random k sat two moments suffice to cross a sharp threshold many np complete constraint satisfaction problems appear to undergo a phase transition from solubility to insolubility when the constraint density passes through a critical threshold in all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment expectation of the number of solutions tends to zero we show that in the case of certain symmetric constraints considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold specifically we prove that the threshold for both random hypergraph 2 colorability property b and random not all equal k sat is 2 k 1 ln 2 o 1 as a corollary we establish that the threshold for random k sat is of order theta 2 k resolving a long standing open problem

exact thresholds for dpll on random xor sat and np complete extensions of xor sat this paper discusses a model of constraint satisfaction problems known as uniquely extendible constraint satisfaction problems this model includes and generalizes xor sat and the model includes an np complete problem that appears to share many of the threshold characteristics of random sat in this paper we find an exact threshold in the behavior of two versions of dpll on random instances of this problem one version uses the unit clause heuristic and the other uses the generalized unit clause heuristic specifically for dpll with the unit clause heuristic we prove that there is a clause density c smaller than the satisfiability threshold such that for random instances with density smaller than this threshold dpll with unit clause will find a satisfying assignment in linear time with uniformly positive probability however for random instances with density larger than this threshold dpll with unit clause will require exponential time with uniformly positive probability to find a satisfying assignment we then find the equivalent threshold density for dpll with the generalized unit clause heuristic we also prove the analog of the 2 p sat conjecture for this class of problems

estimating the covariance matrix by signal subspace averaging an efficient algorithm is presented for estimating a covariance matrix consisting of a low rank signal term and a full rank noise term known apart from a scalar factor for each sample of the vector of sensor outputs the algorithm approximates in the least squares sense a rank one update of the covariance matrix under the side condition that the rank of the signal term remains bounded if the model noise is spatially colored the least squares approximation is preceded by spatial prewhitening it is shown that if the rank of the signal term is small compared to the number of sensors then the proposed algorithm requires substantially less computational work than conventional averaging some simulation results are included indicating that the proposed algorithm reduces the variance of some commonly used spectral estimators in off target directions without impairing their detection and resolution properties

gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

probabilistic analysis of two heuristics for the 3 satisfiability problem an algorithm for the 3 satisfiability problem is presented and a probabilistic analysis is performed the analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics the algorithm assigns values to variables appearing in a given instance of 3 satisfiability one at a time using the unit clause heuristic and a maximum occurring literal selection heuristic at each step a variable is chosen randomly from a subset of variables which is usually large the algorithm runs in polynomial time and it is shown that the algorithm finds a solution to a random instance of 3 satisfiability with probability bounded from below by a constant greater than zero for a range of parameter values the heuristics studied here can be used to select variables in a backtrack algorithm for 3 satisfiability experiments have shown that for about the same range of parameters as above the backtrack algorithm using the heuristics finds a solution in polynomial average time

probabilistic analysis of a generalization of the unit clause literal selection heuristics for the k satisfiability problem abstract two algorithms for the k satisfiability problem are presented and a probabilistic analysis is performed the analysis is based on an instance distribution which is parametrized to simulate a variety of sample characteristics the algorithms assign values to literals appearing in a given instance of k satisfiability one at a time until a solution is found or it is discovered that further assignments cannot lead to finding a solution one algorithm chooses the next literal from a unit clause if one exists and randomly from the set of remaining literals otherwise the other algorithm uses a generalization of the unit clause rule as a heuristic for selecting the next literal at each step a literal is chosen randomly from a clause containing the least number of literals the algorithms run in polynomial time and it is shown that they find a solution to a random instance of k satisfiability with probability bounded below by a constant greater than zero for two different ranges of parameter values it is also shown that the second algorithm mentioned finds a solution with probability approaching one for a wide range of parameter values

fast and stable subspace tracking we consider the problem of adaptive subspace tracking when the rank of the subspace we seek to estimate is assumed to be known starting from the data projection method dpm which constitutes a simple and reliable means for adaptively estimating and tracking subspaces we develop a fast and numerically robust implementation of dpm which comes at a considerably lower computational cost most existing schemes track subspaces corresponding either to the largest or to the smallest singular values while our dpm version can switch from one subspace type to the other with a simple change of sign of its single parameter the proposed algorithm provides orthonormal vector estimates of the subspace basis that are numerically stable since they do not accumulate roundoff errors in fact our scheme constitutes the first numerically stable low complexity algorithm for tracking subspaces corresponding to the smallest singular values regarding convergence towards orthonormality our scheme exhibits the fastest speed among all other subspace tracking algorithms of similar complexity

