the solar dynamic observatory will be launched from cape canaveral in june 2009 / 2010 on an atlas v booster . on board this satellite there will be several instruments dedicated to the study of the solar photospheric and coronal magnetic fields and their relation to the interplanetary medium , space weather and earth climatology .the helioseismic and magnetic imager is an instrument developed at stanford university , lockheed - martin solar and astrophysics laboratory and at the high altitude observatory .hmi consists of a combination of a lyot filter and two michelson interferometers . between the two michelson there is a set of three 10th order retarders : , and .located after these , there is a beam splitter that divides the light between two twin 4096 ccd cameras .each camera will acquire full - disk images of the sun with a pixel size of 0.5 arcsec .the first camera , to be referred to as _ doppler camera _ , will be devoted to the measurement of the right and left circular polarization , , at 6 wavelength positions across the fe i ( =2.5 ) 6173.35  line every 50 seconds or less . with this, full disk maps of the line of sight components of the magnetic field and velocity will be produced with a 50 second cadence for helioseismic studies .the second one , hereafter referred to as _ vector camera _ , will measure the full stokes vector at the same wavelengths as the doppler camera .the cadence in the vector camera will be about 120 seconds .borrero et al .( 2007 ) have conducted a detailed study of the vector camera performance , and concluded that photon noise and p - mode cross - talk between the different stokes parameters makes the 2-minute data inappropriate to determine the magnetic field vector with the desired degree of accuracy .they suggest averaging the observed stokes vector every 10 minutes before it is inverted .  therefore our objective has been to create an inversion code for the radiative transfer equation able to process 13.5 million pixels in 10 minutes : _ 22500 pixels/ second_. our task benefits from the fact that each pixel is treated independently , and therefore our problem can be parallelized in a straight - forward fashion , resulting in a time saving which is directly proportional to the number of cpus employed . consequently , very fast ( _ 3000 - 10000 pixels / second _ ) inversion strategies such a principal component analysis ( pca ; rees et al .2000 ; socas - navarro et al .2001 ) or artificial neural networks ( anns ; socas - navarro 2003 , 2005 ) could meet the hmi requirements with a very small number of cpus . without entering into details about the different methods , their accuracy and robustness ( an interested reader is referred to the recent reviews by del toro iniesta 2003a and bellot rubio 2006 ) ,it is generally acknowledged that these fast inversion techniques retrieve a magnetic field vector that is more suitable for a qualitative analysis such as active region evolution and tracking , global magnetic field appearance .our aim with hmi is to provide the magnetic field vector on the solar surface with an accuracy good enough to carry out more quantitative studies .a further concern , from the point of view of hmi characteristics and scientific objectives , is the fact that current implementations of anns do not allow to obtain a measure of the goodness of the inversion , e.g. .pca does retrieve such a measure in terms of the so - called pca distance .however , principal component analysis is usually employed to reduce the dimensionality of the problem ( e.g. by inverting the first pca or fourier coefficients of the observed stokes vector ) .it is unclear how pca will perform when inverting data from filtergram instruments , such as hmi , where the number of data points is already very small ( six points in wavelength across the four stokes parameters ) .  a good compromise between accuracy and speed is achieved by traditional iterative non - linear squares fitting algorithms , specially if applied to the milne - eddington ( m - e ) solution of the radiative transfer equation ( landolfi & landi deglinnocenti 1982 ) .the reason for this lies in the simplified thermodynamics of this approximation , that avoids tedious iterative calculations and evaluations of the partial pressures , ionization and hydrostatic equilibrium , etc ( ruiz cobo 2006 ) .in addition , the analytical nature of the m - e solution allows to calculate also analytical derivatives , which significantly speeds up the inversion process .the first m - e inversion code that fully takes into account magneto - optical effects dates back , to the best of our knowledge , to auer et al .( 1977 ) ( see also skumanich & lites 1987 ) .this has been for many years the standard inversion technique employed in the analysis of advance stokes polarimeter data ( asp ; previously stokes ii : baur et al . 1981 ) ,and thus it is widely known as the hao / asp inversion code . as a consequence of the increase in the cpus speed since the late 80 s, the hao / asp code has gone from inverting several pixels per hour to about _ 10 pixels / second_. thus , meeting hmi s goal would require more than 2000 cpus .clearly further modifications and refinements over these type of m - e inversion codes are necessary if we are to meet hmi s requirements with a limited number ( ) of cpus .  after a brief introduction to stokes inversion ( section 2 ), we have divided these modifications into three groups : those made on the synthesis module of the code ( section 3 ) ; modifications made on the inversion module of the code ( section 4 ) , and implementation of an accurate initial guess model for the inversion process ( section 5 ) . in section 6we present a study of the inversion code s profile and speed .section 7 shows results obtained from the application of vfisv to hinode / sp data and finally , section 8 summarizes our results .in this section we will describe very briefly how a stokes inversion code typically works .the idea is to familiarize the reader with the common procedures and nomenclature to facilitate the understanding of the following sections .more details about this topic can be found in ( with different levels of complexity ) ruiz cobo & del toro iniesta ( 1992 ) , frutiger ( 2001 ) , del toro iniesta ( 2003b ) , borrero ( 2004 ) and references therein .the basic idea of any stokes inversion code is to iteratively fit the observed stokes vector at each wavelength position is the total intensity , and are the linear polarization profiles and is the circular polarization . ] .this fit is done by producing a synthetic stokes vector , that it is then compared at each wavelength position with the observed one via the of the fit :   ^ 2 \frac{w_{ij}^2}{\sigma_i^2}\ ] ]  where is the number of wavelength points observed ( in hmi s case ) and is the number of free parameters .therefore refers to number of degrees of freedom in the inversion . and refer to each of the 4 stokes parameters ( index ) at each wavelength position ( index ) for both the observed and synthetic stokes vector .section 4.3 discusses in detail the parameters and .the synthetic profiles depend on a series of model parameters with .these parameters define the physical model that we presume as valid : lte , non - lte , multi - component atmospheres , gradients along the line - of - sight , etc . in the case of a milne - eddington atmosphere ,these typically are : .the first 5 of them are the thermodynamic parameters : source function at the base of the photosphere ( ) , gradient of the source function ( ) , center to continuum absorption coefficient ( ) , damping ( ) , and doppler width of the spectral line ( ) .the next three refer to the three components of the magnetic field vector : strength , inclination with respect to the observer , and azimuth of the magnetic field vector in the plane perpendicular to the observer .the following two are kinematic parameters : line - of - sight velocity of the plasma harboring the magnetic field ( ) , and the macroturbulent velocity ( ; used to model unresolved velocity fields ) .the last parameter , , is a geometrical parameter that defines what portion of the resolution element is filled with a magnetized plasma . is computed in the synthesis module of the inversion code .this module also retrieves the derivatives of the synthetic stokes vector with respect to the model parameters ( free parameters ) : .once the synthetic stokes profiles are obtained , we can determine the of the fit ( equation 1 ) .with the derivatives it is possible to compute , which is f - dimensional vector where each of its components can be expressed as :    we can also compute the modified hessian matrix : .this matrix is basically the regular hessian matrix , where second derivatives have been neglected and where the diagonal elements are modified by means of a parameter :   ^ 2\end{aligned}\ ] ]  according to the recipe of the levenberg - mardquart algorithm , using and allows us to determine the perturbations , , that must be applied to the model parameters at iteration , , in order to improve .    to avoid singularities in the inversion of the modified hessian matrix ( equation 5 ) the singular value decomposition method is often employed .once the new model parameters have been determined , the synthesis module is called again to produce the synthetic stokes vector and its derivatives , as depicted in figure 1 .then , a new is computed and the iteration continues .vfisv will be employed for the automatic inversion of data of the fe i ( =2.5 ) 6173.33   line ( see table 1 ) .therefore a straightforward implementation of its zeeman pattern can be hardcoded into the inversion code , avoiding needless extra calculations using the quantum numbers , and of the atomic levels involved ..atomic parameters for the lower and upper levels of the atomic transition originating the fe i 6173.33  spectral line ( from nave et al .the land factors have been calculated in the ls approximation ( see for example eq . 