generating labeled planar graphs uniformly at random we present a deterministic polynomial time algorithm to sample a labeled planar graph uniformly at random our approach uses recursive formulae for the exact number of labeled planar graphs with n vertices and m edges based on a decomposition into 1 2 and 3 connected components we can then use known sampling algorithms and counting formulae for 3 connected planar graphs

a parametric level set approach to simultaneous object identification and background reconstruction for dual energy computed tomography dual energy computerized tomography has gained great interest because of its ability to characterize the chemical composition of a material rather than simply providing relative attenuation images as in conventional tomography the purpose of this paper is to introduce a novel polychromatic dual energy processing algorithm with an emphasis on detection and characterization of piecewise constant objects embedded in an unknown cluttered background physical properties of the objects particularly the compton scattering and photoelectric absorption coefficients are assumed to be known with some level of uncertainty our approach is based on a level set representation of the characteristic function of the object and encompasses a number of regularization techniques for addressing both the prior information we have concerning the physical properties of the object and the fundamental physics based limitations associated with our ability to jointly recover the compton scattering and photoelectric absorption properties of the scene in the absence of an object with appropriate physical properties our approach returns a null characteristic function and thus can be viewed as simultaneously solving the detection and characterization problems unlike the vast majority of methods that define the level set function nonparametrically i e as a dense set of pixel values we define our level set parametrically via radial basis functions and employ a gauss newton type algorithm for cost minimization numerical results show that the algorithm successfully detects objects of interest finds their shape and location and gives an adequate reconstruction of the background

uncertainty characterization of the orthogonal procrustes problem with arbitrary covariance matrices abstract this paper addresses the weighted orthogonal procrustes problem of matching stochastically perturbed point clouds formulated as an optimization problem with a closed form solution a novel uncertainty characterization of the solution of this problem is proposed resorting to perturbation theory concepts which admits arbitrary transformations between point clouds and individual covariance and cross covariance matrices for the points of each cloud the method is thoroughly validated through extensive monte carlo simulations and particularly interesting cases where nonlinearities may arise are further analyzed

least squares estimation of transformation parameters between two point patterns in many applications of computer vision the following problem is encountered two point patterns sets of points x sub i and x sub i i 1 2 n are given in m dimensional space and the similarity transformation parameters rotation translation and scaling that give the least mean squared error between these point patterns are needed recently k s arun et al 1987 and b k p horn et al 1987 presented a solution of this problem their solution however sometimes fails to give a correct rotation matrix and gives a reflection instead when the data is severely corrupted the proposed theorem is a strict solution of the problem and it always gives the correct transformation parameters even when the data is corrupted

controlled non uniform random generation of decomposable structures consider a class of decomposable combinatorial structures using different types of atoms z z 1 z z we address the random generation of such structures with respect to a size n and a targeted distribution in k of its distinguished atoms we consider two variations on this problem in the first alternative the targeted distribution is given by k real numbers m 1 m k such that 0 0 is the number of undistinguished atoms the structures must be generated uniformly among the set of structures of size n that contain exactlyn i atoms z i email protected email protected k we give a o r email protected i 1 kn i 2 mnklogn algorithm for generating m structures which simplifies into a o email protected i 1 kn i mn for regular specifications

globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms we present two types of globally convergent relaxed ordered subsets os algorithms for penalized likelihood image reconstruction in emission tomography modified block sequential regularized expectation maximization bsrem and relaxed os separable paraboloidal surrogates os sps the global convergence proof of the existing bsrem de pierro and yamagishi 2001 required a few a posteriori assumptions by modifying the scaling functions of bsrem we are able to prove the convergence of the modified bsrem under realistic assumptions our modification also makes stepsize selection more convenient in addition we introduce relaxation into the os sps algorithm erdogan and fessler 1999 that otherwise would converge to a limit cycle we prove the global convergence of diagonally scaled incremental gradient methods of which the relaxed os sps is a special case main results of the proofs are from nedic and bertsekas 2001 and correa and lemarechal 1993 simulation results showed that both new algorithms achieve global convergence yet retain the fast initial convergence speed of conventional unrelaxed ordered subsets algorithms