an exactly solvable random satisfiability problem we introduce a new model for the generation of random satisfiability problems it is an extension of the hyper sat model of ricci tersenghi weigt and zecchina which is a variant of the famous k sat model it is extended to q state variables and relates to a different choice of the statistical ensemble the model has an exactly solvable statistic the critical exponents and scaling functions of the sat unsat transition are calculable at zero temperature with no need of replicas also with exact finite size corrections we also introduce an exact duality of the model and show an analogy of thermodynamic properties with the random energy model of disordered spin system theory relations with error correcting codes are also discussed

a novel orthonormalization matrix based fast and stable dpm algorithm for principal and minor subspace tracking we note that the well known fast rayleigh s quotient based adaptive noise subspace frans frans with householder transformation hfrans and fast data projection method fdpm algorithms all inherit from the data projection method dpm algorithm but with different orthonormalization matrices starting from the dpm we analyze the orthonormalization matrices of all these algorithms and develop a novel orthonormalization matrix for our algorithm based on this novel orthonormalization matrix a fast and stable implementation of the dpm algorithm which has the merits of both the frans and fdpm approaches is investigated for principal and minor subspace tracking the proposed algorithm can switch between the principal and minor subspace tracking with a simple sign change of its step size parameter moreover it reaches the 3np lower bound of the dominant complexity and guarantees the orthonormality of the tracked subspace the numerical stability of our algorithm is established theoretically and tested numerically the strengths and weaknesses of the proposed algorithm to some existing subspace tracking algorithms are demonstrated using a de facto benchmark example simulation results are presented to demonstrate the effectiveness of the tracking algorithm advocated

subspace learning and imputation for streaming big data matrices and tensors extracting latent low dimensional structure from high dimensional data is of paramount importance in timely inference tasks encountered with big data analytics however increasingly noisy heterogeneous and incomplete datasets as well as the need for real time processing of streaming data pose major challenges to this end in this context the present paper permeates benefits from rank minimization to scalable imputation of missing data via tracking low dimensional subspaces and unraveling latent possibly multi way structure from incomplete streaming data for low rank matrix data a subspace estimator is proposed based on an exponentially weighted least squares criterion regularized with the nuclear norm after recasting the nonseparable nuclear norm into a form amenable to online optimization real time algorithms with complementary strengths are developed and their convergence is established under simplifying technical assumptions in a stationary setting the asymptotic estimates obtained offer the well documented performance guarantees of the batch nuclear norm regularized estimator under the same unifying framework a novel online adaptive algorithm is developed to obtain multi way decompositions of low rank tensors with missing entries and perform imputation as a byproduct simulated tests with both synthetic as well as real internet and cardiac magnetic resonance imagery mri data confirm the efficacy of the proposed algorithms and their superior performance relative to state of the art alternatives

source coding by efficient selection of ground state clusters we analyze the geometrical structure of clusters of ground states which appear in many frustrated systems over random graphs focusing on the regime of connectivities where the number of clusters is exponential in the size of the problems we identify an appropriate generalization of the survey propagation equations efficiently exploring the geometry the possibility of selecting different clusters has also computational consequences as a proof of concept here we show how a well known physical system can be used to perform nontrivial data compression for which we introduce a unique compression scheme performances are optimized when the number of well separated clusters is maximal in the underlying physical model

near optimal configurations in mean field disordered systems we present a general technique to compute how the energy of a configuration varies as a function of its overlap with the ground state in the case of optimization problems our approach is based on a generalization of the cavity method to a system interacting with its ground state with this technique we study the random matching problem as well as the mean field diluted spin glass as a by product of this approach we calculate the de almeida thouless transition line of the spin glass on a fixed connectivity random graph

paper tracking error bounds of adaptive nonstationary filtering adaptive filtering with error gradient algorithm and constant step size is analyzed for a deterministic time variable optimum filtering vector the unrealistic assumption of independent observations is replaced by a bounded memory model largely justifiable in applications then the mean square tracking deviation msd between the optimum vector and the algorithm output is proved to include two contributions the stationary mode error characteristic of convergence accuracy which is proportional to the step size and the transient mode error reflecting the rapidity of tracking which is proportional to the squared ratio of the maximum optimum estimator increment to the step size this result agrees with the common intuition that there exists an optimum step size which compromises between convergence accuracy and tracking speed

statistical mechanics of maximal independent sets the graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory a maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints it is known that finding minimum and maximum density maximal independent sets are hard optimization problems in this paper we use cavity method of statistical physics and monte carlo simulations to study the corresponding constraint satisfaction problem on random graphs we obtain the entropy of maximal independent sets within the replica symmetric and one step replica symmetry breaking frameworks shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms this is of particular relevance for the application to the study of strategic interactions in social and economic networks where maximal independent sets correspond to pure nash equilibria of a graphical game of public goods allocation