8.30 in del toro iniesta ( 2003b ) . [ cols="^,^",options="header " , ]in this section we present results from the application of vfisv to spectropolarimetric data from hinode / sp . we have used here two different data sets .they correspond to observations carried out on november 14 , 2006 and may 3 , 2007 respectively .both datasets encompass active regions ar 10923 and 10953 , which were located at around .the second dataset has already been used in our experiments in sections 4.1 and 6 .they are comprised by a set of 538 and 1000 fits files , therefore having 550,912 and 1,024,000 stokes profiles , respectively . since hinode / sp observes the fe i lines at 6301.5 and 6302.5  we have trimmed the observations to use only the second spectral line ( see section 3.1 ) .after that , we convolved the original observations with the theoretical filter profiles from hmi , to reduce the number of wavelength positions to 6 .each map has been inverted using a quad - core ( 4-cpus ) intel xeon machine at 2.66 ghz .the first map was inverted in 227.13 seconds , whereas the second one was ready after 405.88 seconds .this translates into a speed of 606.38 and 630.73 profiles / sec / cpu .thus , inverting 22500 profiles / seconds ( as requested by hmi ; see section 1 ) is possible with only 35 - 40 cpus .maps of the magnetic field strength for these two regions are presented in figure 9 . for better visualization only the regions around the two sunspotsare displayed ( the original maps are 86`` '' and 163`` '' ) .we have also tested vfisv with more than 1000 maps available from the hinode / sp database that have been verified for level-1 ( bruce lites , _ private communication _ ) .we have found that vfisv yields adequate results for all kind of structures : network , plage , sunspots , etc .the main issues that we have found are :  * we have observed systematic errors in the umbra of those sunspots , when these are located very close to the limb ( ) . * when inverting using a filling factor ( free ) about 1 - 2 % of pixels in the very quiet sun regions ( e.g. internetwork ) yield inadequate results .these are characterized by very large magnetic field strength , gauss , and very small magnetic filling factors : . * in sunspots where the magnetic field is extremely high in the umbra , gauss , we observe saturation effects in the magnetic field strength whenever we invert only 6 wavelength positions .this effect disappears when using the full spectral profile .we have developed an inversion code for the polarized radiative transfer equation .the code s name is vfisv : very fast inversion of the stokes vector .this code assumes that the properties of the solar photosphere are well described by the milne - eddington approximation .it will be employed to routinely invert polarimetric data from the heliseismic and magnetic imager ( hmi ) that will fly on - board of the solar dynamics observatory ( sdo ) in june 2009/2010 .the code is written in fortran 90 and parallelized with mpich-2 .we have introduced a number of improvements in this code that makes it able to achieve an inversion speed of about 600 pixels per seconds per cpu . at this paceit will be possible to provide solar full - disk maps ( with 1 " resolution ) of the solar magnetic field vector every 10 minutes , using less than 40 cpus .the code has been tested with observations from the spectropolarimeter onboard hinode .it is freely available for download through the community spectro - polarimetric analysis center initiative at the high altitude observatory and national center for atmospheric research ( lites et al .2007 ; http://www.hao.ucar.edu/projects/csac/ ) .auer , l.h . ,house , l.l .& heasley , j.n .1977 : _ solar phys _ , 55 , 47 - 61 baur , t.g . , elmore , d.e . ,lee , r.h , qerfeld , c.w . & rogers , s.r .1981 : _ solar phys _ , 70 , 395 bellot rubio , l.r .2006 : asp conference series , 358 , 107 .proceedings of the solar polarization workshop 4 in boulder , colorado , usa .eds : roberto casini and b.w .lites bishop , c.m .1994 : neural networks for pattern recognition .oxford university press , 1994 .borrero , j.m ., tomczyk , s. , norton , a. , darnell , t. , schou , j. , scherrer , p. , bush , r. & liu , y. 2007 : _ solar phys _ , 240 , 177 borrero , j.m . ,lites , b.w . & solanki , s.k . 2008 : a&a , 481 , l13-l16 borrero , j.m . &solanki , s.k .2008 : apj , 687 , 668 - 677 borrero , j.m . : 2004 , phd thesis , gttingen univertity ( germany ) , published by copernicus gmbh .isbn 3 - 936596 - 33 - 0 del toro iniesta , j.c .2003a : astronomische nachrichten , 324 , no.4 , 383 del toro iniesta , j.c .in introduction to spectropolarimetry . cambridge university press , 2003b .isbn : 0521818273 jefferies , j.t .& mickey , d.l .1991 : apj , 372 , 694 - 702 hui , a.k . ,armstrong , b.h .& wray , a.a . 1977 : jqstr , 18 , 509 - 516 kosugi , t. , matsuzaki , k. , sakuo , t. et al .2007 : _ solar physics _ , 243 , 3 letchworth , k.l . & benner , d.c 2007 : jqsrt , 107 , 173 - 192 lites , b.w ., elmore , d.f .& streander , k.v .2001 : asp conf .series , vol 236 .advanced solar polarimetry .m. sigwarth ( san francisco : asp ) nave , g. , johansson , s. , learner , r.c.m . & thorne , p. 1994 : apjs , 94 , 221 - 459 neckel , h. & labs , d. 1994 : _ solar phys _ , 153 , 91 - 114 landolfi , m. & landi deglinnocenti , e. 1982 : _ solar phys _ , 78 , 355 lites , b. , casini , r. , garcia , j. & socas - navarro , h. 2007 : memorie della societa astronomica italiana , 78 , 148 orozco surez & del toro iniesta , j.c . : 2007 , a&a , 462 , 1137 - 1145 press , w. , flannery , b. , teukolsky , s. & vetterling , w. 1986 : numerical recipes - the art of scientific computing .cambridge : cambridge university press .rees , d.e ., lpez ariste a. , thatcher , j. & semel , m. 2000 : a&a , 355 , 759 .ruiz cobo , b. 2006 : modern solar facilities - advanced solar science , proceedings of a workshop held at gttingen september 27 - 29 , 2006 isbn 978 - 3 - 938616 - 84 - 0 . published by universittsverlag gttingen .skumanich , a. & lites , b.w .1987 : apj , 322 , 473 socas - navarro , h. , lpez ariste , a. & lites , b.w . 2001 : apj , 553 , 949 socas - navarro , h. : 2003 , neural networks , 16 , 355 socas - navarro , h. : 2005 , apj , 621 , 545 unsold , a. : 1955 , physik der sternatmospharen .berlin , springer .westendorp plaza , c. , del toro iniesta , j.c ., ruiz cobo , b. , martnez pillet , v. , lites , b.w . & skumanich , a. 1998 : apj , 494 , 453 wittmann , a. : 1974 , _ solar phys _ , 35

Abstract: in this paper we describe in detail the implementation and main properties of a new inversion code for the polarized radiative transfer equation ( vfisv : very fast inversion of the stokes vector ) . vfisv will routinely analyze pipeline data from the helioseismic and magnetic imager ( hmi ) on - board of the solar dynamics observatory ( sdo ) . it will provide full - disk maps ( 4096 pixels ) of the magnetic field vector on the solar photosphere every 10 minutes . for this reason vfisv is optimized to achieve an inversion speed that will allow it to invert 16 million pixels every 10 minutes with a modest number ( approx . 50 ) of cpus . here we focus on describing a number of important details , simplifications and tweaks that have allowed us to significantly speed up the inversion process . we also give details on tests performed with data from the spectropolarimeter on - board of the hinode spacecraft .   example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore


year 2014 has marked the 50 anniversary of john s. bell s paper  on the einstein - podolski - rosen paradox  , a landmark achievement that has shaped the research in modern quantum mechanics . since then, bell inequalities , which have proven the incompatibility of quantum mechanics with any local realistic theory , have been established as a viable tool for deepening our understanding of the nature of reality and , from a more pragmatic perspective , identify unambiguously correlations of a non - classical nature  .non - locality tests for discrete - variable systems have been formulated , over the years , based on both dichotomic and multichotomic local measurement settings  , showing that bipartite -level entangled states are in conflict with local realistic assumptions more strongly than entangled qubits .inequalities for bipartite -dimensional systems have been put forward  , while various formulation for multipartite systems have been proposed  , and extended to generic arbitrary dimensional systems  .continuous - variable bell - like inequalities have been derived for bipartite states using the phase - space formalism  , and extended to multipartite scenarios recently  .the investigation on non - locality of discrete large- systems is particularly relevant in light of the goal of inferring non - classical effects in increasingly large systems .the possibility to unambiguously signal non - classical correlations , strong enough to violate a bell - like inequality , would provide undoubtable evidence of the relevance of quantum theory in the large - scale domain , and bring new perspectives to the investigation of the quantum - to - classical transition  .first steps along similar lines of investigation have been made by addressing the class of states dubbed as micro - macro