a row action alternative to the em algorithm for maximizing likelihood in emission tomography the maximum likelihood ml approach to estimating the radioactive distribution in the body cross section has become very popular among researchers in emission computed tomography ect since it has been shown to provide very good images compared to those produced with the conventional filtered backprojection fbp algorithm the expectation maximization em algorithm is an often used iterative approach for maximizing the poisson likelihood in ect because of its attractive theoretical and practical properties its major disadvantage is that due to its slow rate of convergence a large amount of computation is often required to achieve an acceptable image here the authors present a row action maximum likelihood algorithm ramla as an alternative to the em algorithm for maximizing the poisson likelihood in ect the authors deduce the convergence properties of this algorithm and demonstrate by way of computer simulations that the early iterates of ramla increase the poisson likelihood in ect at an order of magnitude faster that the standard em algorithm specifically the authors show that from the point of view of measuring total radionuclide uptake in simulated brain phantoms iterations 1 2 3 and 4 of ramla perform at least as well as iterations 45 60 70 and 80 respectively of em moreover the authors show that iterations 1 2 3 and 4 of ramla achieve comparable likelihood values as iterations 45 60 70 and 80 respectively of em the authors also present a modified version of a recent fast ordered subsets em os em algorithm and show that ramla is a special case of this modified os em furthermore the authors show that their modification converges to a ml solution whereas the standard os em does not

average complexity of moore s and hopcroft s algorithms in this paper we prove that for the uniform distribution on complete deterministic automata the average time complexity of moore s state minimization algorithm is o nloglogn where n is the number of states in the input automata and the number of letters in the alphabet is fixed then an unusual family of implementations of hopcroft s algorithm is characterized for which the algorithm will be proved to be always faster than moore s algorithm finally we present experimental results on the usual implementations of hopcroft s algorithm

a hitchhiker s guide to descriptional complexity through analytic combinatorics nowadays increasing attention is being given to the study of the descriptional complexity in the average case although the underlying theory for such a study seems intimidating one can obtain interesting results in this area without too much effort in this gentle introduction we take the reader on a journey through the basic analytical tools of that theory giving some illustrative examples using regular expressions additionally new asymptotic average case results for several e nfa constructions are presented in a unified framework it turns out that asymptotically and in the average case the complexity gap between the several constructions is significantly larger than in the worst case furthermore one of the e nfa constructions approaches the corresponding e free nfa construction asymptotically and on average

on the average state and transition complexity of finite languages we investigate the average case state and transition complexity of deterministic and nondeterministic finite automata when choosing a finite language of a certain size n uniformly at random from all finite languages of that particular size here size means that all words of the language are either of length n or of length at most n it is shown that almost all deterministic finite automata accepting finite languages over a binary input alphabet have state complexity q 2 nn while nondeterministic finite automata are shown to perform better namely the nondeterministic state complexity is in q 2 n interestingly in both cases the aforementioned bounds are asymptotically like in the worst case however the nondeterministic transition complexity is shown to be again q 2 nn the case of unary finite languages is also considered moreover we develop a framework that allows us to investigate the average case complexity of operations like e g union intersection complementation and reversal on finite languages in this setup

stratified generalized procrustes analysis generalized procrustes analysis computes the best set of transformations that relate matched shape data in shape analysis the transformations are usually chosen as similarities while in general statistical data analysis other types of transformation groups such as the affine group may be used generalized procrustes analysis has a nonlinear and nonconvex formulation the classical approach alternates the computation of a so called reference shape and the computation of transformations relating this reference shape to each shape datum in turn we propose the stratified approach to generalized procrustes analysis it first uses the affine transformation group to analyze the data and then upgrades the solution to the sought after group whether euclidean or similarity we derive a convex formulation for each of these two steps and efficient practical algorithms that gracefully handle missing data incomplete shapes extensive experimental results show that our approaches perform well on simulated and real data in particular our closed form solution gives very accurate results for generalized procrustes analysis of euclidean data

regularized image reconstruction algorithms for positron emission tomography we develop algorithms for obtaining regularized estimates of emission means in positron emission tomography the first algorithm iteratively minimizes a penalized maximum likelihood pml objective function it is based on standard de coupled surrogate functions for the ml objective function and de coupled surrogate functions for a certain class of penalty functions as desired the pml algorithm guarantees nonnegative estimates and monotonically decreases the pml objective function with increasing iterations the second algorithm is based on an iteration dependent de coupled penalty function that introduces smoothing while preserving edges for the purpose of making comparisons the mlem algorithm and a penalized weighted least squares algorithm were implemented in experiments using synthetic data and real phantom data it was found that for a fixed level of background noise the contrast in the images produced by the proposed algorithms was the most accurate