many hard examples for resolution for every choice of positive integers c and k such that k 3 and c 2 k 0 7 there is a positive number e such that with probability tending to 1 as n tends to a randomly chosen family of cn clauses of size k over n variables is unsatisfiable but every resolution proof of its unsatisfiability must generate at least 1 e n clauses

rigorous results for random 2 p sat in recent years there has been significant interest in the study of random k sat formulae for a given set of n boolean variables let bk denote the set of all possible disjunctions of k distinct non complementary literals from its variables k clauses a random k sat formula fk n m is formed by selectinguniformly and independently m clauses from bk and takingtheir conjunction motivated by insights from statistical mechanics that suggest a possible relationship between the order of phase transitions and computational complexity monasson and zecchina phys rev e 56 2 1997 1357 proposed the random 2 p sat model for a given p 0 1 a random 2 p sat formula f2 p n m has m randomly chosen clauses over n variables where pm clauses are chosen from b3 and 1 p m from b2 usingthe heuristic replica method of statistical mechanics monasson and zecchina gave a number of non rigorous predictions on the behavior of random 2 p sat formulae in this paper we give the 1rst rigorous results for random 2 p sat including the following surprising fact for p 2 5 with probability 1 o 1 a random 2 p sat formula is satisfiable if its 2 sat subformula is satisfiable that is for p 6 2 5 random 2 p sat behaves like random 2 sat

spanning trees in random satisfiability problems working with tree graphs is always easier than with loopy ones and spanning trees are the closest tree like structures to a given graph we find a correspondence between the solutions of random k satisfiability problem and those of spanning trees in the associated factor graph we introduce a modified survey propagation algorithm which returns null edges of the factor graph and helps us to find satisfiable spanning trees this allows us to study organization of satisfiable spanning trees in the space spanned by spanning trees

application of degenerate perturbation theory to subspace tracking first order perturbation theory has been applied to subspace tracking this previous work did not solve the problem of signal subspace near degeneracy the signal subspace is nearly degenerate when two or more signal subspace eigenvalues are nearly equal which can occur under a number of circumstances failure to correctly handle the near degeneracy leads to errors in the subspace decomposition we demonstrate here the application of degenerate perturbation theory to signal subspace tracking under conditions where the eigenvalues may be degenerate the problems of noise subspace degeneracy and signal subspace near degeneracy are both solved through diagonalization of the perturbation matrix over the degenerate subspace the degenerate subspace diagonalization markedly reduces the subspace error with a minimal increase in computational complexity similarly it is shown that using a second order perturbation update reduces the subspace error by a substantial margin again with a minimal increase in computational complexity the effect of increasing the integration time is also examined by using the second order perturbation update and the degenerate subspace diagonalization the resulting subspace decomposition is extremely accurate robust and fast

fluctuations in the random link matching problem using the replica approach and the cavity method we study the fluctuations of the optimal cost in the random link matching problem by means of replica arguments we derive the exact expression of its variance moreover we study the large deviation function deriving its expression in two different ways namely using both the replica method and the cavity method

descendant invariants and characteristic numbers on a stack of stable maps the cotangent line classes are modified by subtracting certain boundary divisors these modified cotangent line classes are compatible with forgetful morphisms and are well suited to enumerative geometry tangency conditions allow simple expressions in terms of modified cotangent line classes topological recursion relations are established among their top products in genus 0 yielding effective recursions for characteristic numbers of rational curves in any projective homogeneous variety in higher genus the obtained numbers are only virtual due to contributions from spurious components of the space of maps for the projective plane the necessary corrections are determined in genus 1 and 2 to give the characteristic numbers in these cases

a change of coordinates on the large phase space of quantum cohomology the gromov witten invariants of a smooth projective variety v when twisted by the tautological classes on the moduli space of stable maps give rise to a family of cohomological field theories and endow the base of the family with coordinates we prove that the potential functions associated to the tautological classes the large phase space and the classes are related by a change of coordinates which generalizes a change of basis on the ring of symmetric functions our result is a generalization of the work of manin zograf who studied the case where v is a point we utilize this change of variables to derive the topological recursion relations associated to the classes from those associated to the classes

plane rotation based evd updating schemes for efficient subspace tracking we present new algorithms based on plane rotations for tracking the eigenvalue decomposition evd of a time varying data covariance matrix these algorithms directly produce eigenvectors in orthonormal form and are well suited for the application of subspace methods to nonstationary data after recasting evd tracking as a simplified rank one evd update problem computationally efficient solutions are obtained in two steps first a new kind of parametric perturbation approach is used to express the eigenvector update as an unimodular orthogonal transform which is represented in exponential matrix form in terms of a reduced set of small unconstrained parameters second two approximate decompositions of this exponential matrix into products of plane or givens rotations are derived one of which being previously unknown these decompositions lead to new plane rotation based evd updating schemes proteus whose main feature is the use of plane rotations for updating the eigenvectors thereby preserving orthonormality finally the proteus schemes are used to derive new evd trackers whose convergence and numerical stability are investigated via simulations one algorithm can track all the signal subspace evd components in only o lm operations where l and m respectively denote the data vector and signal subspace dimensions while achieving a performance comparable to an exact evd approach and maintaining perfect orthonormality of the eigenvectors the new algorithms show no signs of error buildup