ones  , which have then been extended to macro - macro scenarios  .when tackling the phenomenology of non - classical correlations in large - scale systems , the detrimental effects of improper measurements should be considered on the same footing as non - ideal state preparation .it has been shown that coarse - grained measurements do indeed affect the chances of falsifying local realistic models , even when dealing with otherwise ideal state resources  .on the other hand , a typical assessment of the robustness of bell - like tests is based on the amount of white noise that a given resource state is able to tolerate before losing any discrepancy with local hidden variable ( lhv ) models  .this approach was the basis for the conclusion drawn by kaszilowski _et al . _ on bipartite -level systems  , while such analyses provide useful information on the effects that imperfections and environmental mechanisms have on the revelation of non - classicality , a systematic study of the scaling behavior of non - idealities against the _ size _ of a given system has only been partially addressed  . yet , this is key information for the quest of observing non - classicality at the large - scale limit .for instance , as we address in this paper , bell inequalities formulated for large- systems require a large number of measurement settings : their inadequate arrangement could have rather significant implications for the successful falsification of lhv models even when states endowed with a large degree of quantum correlations are considered . in this sense, our study goes well beyond currently available literature  .our study here addresses precisely this important issue , which we tackle in a careful quantitative manner . by addressing experimentally relevant imperfections in the bell test of -dimensional systems ,we point out , through a careful statistical analysis , the existence of thresholds in the amount of randomness that a given resource state ( or , equivalently , measurement setting ) can tolerate . as we find out, such threshold , which scales unfavorably with the dimensions of the local systems being used , is rather severe and poses significant questions on the suitability of bell - like tests in systems verging to the mesoscopic or macroscopic scale , when evaluated against the actual value taken by . in some conditions , even a maximallyentangled two - qutrit state could fail to violate a bell inequality built from slightly improper measurement settings .the remainder of this paper is organized as follows . in sec .[ sec : genbell ] we introduce the inequalities addressed in this work , setting the notation that will be used throughout our analysis .[ sec : stability ] is devoted to the analysis of the robustness of such bell - like tests for systems of increasing dimensionality and exposed to random perturbations of the otherwise ideal measurement settings required by each of the inequalities that we study . in sec .[ sec : optimization ] we extend our study to the case of random initial states , showing the inherent difficulties related to running bell tests on such resources , and discussing the effectiveness of more sophisticated statistical sampling techniques for ascertaining non - locality in randomly picked states of -dimensional systems . finally , in sec .[ conc ] we draw our conclusions .in this section we review the formulation of bell inequalities for -dimensional systems that are used hereafter .the original formulation of the bell inequalities relates the joint probabilities for local measurements with two possible outcomes ( ) , typically the polarization of photons or a spin-1/2 system ( dimension and the spin quantum number obey the relation ) .however , it is possible to extend ( in a non unique way ) such formulation to systems with higher local dimension  : we consider a coupled bipartite system and two parties ( alice and bob ) each of them having access to a single partition , of dimension , of the system ( either subsystem or ) and able to measure two independent observables , , and , , respectively .the operators and are assumed to have the same spectrum , and , with . clearly , for -dimensional subsystems the measurement can have different outcomes .the generalized bell inequalities are obtained summing the probabilities that alice and bob obtain equal ( or correlated ) outcomes of their measurements . in particular , the first bell expression we will consider reads as where are the probability for alice and bob to have the same measure outcomes : and similarly in general , one can show that for local theories only three of the four equalities in ( [ ibell ] ) can be satisfied at the same time , that is for local theories , while non - local theories could reach a value of .  beside the generalization of the standard bell inequalitypresented above , we also consider a second inequality , which at variance with the previous one is built with a number of terms that grows with the local dimension of the subsystems .explicitly - 1 } \left ( 1 - \frac{2k}{d-1 } \right )  \nonumber \\  & & \times \biggl\ { \bigl [ p(a_1 = b_1 + k ) + p(b_1 = a_2 + k + 1 ) \nonumber  \\  & & + p(a_2 = b_2 + k ) + p(b_2 = a_1 + k ) \bigr ] \nonumber \\  & & - \bigl [ p(a_1 = b_1 -k -1 ) + p(b_1 = a_2 -k ) \nonumber \\  & & + p(a_2 = b_2-k-1 ) + p(b_2 = a_1 -k-1 )  \bigr ] \biggr\ } .\label{idbell}\end{aligned}\ ] ] as before , one can show that the maximal possible value achievable by this quantity is . however , lhv models allow only for .   values for the generalized bell expressions and plotted against the spin quantum number of each subsystem . in order to evaluate and have used the state and the measurement operators given in eqs .( [ eq : optaxes ] ) .bottom ( green ) lines : maximum values achievable through local hidden variable models.,scaledwidth=44.0% ]  generalized maximally entangled ( bell ) quantum states of two -dimensional systems , violate these inequalities for any value of  . it has been shown  that , for , the observables that are suited to test the generalized bell inequalities have associated eigenstates of the form  [ eq : optaxes ] with , ,  .projections of the state of the system onto such states always leads to a violation of and , independently of the particular value of .this is shown in fig .[ fig1 ] , where we report the exact numerical values for both and as a function of the spin quantum number of the system .note that , if the probabilities are computed according to the rules of quantum mechanics , the maximal value of achievable for is  and that numerical data show that monotonically decreases with .histogram of the probabilities to obtain the values for the maximally entangled state given in eq .  with for a perturbation .the red line indicates the border between classical ( ) and non - classical correlations ( ).,scaledwidth=44.0% ]  in this section we characterize as a function of the system size the robustness of the above generalized bell inequalities against perturbations of the projections identified in eqs .( [ eq : optaxes ] ) .this is indeed a fundamental question to address when investigating the possibility to violate a bell - like inequality using a macroscopic quantum system .in fact , any experimental test of this kind will have to cope with non - idealities in the identification of the most suitable projections to implement .our study is thus aimed at determining the tolerance allowed by the bell - like tests addressed above against imperfect measurement settings .upper panel : violation probability as a function of the spin quantum number for ( green triangles , blue squares , black diamonds , magenta circles , red filled circles ) .solid and dashed curves show the fit ( [ eq : errfunctfit ] ) for and , respectively .lower panel : estimate from eq .( [ eq : fit_lstar_epsilon ] ) of the threshold as a function of , for different threshold probabilities ( see legend).,scaledwidth=44.0% ]  in order to attack this problem , we consider the equivalence between an improperly arranged measurement setting [ i.e. , projections of the ideal generalized bell state performed on states that differ from those in eqs .( [ eq : optaxes ] ) , but still being mutually orthogonal ] and the performance of ideal measurements on a state that is locally rotated by an unknown amount and around an unspecified direction .we thus perform a stability analysis in the regime of small perturbations by independently rotating each subsystem by an angle around a random axis .the rotations are performed by applying the bi - local unitary transformation where ( ) is a -dimensional hermitian operator whose entries are randomly chosen from a uniform distribution of values within the range .we have performed an extensive numerical analysis of the value of the bell expressions and within the framework discussed above . in order to obtain the probability distributions for we have accumulated statistics by means of random rotations for each value of and considered in our study .a typical example of the results of this analysis is reported , for , in fig .[ fig3 ] : the distribution is a gaussian - like distribution with a negative skew . in order to provide a quantitative characterization of the robustness of the bell inequalities , we focus on the violation probability where for , respectively .such quantity is the area of the distribution at the rightmost side of the lhv value .we have studied the behavior of the the probability against the dimension of the problem at hand : in fig .