uniform random generation of strings in a context free language let s be the set of all strings of length n generated by a given context free grammar a uniform random generator is one which produces strings from s with equal probability in generating these strings care must be taken in choosing the disjuncts that form the right hand side of a grammar rule so that the produced string will have the specified length uniform random generators have applications in studying the complexity of parsers in estimating the average efficiency of theorem provers for the propositional calculus in establishing a measure of ambiguity of a grammar etc two methods are presented for generating uniform random strings in an unambiguous context free language the first method will generate a random string of length n in linear time but must use a precomputed table of size o n r 1 where r is the number of nonterminals in the grammar used to specify the language the second method precomputes part of the table and calculates the other entries as they are called for it requi

the state complexity of random dfas the state complexity of a deterministic finite state automaton dfa is the number of states in its minimal equivalent dfa we study the state complexity of random n state dfas over a k symbol alphabet drawn uniformly from the set n n k 2 n of all such automata we show that with high probability the latter is k n o n log n for a certain explicit constant k

uniform generation of a motzkin word abstract we present a simple and efficient algorithm which builds randomly and uniformly a motzkin word of size n the average complexity of this algorithm is o n the basic idea is to choose a slightly enlarged probability space where uniformity can be achieved and then to filter out the desired words without destorying uniformity

deciding determinism of caterpillar expressions caterpillar expressions have been introduced by bruggemann klein and wood for applications in markup languages caterpillar expressions provide a convenient formalism for specifying the operation of tree walking automata on unranked trees here we give a formal definition of determinism of caterpillar expressions that is based on the language of instruction sequences defined by the expression we show that determinism of caterpillar expressions can be decided in polynomial time

faster fptases for counting and random generation of knapsack solutions we give faster and simpler fully polynomial time approximation schemes fptases for the p complete problem of counting 0 1 knapsack solutions and for its random generation counterpart our method is based on dynamic programming and discretization of large numbers through floating point arithmetic we improve both deterministic counting fptases in gopalan et al focs 2011 stefankovic et al siam j comput 2012 and the randomized counting and random generation algorithms in dyer stoc 2003 we also improve the complexity of the problem of counting 0 1 knapsack solutions in an arc weighted dag

uniform random generation of decomposable structures using floating point arithmetic the em recursive method formalized by nijenhuis and wilf cite niwi78 and systematized by flajolet van cutsem and zimmermann cite flziva94 is extended here to floating point arithmetic the resulting adz method enables one to generate decomposable data structures both labelled or unlabelled uniformly at random in expected o n 1 epsilon time and space after a preprocessing phase of o n 2 epsilon time which reduces to o n 1 epsilon for context free grammars

convergent incremental optimization transfer algorithms application to tomography no convergent ordered subsets os type image reconstruction algorithms for transmission tomography have been proposed to date in contrast in emission tomography there are two known families of convergent os algorithms methods that use relaxation parameters and methods based on the incremental expectation maximization em approach this paper generalizes the incremental em approach by introducing a general framework incremental optimization transfer the proposed algorithms accelerate convergence speeds and ensure global convergence without requiring relaxation parameters the general optimization transfer framework allows the use of a very broad family of surrogate functions enabling the development of new algorithms this paper provides the first convergent os type algorithm for nonconcave penalized likelihood pl transmission image reconstruction by using separable paraboloidal surrogates sps which yield closed form maximization steps we found it is very effective to achieve fast convergence rates by starting with an os algorithm with a large number of subsets and switching to the new transmission incremental optimization transfer triot algorithm results show that triot is faster in increasing the pl objective than nonincremental ordinary sps and even os sps yet is convergent

tree walking automata do not recognize all regular languages tree walking automata are a natural sequential model for recognizing tree languages it is well known that every tree language recognized by a tree walking automaton is regular we show that the converse does not hold