super solutions of random 3 p sat abstract in this paper we propose a model named random 3 p sat where a random formula n r p contains n variables and rn clauses and 1 p r n of the clauses are 3 clauses and prn of the clauses are 4 clauses this paper studies the 1 0 satisfiability of random 3 p sat and obtains rigorous results that the exact 1 0 satisfiability threshold is r p 1 3 1 p if p 3 7 for p 3 7 we give lower and upper bounds of the 1 0 satisfiability threshold where the lower bound is obtained by using the unit clause algorithm and the upper bound is obtained by using a novel way to count precisely the subset of all 1 0 satisfying assignments that satisfy a locally maximum condition

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

adaptive eigensubspace algorithms for direction or frequency estimation and tracking the authors present an adaptive estimator of the complete noise or signal subspace of a sample covariance matrix as well as the estimator s practical implementations the general formulation of the proposed estimator results from an asymptotic argument which shows the signal or noise subspace computation to be equivalent to a constrained gradient search procedure a highly parallel algorithm denoted the inflation method is introduced for the estimation of the noise subspace the simulation results of these adaptive estimators show that the adaptive subspace algorithms perform substantially better than p a thompson s 1980 adaptive version of v f pisarenko s technique 1973 in estimating frequencies or directions of arrival doa of plane waves for tracking nonstationary parameters the simulation results also show that the adaptive subspace algorithms are better than direct eigendecomposition methods for which computational complexity is much higher than the adaptive versions

chow groups of quotient varieties let g be a reductive algebraic group operating on a scheme x with quotient y x g let ch x be the chow group of x ch y the chow group of y as defined in f in many concrete cases e g binary quantics or certain subspaces of moduli spaces of curves x has a simple structure and we have a lot of information on ch x while ch y is more interesting and much harder to calculate so it is natural to ask what relation is there between ch x and ch y in this article we give a partial answer to this question with the following restrictions we assume that the stabilizer of each point of x is finite and we only consider chow groups with rational coefficients furthermore we assume that x is an algebraic scheme over an algebraically closed field of characteristic 0 set a ch q where q is the field of rational numbers if g is a finite group then a y a x see f example 1 7 61 thus it is only necessary to consider the case in which g is connected theorem 1 says that in this case there is a natural surjective homomorphism a y a x this means that a y is at least as large as a x to be able to bound the kernel of the homomorphism a y a x we make two further assumptions that x is normal and that g cl n or sl n the case g sl n includes the case g pgl n which the one usually found in applications the case of a principal gl n bundle is treated in sga6 exposi ix proposition 4 11 considering the k group of coherent sheaves instead of the chow group what is proved there amounts in the present contest to the following let c c be the chern classes of the rank n vector bundle

experimental results on the crossover point in random 3 sat abstract determining whether a propositional theory is satisfiable is a prototypical example of an np complete problem further a large number of problems that occur in knowledge representation learning planning and other areas of ai are essentially satisfiability problems this paper reports on the most extensive set of experiments to date on the location and nature of the crossover point in satisfiability problems these experiments generally confirm previous results with two notable exceptions first we have found that neither of the functions previously proposed accurately models the location of the crossover point second we have found no evidence of any hard problems in the under constrained region in fact the hardest problems found in the under constrained region were many times easier than the easiest unsatisfiable problems found in the neighborhood of the crossover point we offer explanations for these apparent contradictions of previous results

many hard examples in exact phase transitions this paper analyzes the resolution complexity of two random constraint satisfaction problem csp models i e model rb rd for which we can establish the existence of phase transitions and identify the threshold points exactly by encoding csps into cnf formulas it is proved that almost all instances of model rb rd have no tree like resolution proofs of less than exponential size thus we not only introduce new families of csps and cnf formulas hard to solve which can be useful in the experimental evaluation of csp and sat algorithms but also propose models with both many hard instances and exact phase transitions finally conclusions are presented as well as a detailed comparison of model rb rd with the hamiltonian cycle problem and random 3 sat which respectively exhibit three different kinds of phase transition behavior in np complete problems