[ fig2 ] ( top panel ) we plot as a function of the spin quantum number ( ) , for various choices of and for both and .we first note that for the results for the two bell expressions coincide . on the contrary , from onwards , , namely violation of the bell inequality for , which includes number of terms growing with the dimension of the system ,is more resistant to local perturbations than violation for .it is clear from the top panel of fig .[ fig2 ] that for all values of that we have considered and for a given minimum violation threshold , one can find a threshold value such that for all . in order to quantify , we fit the data of fig .[ fig2 ] ( top panel ) with the function \text{,}\ ] ] characterized by the mean value and the standard deviation .this allows us to estimate as with and for ; and for . in fig .[ fig2 ] ( bottom panel ) we report for several values of .these results clearly indicate that even for small error , the violation probability becomes rapidly negligible when increasing .  besides studying the probability to violate a given inequality , it is interesting to consider whether or not one is actually able to violate a bell inequality in a -dimensional system , for a given and a finite sampling .this is equivalent to estimate the quality of the statistical sampling of the tails of the gaussian - like distribution . in order to do so, we adopt a pragmatic approach according to which we consider it possible to violate inequality for the bell expression if , with a sampling of events , a single instance of violation is found .we then extract the maximum occurred value of the bell expression and study it against for increasing system dimensions .[ fig4 ] shows the results for the case of , highlighting the existence of a value above which the corresponding inequality is no longer violated .although the quantitative analysis is not reported here , we have found that an analogous critical value exists for the case of , too .maximal value for different spin quantum number .red horizontal line indicates the violation bound .,scaledwidth=44.0% ]  the values of both and at given are reported in fig .[ fig5 ] , which also displays the comparison with the fitting function , thus showing a power - law decay of the error threshold with the size of the system , with for and for .consistently with our findings on the probability to violate a given inequality , we find that is more robust than as the corresponding critical value decays with more slowly than .we complete this section by studying the stability of the generalized bell inequalities when global perturbations are used instead of the bi - local ones considered so far . by repeating the same analysis illustrated above under the assumption that the random perturbation now acts on both the spins of our system , thus possibly deteriorating their shared correlations, we find the results reported by the green lines in fig .a global rotation of the state of the system results in a more pronounced decay of the threshold value of that , however , does not affect significantly the scaling with the size of the system .from the same power law fit as above we obtain for and for . for random global rotations, it can be shown analytically ( see appendix [ app : levy ] ) that the asymptotic ( in ) decay of of the violation probability is at least exponentially fast .in this section , we consider the violation of generalized bell inequalities by a bipartite spin- system prepared in an unknown state . in order to make our study more specific and concrete ,we first consider a spin system prepared in a maximally entangled state but defined over random -axes . in such a case , it is no longer possible to identify the form of the optimal measurement operators and , and the measurement settings given in eqs.  would be unsuitable for the successful falsification of a given generalized bell inequality . in order to address this point quantitatively ,we resort to the observation that the picture given above is fully equivalent to that of a spin system prepared in the ideal generalized bell state and subjected to unknown local measurement settings .we thus perform measurements along random directions and check if this procedure allows to detect a violation with a statistically relevant probability .this procedure is not equivalent to the analysis we have done in the previous section , as the _ direction _ of the projections entailed by the measurement steps needed for the construction of the bell function are now fully random and not mutually orthogonal .every measurement operator is rotated individually .log - log plot of critical with corresponding fits ( form ) for ( solid red with squares ) and ( dashed red with circles ) .the solid green line with squares denotes the global critical for global rotations for inequality , the dashed green line with circles is for inequality .,scaledwidth=44.0% ]  in fig .[ fig6 ] we report the resulting distributions of the values taken by for various dimensions of the system . the distributions can be well fitted by gaussian functions specified by mean value and standard deviation decreasing with . specifically , we find the best fit values and .this implies that the probability of measuring a violation is as small as for , and is below machine precision for any .this is an interesting result , as it entails that , for systems with four or more local states , it is virtually impossible to violate a generalized bell inequality , even when operating over a maximally entangled state but with random direction .histograms of the bell inequality for bell state and random measurement operators , , , and , for various spin numbers .,scaledwidth=44.0% ]   optimized values ( upper panel ) and ( lower panel ) for different spin quantum numbers ( blue curve ) and maximally entangled state . in both panelsthe bottom ( green ) line corresponds to the border between classical and non - classical correlations.,scaledwidth=44.0% ]  given that random search fails to detect violation in general , we adopt a more sophisticated strategy and optimize the observables to be measured , employing the nelder - mead simplex algorithm  , which is a general - purpose gradient - free algorithm of vast use in complex multi - variable searches .we repeat the optimization from different randomly chosen initial sets of coefficients for the matrix representation of the observables and and report in fig .[ fig7 ] the maximum achieved value of and . as it can be seen , in both cases a violation of the bell inequalities is found for systems of up to and there is indication ( as it should be ) that for all system sizes it should be possible to find a set of observables capable to detect the violation .however , the computational time of one function evaluation scales as for inequality and as for  . in combination with an increasing number of function evaluations needed for increasing , this once more practically prevents the effectiveness of such protocol for increasing system sizes .  as a final step in our study, we now check how the violation of the generalized bell inequalities behaves for random two - spin states .we considered two cases :  1 .random entangled states whose vectors have elements randomly chosen from a uniform distribution of values within the range  ] , . for and , is the projector onto the d - dimensional subspace wherein , where :  the average of with respect to the uniform , normalized measure on is , because projectors which share the same superscript in and in are unitarily equivalent and the uniform measure is unitarily invariant .then levy s lemma states that  where is the lipschitz constant of , to be presently estimated .first we write : and we note that , for an arbitrary projector and arbitrary normalized states  it follows that :  in the above sum , projectors that share the superscript with different are mutually orthogonal , so is again a projector , and using this in ( [ here ] ) we get in the last line , we used that with the same and different are orthogonal .this inequality shows that . using this in levy s lemma yields the announced estimate .  a sharper boundis obtained by using the following version of levy s lemma : where the median is defined such that prob .while we have no theoretical estimate for , numerical results suggest that , at least with the observables we have chosen , the distribution of is even .this would imply , and then :   while even this stricter bound becomes relevant only for dimensions , bound ( [ main ] ) proves rigorously that the violation probability , upper bounded by the right - hand side of ( [ main ] ) for , decays at least exponentially fast with .n. d. mermin , phys .. lett . * 65 * , 1838 ( 1990 ) ; m. ardehali , phys . rev . a * 46 * , 5375 ( 1992 ) ; a. v. belinskii and d. n. klyshko , phys .36 * , 653 ( 1993 ) ; g. svetlichny , phys .d * 35 * , 3066 ( 1987 ) ; m. seevinck and g. svetlichny , phys .lett . * 89 * , 060401 ( 2002 ) .      c. vitelli , _ et al .a * 81 * , 032123 ( 2010 ) ; f. de martini , f. sciarrino , and c. vitelli , phys . rev . lett . * 100 * , 253601 ( 2008 ) ; p. sekatski , n. brunner , c. branciard , n. gisin , and c. simon , phys . rev. lett . * 103 * , 113601 ( 2009 ) ; b. sanguinetti , e. pomarico , p. sekatski , h. zbinden , and n. gisin , phys .lett . * 105 * , 080503 ( 2010 ) ; n. spagnolo , c. vitelli , m. paternostro , f. de martini , and f. sciarrino , phys .a * 84 * , 032102 ( 2011 ) ; n. bruno , _ et al ._ nature phys .* 9 * , 545 ( 2013 ) ; a. lvovsky , _ et al ._ nature phys .* 9 * , 541 ( 2013 ) .the longest computational time when computing one probability is needed for a matrix - matrix product , scaling with , where . for inequality only , but probabilities have to be calculated , resulting in a behavior .