enumeration and generation with a string automata representation the representation of combinatorial objects is decisive for the feasibility of several enumerative tasks in this work we present a unique string representation for complete initially connected deterministic automata icdfas with n states over an alphabet of k symbols for these strings we give a regular expression and show how they are adequate for exact and random generation allow an alternative way for enumeration and lead to an upper bound for the number of icdfas the exact generation algorithm can be used to partition the set of icdfas in order to parallelize the counting of minimal automata and thus of regular languages a uniform random generator for icdfas is presented that uses a table of pre calculated values based on the same table an optimal coding for icdfas is obtained

on the relation between the isra and the em algorithm for positron emission tomography the image space reconstruction algorithm isra was proposed as a modification of the expectation maximization em algorithm based on physical considerations for application in volume emission computered tomography as a consequence of this modification isra searches for least squares solutions instead of maximizing poisson likelihoods as the em algorithm it is shown that both algorithms may be obtained from a common mathematical framework this fact is used to extend isra for penalized likelihood estimates

enumeration and random generation of accessible automata we present a bijection between the set a n of deterministic and accessible automata with n states on a k letters alphabet and some diagrams which can themselves be represented as partitions of a set of kn 1 elements into n non empty subsets this combinatorial construction shows that the asymptotic order of the cardinality of a n is related to the stirling number knn our bijective approach also yields an efficient random sampler for the uniform distribution of automata with n states its complexity is o n 3 2 using the framework of boltzmann samplers

least squares fitting of two 3 d point sets two point sets pi and p i i 1 2 n are related by p i rpi t ni where r is a rotation matrix t a translation vector and ni a noise vector given pi and p i we present an algorithm for finding the least squares solution of r and t which is based on the singular value decomposition svd of a 3 3 matrix this new algorithm is compared to two earlier algorithms with respect to computer time requirements

parametric random generation of deterministic tree automata uniform random generators deliver a simple empirical means to estimate the average complexity of an algorithm we present a general rejection algorithm that generates sequential letter to letter transducers up to isomorphism we also propose an original parametric random generation algorithm to produce sequential letter to letter transducers with a fixed number of transitions we tailor this general scheme to randomly generate deterministic tree walking automata and deterministic top down tree automata we apply our implementation of the generator to the estimation of the average complexity of a deterministic tree walking automata to nondeterministic top down tree automata construction we also implemented

uniform random generation of words of rational languages the problem of generating uniformly at random words of a given language has been the subject of extensive study in the last few years an important part of that work is devoted to the generation of words of context free languages see e g 6 8 9 12 for a given integer n 0 the words of length n 0 of any unambiguous context free language can be generated uniformly at random by using algorithms derived from the general method which was introduced by wilf 14 15 and systematized by flajolet et al 7 clearly this can be applied to the set of rational languages which constitute an important special case of context free languages most authors use the uniform measure of complexity see 1 in order to compute the complexity of the algorithms of generation this measure is based on the following hypotheses any simple arithmetic operation addition multiplication has time cost 0 1 and a constant amount of memory space is taken by any number thus we know that words of any rational language can be generated by using an algorithm which with respect to the uniform measure of complexity runs in linear time in terms of the length of the words and constant space 9 this measure is realistic only if there is a reasonable bound on the numbers involved in the operations however the classical random generation algorithms involve operations on numbers which grow exponentially in terms of the length of the words to be generated moreover the programs which make use of these algorithms are generally used to generate very large words for example for the purpose of studying the asymptotic behavior of some parameters therefore the uniform measure does not reflect the real behavior of such programs it turns out that the logarithmic measure of complexity is much more realistic one assumes that the space taken by a number k is o log k and that any simple arithmetic operation can be done in time o log k it is with respect to this measure that we will evaluate the performance of algorithms in this paper our goal is to design efficient algorithms in terms of logarithmic complexity to generate uniformly at random words from certain classes of rational languages we consider rational languages defined by their minimal finite deterministic automata when computing complexity neither the size of the automaton nor the cardinality of the alphabet are taken in account in section 2 we present some background on rational languages and their generating series we describe briefly the classical method for generating words of such languages and we study its logarithmic complexity we show that it is at best quadratic for most languages this is due mainly to computations on numbers which grow exponentially with the length of the words to be generated in order to improve significantly the efficiency of the algorithms we must avoid handling of large numbers or at least decrease substantially the frequency of computations on such numbers another alternative briefly discussed in 7 and 12 is to compute with floating point numbers instead of integers in this case the logarithmic complexity is time linear however using floating point numbers leads inevitably to approximations which prevent the exact uniformity of the generation in sections 3 and 4 we show that in some cases we can avoid computations on large numbers entirely or almost entirely while keeping the exact uniformity of the generation we determine two classes of rational languages for which this is the case section 3 concerns languages whose associated generating series have a unique singularity we present a simple version of the classical algorithm which totally avoids handling of large numbers the logarithmic complexity of the method is o n log n in time and o log n in memory space section 4 focuses on languages whose associated generating series have the following property there exists a unique singularity of minimum modulus and this singularity is simple for such languages we give a probabilistic version of the classical algorithm which generates words randomly while avoiding most computations on large numbers this method needs a preprocessing stage which can be done in polynomial time and linear space in terms of the length n of the words following preprocessing any word of length n can be generated in average linear time and space