the gromov witten potential of a point hurwitz numbers and hodge integrals hurwitz numbers which count certain covers of the projective line or equivalently factorizations of permutations into transpositions have been extensively studied for over a century the gromov witten potential f of a point the generating series for descendent integrals on the moduli space of curves is a central object of study in gromov witten theory we define a slightly enriched gromov witten potential g including integrals involving one lambda class and show that after a non trivial change of variables g h in positive genus where h is a generating series for hurwitz numbers we prove a conjecture of goulden and jackson on higher genus hurwitz numbers which turns out to be an analogue of a genus expansion ansatz of itzykson and zuber as consequences we have new combinatorial constraints on f and a much more direct proof of the ansatz of itzykson and zuber we can produce recursions and explicit formulas for hurwitz numbers the algorithm presented proves all such recursions as examples we present surprisingly simple new recursions in genus 0 to 3 similar recursions should exist for all genera as we expect this paper also to be of interest to combinatorialists we have tried to make it as self contained as possible including reviewing some results and definitions well known in algebraic and symplectic geometry and mathematical physics 2000 mathematical subject classification primary 14h10 81t40 secondary 05c30 58d29

is sp bp the survey propagation sp algorithm for solving k sat problems has been shown recently as an instance of the belief propagation bp algorithm in this paper we show that for general constraint satisfaction problems sp may not be reducible from bp we also establish the conditions under which such a reduction is possible along our development we present a unification of the existing sp algorithms in terms of a probabilistically interpretable iterative procedure weighted probabilistic token passing

two solutions to diluted p spin models and xorsat problems we derive analytical solutions for p spin models with finite connectivity at zero temperature these models are the statistical mechanics equivalent of p xorsat problems in theoretical computer science we give a full characterization of the phase diagram location of the phase transitions static and dynamic together with a description of the clustering phenomenon taking place in configurational space we use two alternative methods the cavity approach and a rigorous derivation

the structure of the tautological ring in genus one we prove getzler s claims about the cohomology of the moduli space of stable curves of genus one that is that the even cohomology ring is spanned by the strata classes and that all relations between these classes follow from the witten dijkgraaf verlinde verlinde wdvv relation and getzler s relation in particular the even cohomology ring is isomorphic to the tautological ring

generalized satisfiability problems minimal elements and phase transitions we develop a probabilistic model on the generalized satisfiability problems defined by schaefer in proceedings of the 10th stoc san diego ca usa association for computing machinery new york 1978 pp 216 226 for which the arity of the constraints is fixed in order to study the associated phase transition we establish new results on minimal elements associated with such generalized satisfiability problems these results are the keys of the exploration we conduct on the location and on the nature of the phase transition for generalized satisfiability we first prove that the phase transition occurs at the same scale for every reasonable problem and we provide lower and upper bounds for the associated critical ratio our framework allows one to get these bounds in a uniform way in particular we obtain a lower bound proportional to the number of variables for k sat without analyzing any algorithm finally we reveal the seed of coarseness for the phase transition of generalized satisfiability 2 xor sat

inference by replication in densely connected systems an efficient bayesian inference method for problems that can be mapped onto dense graphs is presented the approach is based on message passing where messages are averaged over a large number of replicated variable systems exposed to the same evidential nodes an assumption about the symmetry of the solutions is required for carrying out the averages here we extend the previous derivation based on a replica symmetric rs like structure to include a more complex one step replica symmetry breaking like 1rsb like ansatz to demonstrate the potential of the approach it is employed for studying critical properties of the ising linear perceptron and for multiuser detection in code division multiple access cdma under different noise models results obtained under the rs assumption in the noncritical regime give rise to a highly efficient signal detection algorithm in the context of cdma while in the critical regime one observes a first order transition line that ends in a continuous phase transition point finite size effects are also observed while the 1rsb ansatz is not required for the original problems it was applied to the cdma signal detection problem with a more complex noise model that exhibits rsb behavior resulting in an improvement in performance

a modified belief propagation algorithm for code word quantization modern coding advances including dirty paper coding and information hiding require quantizing a given binary word to a code word a good solution would approach the rate distortion bound in lossy source compression here we propose a simple variant on belief propagation which is observed to converge to a solution giving respectable rate distortion performance comparisons with other recently proposed source quantization methods reveal that the proposed algorithm holds particular interest in short block length applications as encountered in packet based communication systems

constructing an asymptotic phase transition in random binary constraint satisfaction problems the standard models used to generate random binary constraint satisfaction problems are described at the problem sizes studied experimentally a phase transition is seen as the constraint tightness is varied however achlioptas et al showed that if the problem size number of variables increases while the remaining parameters are kept constant asymptotically almost all instances are unsatisfiable in this paper an alternative scheme for one of the standard models is proposed in which both the number of values in each variable s domain and the average degree of the constraint graph are increased with problem size it is shown that with this scheme there is asymptotically a range of values of the constraint tightness in which instances are trivially satis able with probability at least 0 5 and a range in which instances are almost all unsatisfiable hence there is a crossover point at some value of the constraint tightness between these two ranges this scheme is compared to a similar scheme due to xu and li