Abstract: we explore the challenges posed by the violation of bell - like inequalities by -dimensional systems exposed to imperfect state - preparation and measurement settings . we address , in particular , the limit of high - dimensional systems , naturally arising when exploring the quantum - to - classical transition . we show that , although suitable bell inequalities can be violated , in principle , for any dimension of given subsystems , it is in practice increasingly challenging to detect such violations , even if the system is prepared in a maximally entangled state . we characterize the effects of random perturbations on the state or on the measurement settings , also quantifying the efforts needed to certify the possible violations in case of complete ignorance on the system state at hand .


understanding the complex features of various dynamical systems in the real world continues to be a crucial challenge across the physical and natural sciences . to better understand such complicated interactions it is useful to first transform the system into a new frame of reference .this is the approach we follow here . in this paper , we describe a method to transform a linear model built from a time series representing a ( possibly nonlinear , possibly stochastic ) dynamical system into a network which is capable of representing the time structure of the model to explore undiscovered structure of the original dynamical system .this method enables us to investigate the dynamics of the time series from a network perspective and allows us to explore the globally interrelated structure and the hierarchy .the network transformation provides us with a new representation of a generic model of the original dynamics . by examining the topology of the network ,we can directly obtain better understanding of the structure and the characteristics of the phenomenon represented by the time series modeling .connectivity between different features of the deterministic model structure can now be understood in greater detail than before thanks to this new toolbox from complex network theory .  over the past decadeit has become clear that networks as typified by `` complex networks '' have a vast range of applicability and give a new perspective on various problems in the real world  . in particular , there have been several recent works to combine time series with networks , such as networks through the correlation strength  , recurrence networks  , cycle networks for pseudo - periodic time series  and the horizontal visibility algorithm  .these approaches are proven to be effective in understanding complicated and entangled structure of systems .however , these works still do not address one essential ingredient .although systems in the real world have various different time delay or feedback effects in principle , such effects , which we refer to as `` time structure , '' can not be directly treated or set in the network by current approaches .furthermore , the network structures from time series in the existing approaches may not always be sufficient to capture the salient features of the underlying global interrelation structure , because these structures are essentially built on local information such as the temporal structures on an embedding space and correlations ( similarities ) between each pair of time series among many  .in addition , `` no similarity '' is not equivalent to `` no correlation . '' even when two signals are not similar , these systems can still have some kind of correlated structures . under these circumstances , it is crucial to find a method to translate directly the features of dynamics into network topology .  in this paper , to unravel entangled deterministic dynamical structures , we propose a method of constructing a network based on deterministic model structure from a given time series .time structure is directly set in the network and the network fully reflects the global structure of interrelation and the hierarchy of the components of the model .this is the first attempt to represent a generic deterministic model as a network in order to understand the deterministic dynamics of observed data with its time structure fully incorporated .it is usually difficult to extract the hidden or underlying nature of the structure of a given complicated time dependent phenomenon by directly examining it .however , it is relatively easy to obtain a time series data and build a model from the data . in principle, we can expect that the time series model reflects underlying features of the time series such as entangled time structures in varying degrees and therefore the model can be a good starting point for unravelling the time structure of the system .it should be emphasized that the method for transforming the time series model into network topology presented in this paper offers an approach completely different from existing ones to clearly visualise the hidden time structure of the system .this paper is organized as follows . in section [ sec : algorithm ] , the network construction algorithm from a given time series proposed in this paper is described . to demonstrate a concrete example of this algorithm ,it is applied to a simple `` toy model '' in section [ sec : numerical example ] .the effectiveness of the algorithm is shown in section [ sec : application ] , where the method is applied to two actual time series : annual sunspot numbers and a microtremor data set from an earthquake .we summarise the results in section [ sec : summary ] .to construct networks with time structure , we have to find the underlying time delays in time series as necessary elements and translate them into a network for visualization .an effective first step is the time series modelling . to obtain the precise interrelation of terms with time delays or the essential linear structures included in time series, we use an information theoretic reduction of linear models , the reduced autoregressive ( rar ) model  . unlike a standard ar model ,rar models include only terms that contribute significantly to the model , as assessed by an information criterion .a standard ar model is built up from terms with unit time delay , though some terms ( some time delays ) might not be necessary to capture the essential feature of phenomena .in contrast , rar models include only terms that contribute significantly to the model as assessed by an information criterion .the rar model has proven to be effective in modelling both linear and nonlinear dynamics  , and this model provides a generic method to locally linearize the system under consideration in a way which would be applicable to any smooth vector field , even though the underlying system is not linear .more details about rar models can be found in the literature  .the method described in this paper is composed of two steps : ( i ) building an rar model from a given time series and ( ii ) constructing a network from the model .we first give a brief review of the rar model .  given a time series of observations , an rar model with the largest time delay  can be expressed by where are unknown parameters , and is assumed to be independent and identically distributed gaussian random variables , which are interpreted as a fitting error . the parameters  are chosen to minimize the sum of squares of the fitting errors .among various information criteria used to find the best ( optimal ) model  , we employ the schwartz information criterion ( sic )  .the sic formula is defined by where is the number of data points , is the model size and is the fitting errors .  in building rar models , we need to select necessary terms .firstly , many candidate basis functions are prepared in the form of a dictionary .next we select as many basis functions that can extract the peculiarity of the time series as possible according to various model selection methods  .all of these model selection methods have a common difficulty , which is to identify the globally optimal model , due to trapping into local minima  . for finding the truly optimal model ,all possible combinations of basis functions have to be calculated ( an exhaustive search ) . although an exhaustive search for a dictionary smaller than approximately 30 is usually manageable , the calculation for a large dictionary costs enormous computational time and is unrealistic , as performing an exhaustive search is expected to be an np - hard problem  .it should be noted , however , that we can always resort to other methods for approximately finding the global minimum  , when exhaustive search is computationally impossible .the models obtained by selection algorithms are nearly optimal . to avoid this complexity caused by near optimality, we use exhaustive search in this paper in finding the optimal model that gives the minimum value of sic  . for verification, we also investigated the network structure of nearly optimal models and found that the core structure of the network built up from the optimal model is preserved in these nearly optimal models apart from the other `` leaf '' nodes around the core structure .   after obtaining a model representing the time series under consideration, we transform the model into a directed network ( i ) by representing each term  at time  in the model by a node labelled by the time and ( ii ) by drawing an arrow directed from a node  to the node  , where the time delay term appears in the rar model for the expression of .this arrow represents the influence of on with time delay  .it seems reasonable to assume that a large absolute value of the parameter represents the large influence of on .we treat the influence as a `` distance '' between nodes  and using on the network space ; the larger the absolute value of , the shorter the distance between and . since there have been no previous attempts to translate the values of the parameters into the network space , we introduce the following simple distance based on elementary linear algebra .we should mention the constant parameter  and dynamical noise  in the rar model .they surely play important roles in actual dynamics and both of them should be included in constructing a model .however , these terms contain no time information .since our purpose is to unravel entangled time structure of deterministic dynamics , we do not take them into account in defining the distance between terms with time delay in constructing a network .equation ( [ eq : rar ] ) can be interpreted as a linear combination of the set of linearly independent `` unit vectors , '' , with the coefficients , . by this interpretation, we introduce the `` angle ''  between the directions of and as the distance we introduce should have following properties .firstly , when vectors  and are in the same direction , the angle becomes and the distance  should be .we expect the analyticity of the `` distance '' around and put in this case . secondly , when the vectors and are perpendicular ( ) , the angle  becomes and the distance  should be infinity. thus should be inversely proportional to .finally , the distance must always be a positive real number .regarding all these requirements , we define the distance  between the nodes and as we consider that the concept of `` distance '' introduced here depends on the nature of the system , and other distances such as the inverse of the parameters may be justified in some situations .however , we consider that this distance is appropriate for most cases of linear models , because it reflects the overall balance of the size of parameters in the model .note that the proposed method is independent of the definition of the distance .according to eq .