming that the alphabet of the input string can be identified with numbers from 1 n c for some constant c algorithms with such an approximation guarantee and running time are known however all of them were non trivial and their analyses were involved the here presented algorithm computes the lz77 factorisation and transforms it in phases to a grammar in each phase it maintains an lz77 like factorisation of the word with at most factors as well as additional o letters where was the size of the original lz77 factorisation in one phase in a greedy way by a left to right sweep and a help of the factorisation we choose a set of pairs of consecutive letters to be replaced with new symbols i e nonterminals of the constructed grammar we choose at least 2 3 of the letters in the word and there are o many different pairs among them hence there are o log n phases each of them introduces o nonterminals to a grammar a more precise analysis yields a bound 4 log n as g this yields the desired bound g 4 g log n g

application of kolmogorov complexity and universal codes to identity testing and nonparametric testing of serial independence for time series we show that kolmogorov complexity and such its estimators as universal codes or data compression methods can be applied for hypotheses testing in a framework of classical mathematical statistics the methods for identity testing and nonparametric testing of serial independence for time series are suggested

the complexity of finding minimum length generator sequences abstract the computational complexity of the following problem is investigated given a permutation group specified as a set of generators and a single target permutation which is a member of the group what is the shortest expression for the target permutation in terms of the generators the general problem is demonstrated to be psppace complete and indeed is shown to remain so even when the generator set is restricted to contain only two permutations the restriction on generator set cardinality is the best possible as the problem becomes soluble in polynomial time if the generator set cantains only one permutation an interesting feature of this problem is that it does not fall under the headings of two person games or formal languages which cover the great majority of known pspace complete problems some restricted versions of the problem in which the generator set is fixed rather than being part of the problem instance are also investigated and shown to be computationally tractable one result of this kind is that determining the most compact expression of a permutation in terms of cyclicly adjacent transpositions can be achieved in polynomial time thus from an initial arrangement of distinct objects on a circle one can quickly compute the smallest number of interchanges of adjacent objects required to realise any other arrangement surprisingly this problem appears substantially more difficult to solve than the related one for which a solution has been known for some time in which the objects are arranged on a line segment

local model influence abstract this article develops a general method for assessing the influence of model assumptions in a bayesian analysis we assume that model choices are indexed by a hyperparameter with some given initial choice we use the term model to encompass both the sampling model and the prior distribution we wish to assess the effect of changing the hyperparameter away from the initial choice we are performing a sensitivity analysis with the hyperparameter defining our perturbations we use the kullback leibler divergence to measure the difference between posteriors corresponding to different choices of the hyperparameter we also measure the change in priors if small changes in the priors lead to large changes in posteriors the choice of hyperparameter is influential the second order difference in the kullback leibler divergence is expressed by fisher information matrices the relative change in posteriors compared with priors may be summarized by the relative eigenvalue of the posterior and prior fishe

normalized compression distance of multisets with applications pairwise normalized compression distance ncd is a parameter free feature free alignment free similarity metric based on compression we propose an ncd of multisets that is also metric previously attempts to obtain such an ncd failed for classification purposes it is superior to the pairwise ncd in accuracy and implementation complexity we cover the entire trajectory from theoretical underpinning to feasible practice it is applied to biological stem cell organelle transport and ocr classification questions that were earlier treated with the pairwise ncd with the new method we achieved significantly better results the theoretic foundation is kolmogorov complexity

information distance while kolmogorov 1965 complexity is the accepted absolute measure of information content in an individual finite object a similarly absolute notion is needed for the information distance between two individual objects for example two pictures we give several natural definitions of a universal information metric based on length of shortest programs for either ordinary computations or reversible dissipationless computations it turns out that these definitions are equivalent up to an additive logarithmic term we show that the information distance is a universal cognitive similarity distance we investigate the maximal correlation of the shortest programs involved the maximal uncorrelation of programs a generalization of the slepian wolf theorem of classical information theory and the density properties of the discrete metric spaces induced by the information distances a related distance measures the amount of nonreversibility of a computation using the physical theory of reversible computation we give an appropriate universal antisymmetric and transitive measure of the thermodynamic work required to transform one object in another object by the most efficient process information distance between individual objects is needed in pattern recognition where one wants to express effective notions of pattern similarity or cognitive similarity between individual objects and in thermodynamics of computation where one wants to analyze the energy dissipation of a computation from a particular input to a particular output

information distance in multiples information distance is a parameter free similarity measure based on compression used in pattern recognition data mining phylogeny clustering and classification the notion of information distance is extended from pairs to multiples finite lists we study maximal overlap metricity universality minimal overlap additivity and normalized information distance in multiples we use the theoretical notion of kolmogorov complexity which for practical purposes is approximated by the length of the compressed version of the file involved using a real world compression program