models for random constraint satisfaction problems we introduce a class of models for random constraint satisfaction problems this class includes and generalizes many previously studied models we characterize those models from our class which are asymptotically interesting in the sense that the limiting probability of satisfiability changes significantly as the number of constraints increases we also discuss models which exhibit a sharp threshold for satisfiability in the sense that the limiting probability jumps from 0 to 1 suddenly as the number of constraints increases

bicolouring random hypergraphs we study the problem of bicolouring random hypergraphs both numerically and analytically we apply the zero temperature cavity method to find analytical results for the phase transitions dynamic and static in the one step replica symmetry breaking 1rsb approximation these points appear to be in agreement with the results of the numerical algorithm in the second part we implement and test the survey propagation algorithm for specific bicolouring instances in the so called hard sat phase

noniterative subspace tracking a rank one spherical subspace update that is appropriate for subspace based methods like music and minimum norm is introduced this noniterative highly parallel numerically stabilized subspace update is closely related to rank one eigenstructure updating however a rank one subspace update involves less computation than simple rank one correlation accumulation moreover the frequency tracking capabilities of the noniterative subspace update are virtually identical to and in some case more robust than the more computationally expensive eigen based methods

generating hard satisfiability problems abstract we report results from large scale experiments in satisfiability testing as has been observed by others testing the satisfiability of random formulas often appears surprisingly easy here we show that by using the right distribution of instances and appropriate parameter values it is possible to generate random formulas that are hard that is for which satisfiability testing is quite difficult our results provide a benchmark for the evaluation of satisfiability testing procedures

from zwiebach invariants to getzler relation we introduce the notion of zwiebach invariants that generalize gromov witten invariants and homotopical algebra structures we outline the induction procedure that induces the structure of zwiebach invariants on the sub bicomplex that gives the structure of gromov witten invariants on sub bicomplex with zero differentials we propose to treat hodge dgbv with 1 12 axiom as the simplest set of zwiebach invariants and explicitly prove that it induces wdvv and getzler equations in genera 0 and 1 respectively

hiding quiet solutions in random constraint satisfaction problems we study constraint satisfaction problems on the so called planted random ensemble we show that for a certain class of problems e g graph coloring many of the properties of the usual random ensemble are quantitatively identical in the planted random ensemble we study the structural phase transitions and the easy hard easy pattern in the average computational complexity we also discuss the finite temperature phase diagram finding a close connection with the liquid glass solid phenomenology

the semi classical approximation for modular operads we study the contribution of one loop graphs the semi classical expansion problem in the setting of modular operads as an application we calculate the betti numbers of the deligne mumford knudsen moduli spaces of stable curves of genus 1 with n marked smooth points

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

lower bounds for random 3 sat via differential equations it is widely believed that the probability of satisfiability for random k sat formulae exhibits a sharp threshold as a function of their clauses to variables ratio for the most studied case k 3 there have been a number of results during the last decade providing upper and lower bounds for the threshold s potential location all lower bounds in this vein have been algorithmic i e in each case a particular algorithm was shown to satisfy random instances of 3 sat with probability 1 o 1 if the clauses to variables ratio is below a certain value we show how differential equations can serve as a generic tool for analyzing such algorithms by rederiving most of the known lower bounds for random 3 sat in a simple uniform manner

inference algorithm for finite dimensional spin glasses belief propagation on the dual lattice starting from a cluster variational method and inspired by the correctness of the paramagnetic ansatz at high temperatures in general and at any temperature in the two dimensional 2d edwards anderson ea model we propose a message passing algorithm the dual algorithm to estimate the marginal probabilities of spin glasses on finite dimensional lattices we use the ea models in 2d and 3d as benchmarks the dual algorithm improves the bethe approximation and we show that in a wide range of temperatures compared to the bethe critical temperature our algorithm compares very well with monte carlo simulations with the double loop algorithm and with exact calculation of the ground state of 2d systems with bimodal and gaussian interactions moreover it is usually 100 times faster than other provably convergent methods as the double loop algorithm in 2d and 3d the quality of the inference deteriorates only where the correlation length becomes very large i e at low temperatures in 2d and close to the critical temperature in 3d

replica bounds for diluted non poissonian spin systems in this paper we extend replica bounds and free energy subadditivity arguments to diluted spin glass models on graphs with arbitrary non poissonian degree distribution the new difficulties specific of this case are overcome introducing an interpolation procedure that stresses the relation between interpolation methods and the cavity method as a byproduct we obtain self averaging identities that generalize the ghirlanda guerra ones to the multi overlap case