( [ eq : rar ] ) , the nodes contained in a model are directly connected to .the distance calculated by eq .( [ eq : distance ] ) is referred to as the direct distance ( _ dd _ ) .a pair of nodes , however , can be connected indirectly via some other nodes .the sum of all distances through the path between these two nodes is referred to as the indirect distance ( _ i d _ ) . in a networkwe sometimes find a path with a shorter _ i d _ than the _dd_. in such a case , we can consider that the indirect path that gives the shortest length is the path on which the information is passed through most effectively and that these two nodes are essentially connected through the shortest _i d _ path .we therefore treat the collection of the paths that have the shortest length for any given node pairs as the network of the system .the network constructed in this way reveals the underlying hierarchical structure of the linear model and enables us to know whether the influence of a term may come through other terms .we demonstrate the application of our algorithm and confirm our theoretical argument with a simple example .we begin by the following ( artificial ) rar model :   from the definitions of eqs .( [ eq : angle ] ) and ( [ eq : distance ] ) , we obtain the direct distances ( _ dd_s ) between the nodes as . in fig . [fig : network for simple example ] , we show the overall linkage of the network constructed from eq .( [ eq : simple example ] ) within the time interval from 1 to 15 .   to ) represented by the rar model , eq .( [ eq : simple example ] ) , where the numbers on the nodes are of in eq .( [ eq : simple example ] ) .the first six nodes represented by squares are the initial nodes .the numbers on the arrows are the direct distances between nodes .the positions of the nodes are irrelevant ( only the topology is important ) and the length of the arrows also does not represent the actual scale . from this figure , we can clearly see that there are cases where an indirect distance between a pair of nodes is much shorter than the direct distance .for example , the indirect distance between node and via node is , while their direct distance is .,width=302 ]  the distance between nodes represents the magnitude of influence from the other nodes .according to the model , the nodes  , and are directly connected with the node .for example , the nodes , and are directly connected with the node in fig .[ fig : network for simple example ] . as mentioned in the previous section , however , the _ dd _ is not always the shortest .for example , the optimal ( shortest ) path from to is the one via , while the last node  is directly connected with as fig .[ fig : network for simple example ] shows .since both of the _ dd_s from to and from to are 1.67 , the _ i d _ from to is the summation , , which is much shorter than the _ dd _ from to , , we can thus conclude that the most significant influence of the term to is not the direct one but that comes through the term .  ) .the nodes shown are from to the one with the largest time delay , .note that the node is not directly connected to but connected through .the gray colour corresponds to terms included in the model and the white colour to terms within the largest time delay but not included in the model , respectively .the numbers on the arrows are the direct distances between the nodes .we show all terms within the largest time delay to show the flow of time in the model without gaps and the structure more clearly.,width=264 ]  to show this situation more clearly , we show in fig .[ fig : mst network for simple example ] the set of most optimal paths to node from the nodes within the model ( between and ) .figure [ fig : mst network for simple example ] shows that the node is directly connected with and , and the connection from is indirect via .this is an outcome of the interplay between the sizes of the parameters caused by the interrelation and hierarchy between terms of the model , eq .( [ eq : simple example ] ) . in this way, the network topology constructed by the present method reveals the time structure .note that we can not extract this information by simply examining eq .( [ eq : simple example ] ) .this is the manifestation of the underlying structure of dynamical systems through the network representation .based on the result of these computational studies , we apply the proposed method to two real world data : ( i ) annual sunspot numbers and ( ii ) microtremor data .we first apply the proposed method to annual sunspot numbers from year 1700 to 2008 , and the model is built with time delays up to 15  .( see fig .[ fig : sunspot_numbers ] . )the obtained rar model is three time delay terms , and are selected . and each direct distance to the node  is , and , respectively .figure [ fig : sunspot_network ] shows that the optimal path network of the nodes from to has a simple chain structure .this result indicates that the fundamental mechanism of the oscillation of annual sunspot numbers is simple and the previous sunspot numbers influence the next numbers in series . from this resultwe can see the correspondence between the periodicities and the chain structure of the annual sunspot numbers .  ) , for the annual sunspot numbers .the nodes are from to the largest time delay in the model . for the explanation of notations used in this figure , see fig .( [ fig : mst network for simple example]).,width=302 ]   we next apply the proposed method to microtremor data , which shows rather non - trivial results .these data were taken from the east - west components of the ground velocity signal of the 1982 urakawa - oki ( hokkaido , japan ) earthquake .the data was measured at 50 hz and consists of 2600 data points in total .  to demonstrate how the network structure changes in the course of time development of the oscillation , we show in fig . [fig : earthquake_data ] the network structures corresponding to three different time regions in this microtremor data .three models are built with time delays up to 15  .it should be noted again that , for the calculation of the direct distances to the node , we do not use the constant parameter in the obtained model and dynamic noise .  .note that the positions of the nodes are irrelevant . , width=377 ]  for the time region ( a ) , the obtained rar model has one constant parameter and eight time delay terms , we obtain the following eight direct distances to the node corresponding to each term : .the network for the time region ( a ) in fig . [ fig : earthquake_data ] shows that while the nodes , , , and are directly connected to the node  , the nodes , and are not .the network also indicates another interesting feature of this model .the region ( a ) in fig .( [ fig : earthquake_data ] ) implies that this network structure contains seemingly two `` hubs , '' nodes and , because other nodes flock to them .two chains of nodes separated by the shortest distance , , with time delay 4 can be found , the one of which connects nodes , and and the other of which connects nodes and .we can thus consider from this information that time delay 4 is a major period of this oscillation .we consider that time delays 1 and 3 also play significant roles that can not be ignored , because we can find several chains of nodes with time delay 1 and 3 . for time delay 1 , there are the chain of nodes , and and the chain of nodes and . for time delay 3 , there are the chain of nodes and and the chain of nodes and .it is highly probable that a hub structure appears when an oscillation consists of a main oscillation , where each unit oscillation is furthermore affected by other underlying oscillations with shorter time scales than the main oscillation period .as the absolute parameter value of the term is the largest in the model , one may think that the time difference 4 is the main period and that the nodes and can be `` hubs . ''it is not straightforward , however , to know the significance of time delays 1 and 3 by simply examining the model , eq .( [ eq : p_wave ] ) .the rar model for the time region ( b ) is the direct distances to the node are , respectively .the rar model for the time region ( c ) is the direct distances to the node are , respectively . from figs .[ fig : earthquake_data ] ( a)-(c ) , we notice that the network structure develops from a star - like structure containing hubs to a relatively simple chain structure during the course of oscillation .a chain structure implies that the oscillation at present time is affected by a previous unit oscillation in a simple manner . in this way, our network construction method exhibits the transition in the nature of time series through network structure in addition to the detailed hierarchical relationship between terms in the model .one may think that simple visual inspection of the oscillation profile is sufficient for identifying three regions ( a)-(c ) .it should be emphasized , however , that the simple visual inspection can not reveal the structural differences in the nature of oscillation in these regions .in this paper , we described an algorithm to construct networks with time structure based on deterministic model structure from time series . in this algorithm a linear model containing various terms of different time delays is transformed into network topology .the advantage of this method over the existing ones is that the global structure of the relationship between terms in a time series model is directly translated to the topology of the corresponding network . by extracting the optimal paths for the constructed network , we can find the global structure and hierarchy between terms .the transition in the nature of time dependency is represented by the transformation in the structure of constructed networks . using this correspondence between the nature of oscillation and the structure of the network, we could compare and categorize different model realizations .in addition , networks constructed by the proposed method consists of temporal nodes .it should be emphasized that the insight into the transition in the nature of oscillation through network structure described in this paper can only be gained by network representation of time series with temporal nodes . to the best of our knowledge ,networks with temporal nodes are not well studied and there remains many problems unsolved .our arguments and computational examples show the effectiveness of introducing networks with time structure and that the proposed method has a wide range of applicability and provides us profound insights in investigating time dependent phenomena .the effectiveness and the usefulness of the basic idea presented here should be verified and enlarged by future works . at present , we have two following problems .one is to treat nonlinearity more appropriately and naturally .this requires to find a proper method for translating nonlinear basis functions into network topology .the other is to treat multivariate time series data .though we present our method by univariate time series data in this paper , systems in the real world are not always isolated from their surroundings .each factor or element in those systems is interconnected or interrelated in some way or another to varying degrees .hence , it is important to extend our method to be applicable to multivariate data .we would like to thank the anonymous referees for their very valuable remarks .x. xu , j. zhang , m. small , superfamily phenomena and motifs of networks induced from time series , proceedings of the national academy of sciences of the united states of america 105 ( 2008 ) 1960119605 .