the subsequence composition of a string words that appear as constrained subsequences in a text string are considered as possible indicators of the host string structure hence also as a possible means of sequence comparison and classification the constraint consists of imposing a bound on the number w of positions in the text that may intervene between any two consecutive characters of a subsequence a subset of such w sequences is then characterized that consists in intuitive terms of sequences that could not be enriched with more characters without losing some occurrence in the text a compact spatial representation is then proposed for these representative sequences within which a number of parameters can be defined and measured in the final part of the paper such parameters are empirically analyzed on a small collection of text strings endowed with various degrees of structure

freely independent random variables with non atomic distributions 2015 american mathematical society we examine the distributions of non commutative polynomials of non atomic freely independent random variables in particular we obtain an analogue of the strong atiyah conjecture for free groups thus proving that the measure of each atom of any n n matricial polynomial of nonatomic freely independent random variables is an integer multiple of n 1 in addition we show that the cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real analytic except at a finite number of points

sequence similarity measures based on bounded hamming distance a growing number of measures of sequence similarity are being based on some underlying notion of relative compressibility within this paradigm similar sequences are expected to share a large number of common substrings or subsequences or more complex patterns or motifs and so on in this paper measures of sequence similarity are introduced and studied in which patterns in a pair are considered similar if they coincide up to a preset number of mismatches that is within a bounded hamming distance it is shown here that for some such measures bounds are achievable that are slightly better than o n 2 preliminary experiments demonstrate the potential applicability to phylogeny and classification of these similarity measures

the trace on the k theory of group c algebras the canonical trace on the reduced c algebra of a discrete group gives rise to a homomorphism from the k theory of this c algebra to the real numbers this paper studies the range of this homomorphism for torsion free groups the baum connes conjecture together with atiyah s l 2 index theorem implies that the range consists of the integers we give a direct and elementary proof that if g acts on a tree and admits a homomorphism to another group h whose restriction jgv to every stabilizer group of a vertex is injective then

integrality of l 2 betti numbers the atiyah conjecture predicts that the l 2 betti numbers of a finite cw complex with torsion free fundamental group are integers we establish the atiyah conjecture under the condition that it holds for g and that h lhd g is a normal subgroup for amalgamated free products g h h rtimes f here f is a free group and h rtimes f is an arbitrary semi direct product this includes free products g f and semi direct products g rtimes f we also show that the atiyah conjecture holds with an additional technical condition for direct and inverse limits of groups for which it is true as a corollary it holds for positive 1 relator groups with torsion free abelianization putting everything together we establish a new bigger class of groups for which the atiyah conjecture holds which contains all free groups and in particular is closed under taking subgroups direct sums free products extensions with torsion free elementary amenable quotient or with free quotient and under certain direct and inverse limits

sequential vs parallel complexity in simple gene assembly we investigate some differences between the general intramolecular model for gene assembly and its restricted simple model although both models satisfactorily sort all current experimental data we show that the general model offers assembly strategies for a given string that vary in both assembly length and the operations used while the simple model will always use the same number of each type of operation to sort a gene when simple operations are applied in parallel this is given a new twist we prove that for any n 1 there exists a string having maximally parallel assemblies of any length between n and 2n

extracting kolmogorov complexity with applications to dimension zero one laws we apply results on extracting randomness from independent sources to extract kolmogorov complexity for any a e 0 given a string x with k x a x we show how to use a constant number of advice bits to efficiently compute another string y y w x with k y 1 e y this result holds for both unbounded and space bounded kolmogorov complexity we use the extraction procedure for space bounded complexity to establish zero one laws for the strong dimensions of complexity classes within espace the unbounded extraction procedure yields a zero one law for the constructive strong dimensions of turing degrees

information theoretic measures for clusterings comparison variants properties normalization and correction for chance information theoretic measures form a fundamental class of measures for comparing clusterings and have recently received increasing interest nevertheless a number of questions concerning their properties and inter relationships remain unresolved in this paper we perform an organized study of information theoretic measures for clustering comparison including several existing popular measures in the literature as well as some newly proposed ones we discuss and prove their important properties such as the metric property and the normalization property we then highlight to the clustering community the importance of correcting information theoretic measures for chance especially when the data size is small compared to the number of clusters present therein of the available information theoretic based measures we advocate the normalized information distance nid as a general measure of choice for it possesses concurrently several important properties such as being both a metric and a normalized measure admitting an exact analytical adjusted for chance form and using the nominal 0 1 range better than other normalized variants