survey propagation at finite temperature application to a sourlas code as a toy model in this paper we investigate a finite temperature generalization of survey propagation by applying it to the problem of finite temperature decoding of a biased finite connectivity sourlas code for temperatures lower than the nishimori temperature we observe that the result is a shift of the location of the dynamical critical channel noise to larger values than the corresponding dynamical transition for belief propagation as suggested recently by migliorini and saad for ldpc codes we show how the finite temperature 1rsb sp gives accurate results in the regime where competing approaches fail to converge or fail to recover the retrieval state

steady state statistical performance analysis of subspace tracking methods subspace tracking using low complexity methods is a common research area in signal processing while numerical properties of available methods are extensively addressed in literature there is a paucity of statistical results on limiting behavior of subspace tracking methods in this contribution we will study steady state error of dpm malase champagne s pa karasalo and past methods we show that dpm malase and past are approximations of the pa method while karasalo s method is equivalent to the pa method in convergence region a tradeoff between steady state error of methods and their local convergence rate is demonstrated a nonstationary signal model is assumed in which true subspace is slowly and randomly varying simulation results show pa and past are asymptotically close to the optimum sample covariance matrix scm while this is not true for dpm and malase in stationary signal case in the steady state these methods can be expressed with a unified approach using stochastic first order difference equations whose fixed points are calculated the results can be used to determine optimal step size in terms of tolerable total signal subspace projection error simulation results confirm the validity of theoretical error calculations

stable subspace tracking algorithm based on a signed urv decomposition subspace estimation and tracking are of fundamental importance in many signal processing algorithms the class of schur subspace estimators provides a complete parametrization of all principal subspace estimates defined as the column spans of corresponding low rank matrix approximants that lie within a specified 2 norm distance of a given matrix the parametrization is found in terms of a two sided hyperbolic decomposition hyperbolic urv or hurv which can be computed using hyperbolic rotations unfortunately such rotations are commonly associated with numerical instabilities in this paper we present a numerically stable non iterative algorithm to compute the hurv called the signed urv surv algorithm we show that this algorithm implicitly imposes certain constraints on the hurv such that important norm bounds that guarantee stability are satisfied the constraints also restrict the parametrization of the subspace estimate such that it becomes close to the principal subspace provided by the svd which is a special case within this class the complexity of the algorithm is of the same order as that of a qr update updating and downdating are of the same complexity and are both numerically stable surv is proven to provide rank estimates consistent with the svd with the same rank threshold it can replace an svd where only subspace estimation is needed typical applications would e g be the detection of the number of signals in array signal processing and subspace estimation for source separation and interference mitigation such as the first step in music and esprit type algorithms simulation results demonstrate the numerical stability and confirm that this algorithm provides exact rank estimates and good principal subspace estimates as compared to the svd

on the asymptotic convergence and numerical stability of the proteus evd trackers the asymptotic convergence and numerical stability of the previously introduced subspace tracking algorithms proteus 1 and 2 are investigated by means of the ode method it is shown that 1 under weak conditions both algorithms globally converge with probability one to the desired eigenvalue decomposition evd components of the data covariance matrix and 2 they have a built in mechanism that prevents deviation from orthonormality in the eigenvector estimates over long periods of operation i e numerical stability

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

fast approximated power iteration subspace tracking this paper introduces a fast implementation of the power iteration method for subspace tracking based on an approximation that is less restrictive than the well known projection approximation this algorithm referred to as the fast approximated power iteration api method guarantees the orthonormality of the subspace weighting matrix at each iteration moreover it outperforms many subspace trackers related to the power iteration method such as past nic np3 and opast while having the same computational complexity the api method is designed for both exponential windows and sliding windows our numerical simulations show that sliding windows offer a faster tracking response to abrupt signal variations

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

the resolution complexity of random constraint satisfaction problems we consider random instances of constraint satisfaction problems where each variable has domain size d and each constraint contains t restrictions on k variables for each d k t we determine whether the resolution complexity is a s constant polynomial or exponential in the number of variables for a particular range of d k t we determine a sharp threshold for resolution complexity where the resolution complexity drops from a s exponential to a s polynomial when the clause density passes a specific value

on the relative complexity of resolution refinements and cutting planes proof systems an exponential lower bound for the size of tree like cutting planes refutations of a certain family of conjunctive normal form cnf formulas with polynomial size resolution refutations is proved this implies an exponential separation between the tree like versions and the dag like versions of resolution and cutting planes in both cases only superpolynomial separations were known a urquhart bull symbolic logic 1 1995 pp 425 467 j johannsen inform process lett 67 1998 pp 37 41 p clote and a setzer in proof complexity and feasible arithmetics amer math soc providence ri 1998 pp 93 117 in order to prove these separations the lower bounds on the depth of monotone circuits of raz and mckenzie in combinatorica 19 1999 pp 403 435 are extended to monotone real circuits an exponential separation is also proved between tree like resolution and several refinements of resolution negative resolution and regular resolution actually this last separation also provides a separation between tree like resolution and ordered resolution and thus the corresponding superpolynomial separation of a urquhart bull symbolic logic 1 1995 pp 425 467 is extended finally an exponential separation between ordered resolution and unrestricted resolution also negative resolution is proved only a superpolynomial separation between ordered and unrestricted resolution was previously known a goerdt ann math artificial intelligence 6 1992 pp 169 184