Abstract: we propose a method of constructing a network , in which its time structure is directly incorporated , based on a deterministic model from a time series . to construct such a network , we transform a linear model containing terms with different time delays into network topology . the terms in the model are translated into temporal nodes of the network . on each link connecting these nodes , we assign a positive real number representing the strength of relationship , or the `` distance , '' between nodes specified by the parameters of the model . the method is demonstrated by a known system and applied to two actual time series .   time series modelling ; networks ; minimum spanning trees


the study of dynamics of opinion formation is nowadays a hot topic in the statistical physics of complex systems , with a considerable amount of papers published in the last years ( see and references therein ) .even simple models can exhibit an interesting collective behavior that emerges from the microscopic interaction among indivuals or agents in a given social network .usually those models exhibit nonequilibrium phase transition and rich critical phenomena , which justifies the interest of physicists in the study of opinion dynamics .the majority - vote model have been extensively studied in the last 20 years . in the standard model , each site of a square latticehas an ising - spin variable whose state may be associated with the opinion of an individual in a given social community .the time evolution of the model is governed by an inflow dynamics , where the center spin is influenced by its nearest neighbors : an individual located at the central site adopts the minority sign of the spins in its neighborhood with probability and the majority sign with probability .numerical results indicate that the model undergoes a nonequilibrium phase transition at a critical noise , and the critical exponents were found to be the same as those of the equilibrium 2d ising model , i.e. , , and  .the transition separates an ordered phase , where one of the opinions dominates the population , from a disordered one , where the two opinions coexist in equal proportions .after that , many extensions of the original model were proposed .for example , we can highlight the interaction between two different classes of agents , spins with three - state variables , noise distributed according to a bimodal distribution , diffusion and heterogeneous agents .the influence of topology was also considered , with studies on random , hypercubic and hyperbolic lattices , random graphs , small - world and scale - free networks , among others .some new universality classes were found for different topologies , but in some of the above - mentioned modifications the critical exponents are ising - like , i.e , they are the same as those of the original majority - vote model .the consideration of the usual noise in the majority - vote model produces an effect similar to the introduction of the _ contrarians _ in the population , individuals that adopt the choice opposite to the prevailing choice of the others , whatever this choice is .indeed , is the probability of an agent assume the opposite opinion shared by the local majority .the contrarian effect is called _ anticonformism _ in the language of social sciences , and it is one kind of nonconformism .another kind of nonconformist is _ independence _, where the agent also take cognizance of the group norm , but he / she decides to take one of the possible opinions ( in the case of the majority - vote model ) independently of the majority or the minority opinion in the group .effects of independence on phase transitions in opinion models were considered recently .  in this workwe consider the two mentioned kinds of nonconformity , anticonformity and independence .the last is introduced in the system as a probability , in a way that a given agent chooses one of the two possible opinions with equal probability ( ) .thus , the population evolves under the presence of two types of noises , represented by the parameters and .we consider agents placed on fully - connected networks and on square lattices , and our interest is to study the effects of the two kinds of stochastic driving in the phase transitions of the model .this work is organized as follows . in section 2we present the microscopic rules that define the model , and the analytical and numerical results are discussed in two distinct subsections , considering the mean - field approximation and the square lattice .finally , our conclusions are presented in section 3 .our model is based on the majority - vote model .every site of a given lattice with sites is occupied by an agent , and to each site we assign a random opinion with equal probability , corresponding to the two possible opinions in a certain subject . in the original model , a randomly chosen individual follows the opinion of the minority of its nearest neighbors with probability and adopts the majority sign of the spins in its neighborhood with probability . in this work we will consider the effects of the independent behavior , where a randomly chosen individual acts independently of their neighbors with probability . in this case ,he / she chooses one of the two possible opinions with equal probability .on the other hand , with the complementary probability we apply the original rule of the majority - vote model . summarizing , considering the effects of the two noises and , a given spin flipped with probability + \frac{p}{2 } ~,\ ] ] where , if , and the summation is over the nearest neighbors . observe that in the absence of the independent behavior ( ) we recover the standard flip probability of the majority - vote model .thus , our model presents two distinct types of stochastic driving , governed by the two noises and . in the language of social sciences, we are considering two kinds of nonconformity , namely anticonformity ( contrarian effect , parameter ) and independence ( parameter ) .we are interested in the critical behavior of the model .as the parameter is the novelty of the model , we will consider the quantities of interest as functions of , for typical values of .these quantities are magnetization per spin , the susceptibility and the binder cumulant , given by respectively . in eqs .( [ eq2 ] ) , ( [ eq3 ] ) and ( [ eq4 ] ) , denotes a configurational average taken at steady states .  in this workwe will consider two distinct topologies for the model , the fully - connected network and the square lattice .these distinct cases will be treated separately in the following subsections .   in this sectionwe consider that each agent can interact with all others , which corresponds to a mean - field limit .following ref . , one can derive analytically the behavior of the stationary magnetization .a given configuration of the system can be denoted by .the time evolution of the probability to found the system in the state at a time is governed by the master equation  ~.\ ] ] in the above equation , the state can be obtained from the state by the flip of the spin on site , i.e. , , and the factor can be interpreted as the flip rate of the -th site ( ) , given by eq .( [ eq1 ] ) .the summation is over all the sites . from eq .( [ eq5 ] ) , one can get the time evolution of the average , considering our modified transition rate given by eq .( [ eq1 ] ) , eq .( [ eq6 ] ) gives us in the mean - field limit , we choose a random site , that will be the `` central '' site , and its `` nearest '' neighbors are also randomly chosen .if these neighbors are labeled as , , and , one can write  the mean - field approximation disregards all correlations , and one can write thus , using eqs .( [ eq8 ] ) , ( [ eq9 ] ) and ( [ eq10 ] ) in eq .( [ eq7 ] ) , one obtains where   at the stationary state we have , and eq .( [ eq11 ] ) give us two distinct solutions .the first one is the disordered state solution , valid for , and the second one is the ordered state solution , valid for .the last solution , using the definition and eq .( [ eq12 ] ) , gives us or in the usual form , where   as a function of the independence probability for typical values of the noise , considering the mean - field formulation of the model .the symbols are numerical results for population size , averaged over independent simulations , and the full lines are the analytical prediction given by eq .( [ eq13 ] ) .observe that for sufficiently large values of and the system is disordered , which corresponds to the analytical solution .,scaledwidth=40.0% ]  and we found a typical mean - field exponent , as expected , indicating that the model should belong to the mean - field ising universality class .( [ eq14 ] ) give us for , in agreement with the standard mean - field majority - vote model , i.e. , in the absence of independence .in addition , this result shows that there is another transition due to independence , , i.e. , even in the absence of the usual noise the system undergoes an order - disorder transition due to the presence of the independent behavior . summarizing, there is an order - disorder phase transition in the plane versus for and ; otherwise , the only valid solution is the disordered one , .in other words , for sufficient large independence probability ( ) , the phase transition is suppressed , and the system is disordered for all values of . finally , another important result comes from eq .( [ eq13 ] ) : the consensus states with are obtained only for , in the absence of the two noises .  to verify the above calculations, we performed computer simulations of the majority - vote model considering a fully - connected graph . in this case , all the agents considering during an interaction are randomly chosen . in fig .[ fig1 ] we show the magnetization per spin as a function of , estimated numerically from eq .( [ eq2 ] ) and a comparison with the derived eq .( [ eq13 ] ) , for typical values of .notice that for there is no transition anymore , for all values of , in agreement with the analytical prediction that give us ( see the curve for in fig .[ fig1 ] ) .in addition , one can see a transition even in the absence of the usual noise , generated by the independent behavior of the agents .( independence probability ) versus ( usual noise ) .one can see the ordered and the disordered phases .the squares are the numerical estimates of the critical points , and the full line is the analytical prediction , eq .