approximating l2 invariants and the atiyah conjecture let g be a torsion free discrete group and let denote the field of algebraic numbers in we prove that g fulfills the atiyah conjecture if g lies in a certain class of groups d which contains in particular all groups that are residually torsion free elementary amenable or are residually free this result implies that there are no nontrivial zero divisors in g the statement relies on new approximation results for l2 betti numbers over g which are the core of the work done in this paper another set of results in the paper is concerned with certain number theoretic properties of eigenvalues for the combinatorial laplacian on l2 cochains on any normal covering space of a finite cw complex we establish the absence of eigenvalues that are transcendental numbers whenever the covering transformation group is either amenable or in the linnell class we also establish the absence of eigenvalues that are liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class 2003 wiley periodicals inc

brauer induction for g0 of certain infinite groups f quinn has recently proven a brauer induction theorem for k of certain infinite groups 9 f t farrell points out 3 that the theorem does not extend to more general coefficient rings arguments of formanek have lead to such an extension on the level of trace functions which requires an artin exponent s we begin by analyzing the structure of a noetherian ring u graded by a virtually polycyclic group r and with units in all degrees for each such u and i and each finite h c f denote by u the subring of u supported on h then we prove surjectivity of the induction map

diagnostic measures for model criticism abstract we discuss the problem of model criticism with emphasis on developing summary diagnostic measures we approach model criticism by identifying possible troublesome features of the currently entertained model embedding the model in an elaborated model and measuring the value of elaborating this requires three elements a model elaboration a prior distribution and a utility function each triplet generates a different diagnostic measure we focus primarily on the measure given by a kullback leibler divergence between the marginal prior and posterior distributions on the elaboration parameter we also develop a linearized version of this diagnostic and use it to show that our procedure is related to other tools commonly used for model diagnostics such as bayes factors and the score function one attraction of this approach is that it allows model criticism to be performed jointly with parameter inference and prediction also this diagnostic approach aims at maintaining an exploratory nature t

sorting permutations and binary strings by length weighted rearrangements abstract comparative genomics is a line of study which aims at determining dis similarities between genomes one way of doing this is to infer the large scale mutations also called genome rearrangements that are responsible for transforming one genome into another under the parsimony principle numerous efforts have been made over the past years regarding the traditional approach in which all genome rearrangements have the same unit cost however since long rearrangements are more likely to disturb the organism some works considered the length weighted approach in which the cost of each rearrangement is a function of its length the problem is thus modeled as follows given two genomes g 1 and g 2 and a set of allowed genome rearrangements each with a given cost determine their distance i e the total cost of a minimum cost sequence of rearrangements from that transforms g 1 into g 2 in this paper we study length weighted reversals and transpositions mainly considering their prefix variants in which the leftmost part of the genome must always be included we present approximation algorithms and bounds on the distances and diameters i e maximum possible distance over any two genomes of same length for eight problems when genomes are represented as permutations and another eight problems when genomes are represented as binary strings in each case considering two different cost functions

sorting by reversals and the theory of 4 regular graphs abstract we show that the theory of sorting by reversals fits into the well established theory of circuit partitions of 4 regular multigraphs which also involves the combinatorial structures of circle graphs and delta matroids in this way we expose strong connections between the two theories that have not been fully appreciated before we also discuss a generalization of sorting by reversals involving the double cut and join dcj operation finally we also show that the theory of sorting by reversals is closely related to that of gene assembly in ciliates

image registration by compression abstract image registration consists in finding the transformation that brings one image into the best possible spatial correspondence with another image in this paper we present a new framework for image registration based on compression the basic idea underlying our approach is the conjecture that two images are correctly registered when we can maximally compress one image given the information in the other the contribution of this paper is twofold first we show that image registration can be formulated as a compression problem second we demonstrate the good performance of the similarity metric introduced by li et al in image registration two different approaches for the computation of this similarity metric are described the kolmogorov version computed using standard real world compressors and the shannon version calculated from an estimation of the entropy rate of the images

quick greedy computation for minimum common string partition in the minimum common string partition problem one is given two strings s and t with the same character statistics and one seeks for the smallest partition of s into substrings so that t can also be partitioned into the same substring multiset the problem is fundamental in several variants of edit distance with block operations e g signed reversal distance with duplicates and edit distance with moves the minimum common string partition problem is known to be np complete and the best approximation algorithm known has an approximation factor of o lognlog n since the minimum common string partition problem is of utmost practical importance one seeks a heuristic that will 1 usually have a low approximation factor and 2 will run fast a simple greedy algorithm is known which iteratively choose non overlapping longest common substrings of the input strings this algorithm has been well studied from an approximation point of view and it has been shown to have a bad worst case approximation factor however all the bad approximation factors presented so far stem from complicated recursive construction in practice the greedy algorithm seems to have small approximation factors however the best current implementation of greedy runs in quadratic time we propose a novel method to implement greedy in linear time