regularized image reconstruction algorithms for dual isotope myocardial perfusion spect mps imaging using a cross tracer prior in simultaneous dual isotope myocardial perfusion spect mps imaging data are simultaneously acquired to determine the distributions of two radioactive isotopes the goal of this work was to develop penalized maximum likelihood pml algorithms for a novel cross tracer prior that exploits the fact that the two images reconstructed from simultaneous dual isotope mps projection data are perfectly registered in space we first formulated the simultaneous dual isotope mps reconstruction problem as a joint estimation problem a cross tracer prior that couples voxel values on both images was then proposed we developed an iterative algorithm to reconstruct the mps images that converges to the maximum a posteriori solution for this prior based on separable surrogate functions to accelerate the convergence we developed a fast algorithm for the cross tracer prior based on the complete data os em cosem framework the proposed algorithm was compared qualitatively and quantitatively to a single tracer version of the prior that did not include the cross tracer term quantitative evaluations included comparisons of mean and standard deviation images as well as assessment of image fidelity using the mean square error we also evaluated the cross tracer prior using a three class observer study with respect to the three class mps diagnostic task i e classifying patients as having either no defect reversible defect or fixed defects for this study a comparison with conventional ordered subsets expectation maximization os em reconstruction with postfiltering was performed the comparisons to the single tracer prior demonstrated similar resolution for areas of the image with large intensity changes and reduced noise in uniform regions the cross tracer prior was also superior to the single tracer version in terms of restoring image fidelity results of the three class observer study showed that the proposed cross tracer prior and the convergent algorithms improved the image quality of dual isotope mps images compared to os em

a calculus for the random generation of labelled combinatorial structures a systematic approach to the random generation of labelled combinatorial objects is presented it applies to structures that are decomposable i e formally specifiable by grammars involving set sequence and cycle constructions a general strategy is developed for solving the random generation problem with two closely related types of methods for structures of size n the boustrophedonic algorithms exhibit a worst case behaviour of the form o n log n the sequential algorithms have worst case o n2 while offering good potential for optimizations in the average case the complexity model is in terms of arithmetic operations and both methods appeal to precomputed numerical table of linear size that can be computed in time o n2 a companion calculus permits systematically to compute the average case cost of the sequential generation algorithm associated to a given specification using optimizations dictated by the cost calculus several random generation algorithms of the sequential type are developed most of them have expected complexity 1 2n log n and are thus only slightly superlinear the approach is exemplified by the random generation of a number of classical combinatorial structures including cayley trees hierarchies the cycle decomposition of permutations binary trees functional graphs surjections and set partitions

on the estimation of rigid body rotation from noisy data we derive an exact solution to the problem of estimating the rotation of a rigid body from noisy 3d image data our approach is based on total least squares tls but unlike previous work involving tls we include the constraint that the transformation matrix should be orthonormal it turns out that the solution to the estimation problem has the same form as if the data are not noisy and thus the solution to the standard procrustes problem can be applied

sampling different kinds of acyclic automata using markov chains we propose algorithms that use markov chain techniques to generate acyclic automata uniformly at random we first consider deterministic accessible and acyclic automata then focus on the class of minimal acyclic automata in each case we explain how to define random local transformations that describe an ergodic and symmetric markov chain the distribution of the automaton obtained after t random steps in this markov chain tends to the uniform distribution as t tends to infinity