on the satisfiability threshold of formulas with three literals per clause in this paper we present a new upper bound for randomly chosen 3 cnf formulas in particular we show that any random formula over n variables with a clauses to variables ratio of at least 4 4898 is as n grows large asymptotically almost surely unsatisfiable the previous best such bound due to dubois in 1999 was 4 506 the first such bound independently discovered by many groups of researchers since 1983 was 5 19 several decreasing values between 5 19 and 4 506 were published in the years between we believe that the probabilistic techniques we use for the proof are of independent interest

adaptive eigendecomposition of data covariance matrices based on first order perturbations in this paper new algorithms for adaptive eigendecomposition of time varying data covariance matrices are presented the algorithms are based on a first order perturbation analysis of the rank one update for covariance matrix estimates with exponential windows different assumptions on the eigenvalue structure lead to three distinct algorithms with varying degrees of complexity a stabilization technique is presented and both issues of initialization and computational complexity are discussed computer simulations indicate that the new algorithms can achieve the same performance as a direct approach in which the exact eigendecomposition of the updated sample covariance matrix is obtained at each iteration previous algorithms with similar performance require o lm sup 2 complex operations per iteration where l and m respectively denote the data vector and signal subspace dimensions and involve either some form of gram schmidt orthogonalization or a nonlinear eigenvalue search the new algorithms have parallel structures sequential operation counts of order o lm or less and do not involve any of the above steps one particular algorithm can be used to update the complete signal subspace eigenstructure in 5lm complex operations this represents an order of magnitude improvement in computational complexity over existing algorithms with similar performance finally a simplified local convergence analysis of one of the algorithms shows that it is stable and converges in the mean to the true eigendecomposition the convergence is geometrical and is characterized by a single time constant

short proofs are narrow resolution made simple the width of a resolution proof is defined to be the maximal number of literals in any clause of the proof in this paper we relate proof width to proof length size in both general resolution and its tree like variant the following consequences of these relations reveal width as a crucial resource of resolution proofs in one direction the relations allow us to give simple unified proofs for almost all known exponential lower bounds on size of resolution proofs as well as several interesting new ones they all follow from width lower bounds and we show how these follow from natural expansion property of clauses of the input tautology in the other direction the width size relations naturally suggest a simple dynamic programming procedure for automated theorem proving one which simply searches for small width proofs this relation guarantees that the runnuing time and thus the size of the produced proof is at most quasi polynomial in the smallest tree like proof this algorithm is never much worse than any of the recursive automated provers such as dll used in practice in contrast we present a family of tautologies on which it is exponentially faster

projection approximation subspace tracking subspace estimation plays an important role in a variety of modern signal processing applications we present a new approach for tracking the signal subspace recursively it is based on a novel interpretation of the signal subspace as the solution of a projection like unconstrained minimization problem we show that recursive least squares techniques can be applied to solve this problem by making an appropriate projection approximation the resulting algorithms have a computational complexity of o nr where n is the input vector dimension and r is the number of desired eigencomponents simulation results demonstrate that the tracking capability of these algorithms is similar to and in some cases more robust than the computationally expensive batch eigenvalue decomposition relations of the new algorithms to other subspace tracking methods and numerical issues are also discussed

instability of one step replica symmetry broken phase in satisfiability problems we reconsider the one step replica symmetry breaking 1rsb solutions of two random combinatorial problems k xorsat and k sat we present a general method for establishing the stability of these solutions with respect to further steps of replica symmetry breaking our approach extends the ideas of montanari and ricci tersenghi 2003 eur phys j b 33 339 to more general combinatorial problems it turns out that 1rsb is always unstable at sufficiently small clause density or high energy in particular the recent 1rsb solution to 3 sat is unstable at zero energy for m with m 4 153 on the other hand the sat unsat phase transition seems to be correctly described within 1rsb

the κ ring of the moduli of curves of compact type the subalgebra of the tautological ring of the moduli of curves of compact type generated by the classes is studied in all genera relations constructed via the virtual geometry of the moduli of stable quotients are used to obtain minimal sets of generators bases and betti numbers of the rings are computed a universality property relating the higher genus rings to the genus 0 rings is proven using the virtual geometry of the moduli space of stable maps the g formula for hodge integrals arises as the simplest consequence