p values for composite null models abstract the problem of investigating compatibility of an assumed model with the data is investigated in the situation when the assumed model has unknown parameters the most frequently used measures of compatibility are p values based on statistics t for which large values are deemed to indicate incompatibility of the data and the model when the null model has unknown parameters p values are not uniquely defined the proposals for computing a p value in such a situation include the plug in and similar p values on the frequentist side and the predictive and posterior predictive p values on the bayesian side we propose two alternatives the conditional predictive p value and the partial posterior predictive p value and indicate their advantages from both bayesian and frequentist perspectives

a 2 approximation algorithm for genome rearrangements by reversals and transpositions recently a new approach to analyze genomes evolving which is based on comparision of gene orders versus traditional comparision of dna sequences was proposed sankoff et al 1992 the approach is based on the global rearrangements e g inversions and transpositions of fragments analysis of genomes evolving by inversions and transpositions leads to a combinatorial problem of sorting by reversals and transpositions i e sorting of a permutation using reversals and transpositions of arbitrary fragments we study sorting of signed permutations by reversals and transpositions a problem which adequately models genome rearrangements as the genes in dna are oriented we establish a lower bound and give a 2 approximation algorithm for the problem

the strong atiyah conjecture for virtually cocompact special groups we provide new conditions for the strong atiyah conjecture to lift to finite group extensions in particular we show that fundamental groups of compact special cube complexes satisfy these conditions so the conjecture holds for finite extensions of these groups

compact group automorphisms addition formulas and fuglede kadison determinants for a countable amenable group and an element f in the integral group ring z being invertible in the group von neumann algebra of we show that the entropy of the shift action of on the pontryagin dual of the quotient of z by its left ideal generated by f is the logarithm of the fuglede kadison determinant of f for the proof we establish an p version of rufus bowen s denition of topological entropy addition formulas for group extensions of countable amenable group actions and an approximation formula for the fuglede kadison determinant of f in terms of the determinants of perturbations of the compressions of f

on the mathematical properties of the structural similarity index since its introduction in 2004 the structural similarity ssim index has gained widespread popularity as a tool to assess the quality of images and to evaluate the performance of image processing algorithms and systems there has been also a growing interest of using ssim as an objective function in optimization problems in a variety of image processing applications one major issue that could strongly impede the progress of such efforts is the lack of understanding of the mathematical properties of the ssim measure for example some highly desirable properties such as convexity and triangular inequality that are possessed by the mean squared error may not hold in this paper we first construct a series of normalized and generalized vector valued metrics based on the important ingredients of ssim we then show that such modified measures are valid distance metrics and have many useful properties among which the most significant ones include quasi convexity a region of convexity around the minimizer and distance preservation under orthogonal or unitary transformations the groundwork laid here extends the potentials of ssim in both theoretical development and practical applications

causal inference using the algorithmic markov condition inferring the causal structure that links n observables is usually based upon detecting statistical dependences and choosing simple graphs that make the joint measure markovian here we argue why causal inference is also possible when the sample size is one we develop a theory how to generate causal graphs explaining similarities between single objects to this end we replace the notion of conditional stochastic independence in the causal markov condition with the vanishing of conditional algorithmic mutual information and describe the corresponding causal inference rules we explain why a consistent reformulation of causal inference in terms of algorithmic complexity implies a new inference principle that takes into account also the complexity of conditional probability densities making it possible to select among markov equivalent causal graphs this insight provides a theoretical foundation of a heuristic principle proposed in earlier work we also sketch some ideas on how to replace kolmogorov complexity with decidable complexity criteria this can be seen as an algorithmic analog of replacing the empirically undecidable question of statistical independence with practical independence tests that are based on implicit or explicit assumptions on the underlying distribution