fast em like methods for maximum a posteriori estimates in emission tomography the maximum likelihood ml approach in emission tomography provides images with superior noise characteristics compared to conventional filtered backprojection fbp algorithms the expectation maximization em algorithm is an iterative algorithm for maximizing the poisson likelihood in emission computed tomography that became very popular for solving the ml problem because of its attractive theoretical and practical properties recently browne and depierro 1996 and hudson and larkin 1991 block sequential versions of the em algorithm that take advantage of the scanner s geometry have been proposed in order to accelerate its convergence in hudson and larkin 1991 the ordered subsets em os em method was applied to the hit problem and a modification os gp to the maximum a posteriori map regularized approach without showing convergence in browne and depierro 1996 we presented a relaxed version of os em ramla that converges to an ml solution in this paper we present an extension of ramla for map reconstruction we show that if the sequence generated by this method converges then it must converge to the true map solution experimental evidence of this convergence is also shown to illustrate this behavior we apply the algorithm to positron emission tomography simulated data comparing its performance to os gp

transitive closure logic nested tree walking automata and xpath we study fo mtc first order logic with monadic transitive closure a logical formalism in between fo and mso on trees we characterize the expressive power of fo mtc in terms of nested tree walking automata using the latter we show that fo mtc is strictly less expressive than mso solving an open problem we also present a temporal logic on trees that is expressively complete for fo mtc in the form of an extension of the xml document navigation language xpath with two operators the kleene star for taking the transitive closure of path expressions and a subtree relativisation operator allowing one to restrict attention to a specific subtree while evaluating a subexpression we show that the expressive power of this xpath dialect equals that of fo mtc for boolean unary and binary queries we also investigate the complexity of the automata model as well as the xpath dialect we show that query evaluation be done in polynomial time combined complexity but that emptiness or satisfiability is 2exptime complete

a modified expectation maximization algorithm for penalized likelihood estimation in emission tomography the maximum likelihood ml expectation maximization em approach in emission tomography has been very popular in medical imaging for several years in spite of this no satisfactory convergent modifications have been proposed for the regularized approach here a modification of the em algorithm is presented the new method is a natural extension of the em for maximizing likelihood with concave priors convergence proofs are given

a large deviations principle for the maki thompson rumour model abstract we consider the stochastic model for the propagation of a rumour within a population which was formulated by maki and thompson 20 sudbury 22 established that as the population size tends to infinity the proportion of the population never hearing the rumour converges in probability to 0 2032 watson 23 later derived the asymptotic normality of a suitably scaled version of this proportion we prove a corresponding large deviations principle with an explicit formula for the rate function

multi material decomposition using statistical image reconstruction for spectral ct spectral computed tomography ct provides information on material characterization and quantification because of its ability to separate different basis materials dual energy de ct provides two sets of measurements at two different source energies in principle two materials can be accurately decomposed from dect measurements however many clinical and industrial applications require three or more material images for triple material decomposition a third constraint such as volume conservation mass conservation or both is required to solve three sets of unknowns from two sets of measurements the recently proposed flexible image domain id multi material decomposition method assumes each pixel contains at most three materials out of several possible materials and decomposes a mixture pixel by pixel we propose a penalized likelihood pl method with edge preserving regularizers for each material to reconstruct multi material images using a similar constraint from sinogram data we develop an optimization transfer method with a series of pixel wise separable quadratic surrogate pwsqs functions to monotonically decrease the complicated pl cost function the pwsqs algorithm separates pixels to allow simultaneous update of all pixels but keeps the basis materials coupled to allow faster convergence rate than our previous proposed material and pixel wise sqs algorithms comparing with the id method using 2 d fan beam simulations the pl method greatly reduced noise streak and cross talk artifacts in the reconstructed basis component images and achieved much smaller root mean square errors

convergence theorems for generalized alternating minimization procedures the em algorithm is widely used to develop iterative parameter estimation procedures for statistical models in cases where these procedures strictly follow the em formulation the convergence properties of the estimation procedures are well understood in some instances there are practical reasons to develop procedures that do not strictly fall within the em framework we study em variants in which the e step is not performed exactly either to obtain improved rates of convergence or due to approximations needed to compute statistics under a model family over which e steps cannot be realized since these variants are not em procedures the standard g em convergence results do not apply to them we present an information geometric framework for describing such algorithms and analyzing their convergence properties we apply this framework to analyze the convergence properties of incremental em and variational em for incremental em we discuss conditions under these algorithms converge in likelihood for variational em we show how the e step approximation prevents convergence to local maxima in likelihood