rate distortion and denoising of individual data using kolmogorov complexity we examine the structure of families of distortion balls from the perspective of kolmogorov complexity special attention is paid to the canonical rate distortion function of a source word which returns the minimal kolmogorov complexity of all distortion balls containing that word subject to a bound on their cardinality this canonical rate distortion function is related to the more standard algorithmic rate distortion function for the given distortion measure examples are given of list distortion hamming distortion and euclidean distortion the algorithmic rate distortion function can behave differently from shannon s rate distortion function to this end we show that the canonical rate distortion function can and does assume a wide class of shapes unlike shannon s we relate low algorithmic mutual information to low kolmogorov complexity and consequently suggest that certain aspects of the mutual information formulation of shannon s rate distortion function behave differently than would an analogous formulation using algorithmic mutual information we explore the notion that low kolmogorov complexity distortion balls containing a given word capture the interesting properties of that word which is hard to formalize in shannon s theory and this suggests an approach to denoising

extracting information is hard a turing degree of non integral effective hausdorff dimension abstract we construct a 2 0 infinite binary sequence with effective hausdorff dimension 1 2 that does not compute a sequence of higher dimension introduced by lutz effective hausdorff dimension can be viewed as a measure of the information density of a sequence in particular the dimension of a 2 is the lim inf of the ratio between the information content and length of initial segments of a thus the main result demonstrates that it is not always possible to extract information from a partially random source to produce a sequence that has higher information density

genome rearrangements and sorting by reversals sequence comparison in molecular biology is in the beginning of a major paradigm shift a shift from gene comparison based on local mutations i e insertions deletions and substitutions of nucleotides to chromosome comparison based on global rearrangements i e inversions and transpositions of fragments the classical methods of sequence comparison do not work for global rearrangements and little is known in computer science about the edit distance between sequences if global rearrangements are allowed in the simplest form the problem of gene rearrangements corresponds to sorting by reversals i e sorting of an array using reversals of arbitrary fragments recently kececioglu and sankoff gave the first approximation algorithm for sorting by reversals with guaranteed error bound 2 and identified open problems related to chromosome rearrangements one of these problems is gollan s conjecture on the reversal diameter of the symmetric group this paper proves the conjecture further the problem of expected reversal distance between two random permutations is investigated the reversal distance between two random permutations is shown to be very close to the reversal diameter thereby indicating that reversal distance provides a good separation between related and nonrelated sequences in molecular evolution studies the gene rearrangement problem forces us to consider reversals of signed permutations as the genes in dna could be positively or negatively oriented an approximation algorithm for signed permutation is presented which provides a performance guarantee of 3 over 2 finally using the signed permutations approach an approximation algorithm for sorting by reversals is described which achieves a performance guarantee of 7 over 4

generative models for similarity based classification a maximum entropy approach to generative similarity based classifiers model is proposed first a descriptive set of similarity statistics is assumed to be sufficient for classification then the class conditional distributions of these descriptive statistics are estimated as the maximum entropy distributions subject to empirical moment constraints the resulting exponential class conditional distributions are used in a maximum a posteriori decision rule forming the similarity discriminant analysis sda classifier simulated and real data experiments compare performance to the k nearest neighbor classifier the nearest centroid classifier and the potential support vector machine psvm

a simpler and faster 1 5 approximation algorithm for sorting by transpositions an important problem in genome rearrangements is sorting permutations by transpositions the complexity of the problem is still open and two rather complicated 1 5 approximation algorithms for sorting linear permutations are known bafna and pevzner 98 and christie 99 the fastest known algorithm is the quadratic algorithm of bafna and pevzner in this paper we observe that the problem of sorting circular permutations by transpositions is equivalent to the problem of sorting linear permutations by transpositions hence all algorithms for sorting linear permutations by transpositions can be used to sort circular permutations our main result is a new o n3 2 log n 1 5 approximation algorithm which is considerably simpler than the previous ones and whose analysis is significantly less involved

symmetric diagnostics for the analysis of the residuals in regression models summary typical alternative hypotheses in the analysis of residuals of a standard regression model are considered and for each one a bayesian diagnostic based on a symmetric form of the kullback leibler divergence is determined the results include an explicit expression for the diagnostic when the alternative hypothesis is that the errors are generated by an unknown distribution function with a dirichlet process prior this expression is immediately interpretable exactly computable and endowed with important asymptotic connections a linear approximation of the diagnostic reveals close links with the class of lagrange multiplier test statistics when the alternative hypothesis is that the errors are generated by an autoregressive process the linear approximation is proportional to the box pierce statistic or to the ljung box statistic according to the characteristics of the prior if the observations have zero mean it depends on the durbin watson statistic if the errors are first order autoregressive and it is related to the cliff ord statistic if they are generated by a first order spatial autoregression the sensitivity to the prior of the diagnostic and of its linear approximation is also discussed

k theoretic amenability for sl2 qp and the action on the associated tree abstract it is proved that if a locally compact group g acts simplicially on a tree in such a way that the stabilizers of the vertices are amenable then g is k amenable in particular the canonical map from the full c algebra onto the reduced c algebra of g induces isomorphisms in k theory the main corollary of our result is that sl 2 q p and some other groups over local fields are k amenable

the large sample correspondence between classical hypothesis tests and bayesian posterior odds tests this paper establishes a correspondence in large samples between classical hypothesis tests and bayesian posterior odds tests for models without trends more specifically tests of point null hypotheses and one or two sided alternatives are considered where nuisance parameters may be present under both hypotheses it is shown that for certain priors the bayesian posterior odds test is equivalent in large samples to classical wald lagrange multiplier and likelihood ratio tests for some significance level and vice versa the priors considered under the alternative hypothesis are taken to shrink to the null hypothesis at rate n superscript 1 2 as the sample size n increases copyright 1994 by the econometric society

successor rules for flipping pancakes and burnt pancakes a stack of n pancakes can be rearranged in all n ways by a sequence of n 1 flips and a stack of n burnt pancakes can be rearranged in all 2 n n ways by a sequence of 2 n n 1 flips in both cases a computer program can efficiently generate suitable solutions we approach these tasks instead from a human perspective how can we determine the next flip directly from the current stack how can we flip the minimum or maximum number of burnt pancakes overall what if we are only allowed to flip the top n 2 n 1 or n burnt pancakes we answer the first question with simple successor rules that take worst case o n time and amortized o 1 time then we answer the second question exactly for minimization and provide conjectures for maximization for the third question we prove that solutions almost certainly exist for pancakes and burnt pancakes using only these three flips more broadly we discuss how efficiency and optimality can shape iterative solutions to hamilton cycle problems in highly symmetric graphs

discussion on the meeting on statistical modelling and analysis of genetic data david j balding imperial college school of medicine london i extendmy apologies to the authors that an unavoidable commitment arising unexpectedly in the 18 hours before the meeting robbed me of my final preparation so that my comments at the meeting were poorly presented i shall try to do a better job in this written version and to leave enough space i shall omit my introductory comments about the role of statisticians in bioinformatics

some remarks on the entropy for algebraic actions of amenable groups in this short note we study the entropy for algebraic actions of certain amenable groups the possible values for this entropy are studied various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant we provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to these results extend previous work by lind and ward on p adic entropy

the similarity metric a new class of distances appropriate for measuring similarity relations between sequences say one type of similarity per distance is studied we propose a new normalized information distance based on the noncomputable notion of kolmogorov complexity and show that it is in this class and it minorizes every computable distance in the class that is it is universal in that it discovers all computable similarities we demonstrate that it is a metric and call it the similarity metric this theory forms the foundation for a new practical tool to evidence generality and robustness we give two distinctive applications in widely divergent areas using standard compression programs like gzip and gencompress first we compare whole mitochondrial genomes and infer their evolutionary history this results in a first completely automatic computed whole mitochondrial phylogeny tree secondly we fully automatically compute the language tree of 52 different languages

an extension of the burrows wheeler transform we describe and highlight a generalization of the burrows wheeler transform bwt to a multiset of words the extended transformation denoted by ebwt is reversible moreover it allows to define a bijection between the words over a finite alphabet a and the finite multisets of conjugacy classes of primitive words in a besides its mathematical interest the extended transform can be useful for applications in the context of string processing in the last part of this paper we illustrate one such application providing a similarity measure between sequences based on ebwt

automatic upright orientation and good view recognition for 3d man made models many graphics applications require that 3d models automatically be presented upright and from a good view we propose a method that simultaneously recognizes upright orientation and good view for 3d man made models the strategy is to determine the best base on which a 3d model can stand upright from a small set of candidate bases every candidate base is composed of clustered facets of the simplified convex hull of the given 3d model next a proposed uv measurement selects the best base from the candidate bases using weighted feature based evaluation functions based on geometrical physical and visual aspects our method has been tested using a public 3d model database and compared with previous methods as experimental results show our method outperforms previous work in both efficiency and accuracy

exact expression for information distance information distance can be defined not only between two strings but also in a finite multiset of strings of cardinality greater than two we determine a best upper bound on the information distance it is exact since the upper bound on the information distance for all multisets is the same as the lower bound for infinitely many multisets of each of infinitely many cardinalities up to a constant additive term

hyperlinearity essentially free actions and l2 invariants the sofic property we prove that connes embedding conjecture holds for the von neumann algebras of sofic groups that is sofic groups are hyperlinear hence we provide some new examples of hyperlinearity we also show that the determinant conjecture holds for sofic groups as well we introduce the notion of essentially free actions and amenable actions and study their properties

conditional complexity and codes let x and y be binary strings we prove that there exists a program p of size about k x y that maps y tox and has small complexity when x is known k p x 0 8776 0 having in mind the parallelism between shannon information theory and algorithmic information theory one can say that this result is parallel to wolf slepian and korner csiszar marton theorems see i csiszar and j korner information theory coding theorems for discrete memoryless systems akademiai kiado budapest 1981 we show also that for any three stringsx y z of length at most n the length of the shortest program p that maps both y and z to x i e p y p z x equals max k x y k x z o logn

universal divergence estimation for finite alphabet sources this paper studies universal estimation of divergence from the realizations of two unknown finite alphabet sources two algorithms that borrow techniques from data compression are presented the first divergence estimator applies the burrows wheeler block sorting transform to the concatenation of the two realizations consistency of this estimator is shown for all finite memory sources the second divergence estimator is based on the context tree weighting method consistency is shown for all sources whose memory length does not exceed a known bound experimental results show that both algorithms perform similarly and outperform string matching and plug in methods

asymptotics of discrete mdl for online prediction minimum description length mdl is an important principle for induction and prediction with strong relations to optimal bayesian learning this paper deals with learning processes which are independent and identically distributed i i d by means of two part mdl where the underlying model class is countable we consider the online learning framework i e observations come in one by one and the predictor is allowed to update its state of mind after each time step we identify two ways of predicting by mdl for this setup namely a static and a dynamic one a third variant hybrid mdl will turn out inferior we will prove that under the only assumption that the data is generated by a distribution contained in the model class the mdl predictions converge to the true values almost surely this is accomplished by proving finite bounds on the quadratic the hellinger and the kullback leibler loss of the mdl learner which are however exponentially worse than for bayesian prediction we demonstrate that these bounds are sharp even for model classes containing only bernoulli distributions we show how these bounds imply regret bounds for arbitrary loss functions our results apply to a wide range of setups namely sequence prediction pattern classification regression and universal induction in the sense of algorithmic information theory among others

transforming cabbage into turnip polynomial algorithm for sorting signed permutations by reversals genomes frequently evolve by reversals r i j that transform a gene order 1 i i 1 j 1 j n into 1 i j 1 i 1 j n reversal distance between permutations and sis the minimum number of reversals to transform into a analysis of genome rearrangements in molecular biology started in the late 1930 s when dobzhansky and sturtevant published a milestone paper presenting a rearrangement scenario with 17 inversions between the species of drosophilia analysis of genomes evolving by inversions leads to a combinatorial problem of sorting by reversals studied in detail recently we study sorting of signed permutations by reversals a problem that adequately models rearrangements in a small genomes like chloroplast or mitochondrial dna the previously suggested approximation algorithms for sorting signed permutations by reversals compute the reversal distance between permutations with an astonishing accuracy for both simulated and biological data we prove a duality theorem explaining this intriguing performance and show that there exists a hidden parameter that allows one to compute the reversal distance between signed permutations in polynomial time

independent minimum length programs to translate between given strings a string p is called a program to compute y given x if u p x y where u denotes universal programming language kolmogorov complexity k y x of y relative to x is defined as minimum length of a program to compute y given x let k x denote k x emptystring kolmogorov complexity of x and let i x y k x k y k x y the amount of mutual information in x y in the present paper we answer in the negative the following question posed in bennett et al ieee trans inform theory 44 4 1998 1407 1423 is it true that for any strings x y there are independent minimum length programs p q to translate between x y that is is it true that for any x y there are p q such that u p x y u q y x the length of p is k y x the length of q is k x y and i p q 0 where the last three equalities hold up to an additive o log k x y k y x term

an efficient algorithm for one sided block ordering problem under block interchange distance abstract in this work we study the one sided block ordering problem under block interchange distance given two signed permutations and of size n where represents a partially assembled genome consisting of several blocks i e contigs and represents a completely assembled genome the one sided block ordering problem under block interchange distance is to order i e assemble the blocks of such that the block interchange distance between the assembly of and is minimized the one sided block ordering problem is useful in genome resequencing because its algorithms can be used to assemble the contigs of partially assembled resequencing genomes based on their completely assembled genomes by using permutation groups in algebra we design an efficient algorithm to solve the one sided block ordering problem under block interchange distance in o n log n time moreover we show that the assembly of can be done in o n time and its block interchange distance from can also be calculated in advance in o n time

the normalized compression distance is resistant to noise this correspondence studies the influence of noise on the normalized compression distance ncd a measure based on the use of compressors to compute the degree of similarity of two files this influence is approximated by a first order differential equation which gives rise to a complex effect which explains the fact that the ncd may give values greater than 1 observed by other authors the model is tested experimentally with good adjustment finally the influence of noise on the clustering of files of different types is explored finding that the ncd performs well even in the presence of quite high noise levels

erratum to integrality of l2 betti numbers a gap has been found in the proof of proposition 3 of 3 consequently this result and the consequences drawn in the paper have to be considered as conjectures in particular it is not proved that the atiyah conjecture is stable under extensions with free quotient groups and under free products with free groups

a faster and simpler algorithm for sorting signed permutations by reversals we give a quadratic time algorithm for finding the minimum number of reversals needed to sort a signed permutation our algorithm is faster than the previous algorithm of hannenhalli and pevzner and its faster implementation by berman and hannenhalli the algorithm is conceptually simple and does not require special data structures our study also considerably simplifies the combinatorial structures used by the analysis

a fast quartet tree heuristic for hierarchical clustering the minimum quartet tree cost problem is to construct an optimal weight tree from the 3 n4 weighted quartet topologies on n objects where optimality means that the summed weight of the embedded quartet topologies is optimal so it can be the case that the optimal tree embeds all quartets as nonoptimal topologies we present a monte carlo heuristic based on randomized hill climbing for approximating the optimal weight tree given the quartet topology weights the method repeatedly transforms a dendrogram with all objects involved as leaves achieving a monotonic approximation to the exact single globally optimal tree the problem and the solution heuristic has been extensively used for general hierarchical clustering of nontree like non phylogeny data in various domains and across domains with heterogeneous data we also present a greatly improved heuristic reducing the running time by a factor of order a thousand to ten thousand all this is implemented and available as part of the complearn package we compare performance and running time of the original and improved versions with those of upgma bionj and nj as implemented in the splitstree package on genomic data for which the latter are optimized

on the complexity of unsigned translocation distance translocation is one of the basic operations for genome rearrangement translocation distance is the minimum number of translocations required to transform one genome into the other in this paper we show that computing the translocation distance for unsigned genomes is np hard moreover we show that approximating the translocation distance for unsigned genomes within ratio 1 00017 is np hard

the evolution of the random reversal graph genomes are typically represented as signed permutations where each integer corresponds to a single gene and its sign indicates the orientation of the gene in this paper we study genome rearrangements via reversals sankoff and blanchette 1999 20 flipping entire segments of genes and changing their orientations this abstraction leads to the notion of the reversal graph in which two signed permutations are neighbors if they differ by one reversal we identify the evolution of multiple genomes within the random reversal graph i e the probability space consisting of subgraphs over signed permutations obtained by selecting edges reversals with independent probability n which is an important framework for the analysis of genome rearrangements in the sense of relating the reversal rate n with the global evolution of genomes our main result shows that the structure of the random reversal graph changes dramatically at n 1 n 1 2 for n 1 n 1 2 the random graph consists of components of size at most o n ln n a s and for 1 n 1 2 there emerges a unique largest component of size 2 n n a s where is the survival probability of a certain branching process this giant component is furthermore dense in the reversal graph

image comparison by compound disjoint information with applications to perceptual visual quality assessment image registration and tracking in this paper we study normalized disjoint information as a metric for image comparison and its applications to perceptual image quality assessment image registration and video tracking disjoint information is the joint entropy of random variables excluding the mutual information this measure of statistical dependence and information redundancy satisfies more rigorous metric conditions than mutual information including self similarity minimality symmetry and triangle inequality it is applicable to two or more random variables and can be computed by vector histogramming vector parzen window density approximation and upper bound approximation involving fewer variables we show such a theoretic advantage does have implications in practice in the domain of digital image and video multiple visual features are extracted and normalized compound disjoint information is derived from a set of marginal densities of the image distributions thus enriching the vocabulary of content representation the proposed metric matching functions are applied to several domain applications to demonstrate their efficacy

polynomial time sortable stacks of burnt pancakes pancake flipping a famous open problem in computer science can be formalised as the problem of sorting a permutation of positive integers using as few prefix reversals as possible in that context a prefix reversal of length k reverses the order of the first k elements of the permutation the burnt variant of pancake flipping involves permutations of signed integers and reversals in that case not only reverse the order of elements but also invert their signs although three decades have now passed since the first works on these problems neither their computational complexity nor the maximal number of prefix reversals needed to sort a permutation is yet known in this work we prove a new lower bound for sorting burnt pancakes and show that an important class of permutations known as simple permutations can be optimally sorted in polynomial time