( [ eq14 ] ) .the error bars are smaller than data points.,scaledwidth=40.0% ]  as a final discussion , we performed a finite - size scaling ( fss ) analysis ( not shown ) for typical values of in order to obtain numerical estimates for the critical exponents of the model , as well as for the critical points .based on the standard fss equations , we obtained , and , for all values of , indicating that the model belongs to the mean - field ising universality class , as suggested by the analytical calculations .in addition , the fss analysis allows us to identify the critical points by the crossing of the binder cumulant curves for distinct population sizes .considering the estimated values of for typical values of , we exhibit in fig .[ fig2 ] the phase diagram of the model in the plane versus .we also plot the analytical prediction , eq .( [ eq14 ] ) , and one can see that we have a good agreement among the analytical and numerical results .   in this subsectionwe simulate our model on square lattices of distinct sizes , in order to verify if the mean - field predictions are at least qualitatively valid when we consider a neighborhood .  in fig .[ fig3 ] we exhibit results for the magnetization per spin as a function of , for and typical values of the noise .one can observe an order - disorder phase transition even for , and for sufficiently large values of there is no transition anymore .this result is in agreement with the numerical estimate of the critical point  . as in the previous case , for sufficient large values of ( , as we will see in the following ) ,the phase transition is suppressed , and the system is disordered for all values of .   as a function of the independence probability for typical values of the noise , considering the model defined on a 2d square lattice of linear size .observe that for sufficiently large values of and the system is disordered .each point is averaged over independent simulations , and the lines are just guides to the eye.,scaledwidth=40.0% ]  one can estimate the critical points and exponents considering the fss eqs .( [ eq15 ] ) to ( [ eq18 ] ) , considering the change .one example is exhibited in fig .[ fig4 ] , for .we obtained the same critical exponents for all values of , , and , i.e. , the model on the square lattice belongs to the 2d ising model universality class .in other words , the critical exponents are not affected by the introduction of another noise in the model , namely independence .  .the best collapse of data was found for , , and .,title="fig:",scaledwidth=30.0% ] .the best collapse of data was found for , , and .,title="fig:",scaledwidth=30.0% ] + .the best collapse of data was found for , , and .,title="fig:",scaledwidth=30.0% ]  thus , the mean - field formulation presented in the previous subsection predicts some important behaviors : ( i ) there is an order - disorder transition even for , generated by the independent behavior ; ( ii ) the model is in the ising model universality class ; ( iii ) the consensus states are obtained only for . despite the quantitative differences between the two formulations , mean field and square lattice , the above results are in agreement with the simulations of the model on the square lattice .considering the critical points estimated from the crossing of the binder cumulant curves , we exhibit in fig .[ fig5 ] the phase diagram of the model on the square lattice , in the plane versus . based on eq .( [ eq14 ] ) , we propose a qualitative description of the phase boundary that separates the ordered and the disordered phase , based on the relation with 4 parameters , , and .fitting data , we obtained the estimates .this heuristic description presents a good agreement with the critical points obtained from fss analysis , as shown in fig .in particular , for we have and for , and the last result is in agreement with the result obtained in the original majority - vote model .   versus , separating the ordered and the disordered phases .the squares are the numerical estimates of the critical points , obtained by the crossing of the binder cumulant curves .the error bars are smaller than data points , and the dashed line is a qualitative description of the phase boundary , as discussed in the text.,scaledwidth=40.0% ]in this work , we have studied the majority - vote model in the presence of two noises .the first one is the usual noise , representing the hesitance of an agent to follow a local majority opinion . in this case , this behavior is similar to the galam s contrarians . on the other hand , we considered the independent behavior , governed by a probability that an agent chooses his / her own opinion or with equal probability , independent of the local majority or minority opinion .we considered the model on a fully - connected network ( mean - field formulation ) and on a square lattice , and despite the quantitative differences between the two formulations , as expected , the results in both cases are in qualitative agreement .we performed monte carlo simulations of the model for distinct population sizes . in this case , the above - mentioned two noises contribute to disorder the system . for both cases ( mean field and 2d ) ,consensus states are only obtained in the total absence of noise , i.e. , for .on the other hand , we have an order - disorder transition even in the absence of the usual noise , due to presence of the independent behavior in the population .furthermore , the phase transition is in the same universality class of the equilibrium ising model , i.e. , the critical exponents are not affected by the presence of a second noise ( independence ) .this result is in accordance to the grinstein s criterion that states that nonequilibrium stochastic spin - like systems with up - down symmetry in regular lattices fall into the same universality class of the equilibrium ising model . at mean - field level , the results were complemented by analytical calculations , based on the master equation .we also verified that there is a threshold value above which the system is disordered for all values of . in terms of a public debate with two distinct choices ,it means that if at least a fraction of the population takes independent decisions , there is no final decision in the debate , independent of the presence of contrarians .otherwise , there can be a majority opinion dominating the population , which means a decision on the debate . in the mean - field approximation of the model, we showed analytically that the mentioned threshold value is .as an extension of this work , it would be interesting to verify how results change with the number of neighbors .it could occur , similarly as in the q - voter model , that the phase transition changes its nature from continuous to discontinuous with the increase of the number of neighbors .the authors acknowledge financial support from the brazilian funding agency cnpq .

Abstract: in this work we study the majority - vote model with the presence of two distinc noises . the first one is the usual noise , that represents the probability that a given agent follows the minority opinion of his / her social contacts . on the other hand , we consider the independent behavior , such that an agent can choose his / her own opinion or with equal probability , independent of the group s norm . we study the impact of the presence of such two kinds of stochastic driving in the phase transitions of the model , considering the mean field and the square lattice cases . our results suggest that the model undergoes a nonequilibrium order - disorder phase transition even in the absence of the noise , due to the independent behavior , but this transition may be suppressed . in addition , for both topologies analyzed , we verified that the transition is in the same universality class of the equilibrium ising model , i.e. , the critical exponents are not affected by the presence of the second noise , associated with independence .       keywords : dynamics of social systems , collective phenomena , computer simulations , phase transitions


the primary purpose of this paper is to demonstrate the result stated in the title :  [ thm : main ] the maximum likelihood estimators of gaussian mixture models are transcendental functions .more precisely , there exist rational samples in whose maximum likelihood parameters for the mixture of two -dimensional gaussians are not algebraic numbers over .the principle of maximum likelihood ( ml ) is central to statistical inference .most implementations of ml estimation employ iterative hill - climbing methods , such as expectation maximization ( em ) .these can rarely certify that a globally optimal solution has been reached .an alternative paradigm , advanced by algebraic statistics  , is to find the ml estimator ( mle ) by solving the likelihood equations .this is only feasible for small models , but it has the benefit of being exact and certifiable . an important notion in this approach is the _ ml degree _ , which is defined as the algebraic degree of the mle as a function of the data .this rests on the premise that the likelihood equations are given by polynomials .many models used in practice , such as exponential families for discrete or gaussian observations , can be represented by polynomials .hence , they have an ml degree that serves as an upper bound for the number of isolated local maxima of the likelihood function , independently of the sample size and the data .the ml degree is an intrinsic invariant of a statistical model , with interesting geometric and topological properties .the notion has proven useful for characterization of when mles admit a ` closed form ' . when the ml degree is moderate , these exact tools are guaranteed to find the optimal solution to the ml problem .however , the ml degree of a statistical model is only defined when the mle is an algebraic function of the data .theorem [ thm : main ] means that there is no ml degree for gaussian mixtures .it also highlights a fundamental difference between likelihood inference and the method of moments .the latter is a computational paradigm within algebraic geometry , that is , it is based on solving polynomial equations .ml estimation being transcendental means that likelihood inference in gaussian mixtures is outside the scope of algebraic geometry .  the proof of theorem [ thm : main ] will appear in section [ sec2 ] . in section [ sec3 ] we shed further light on the transcendental nature of gaussian mixture models .we focus on mixtures of two univariate gaussians , the model given in ( [ eq : pdf - uni ] ) below , and we present a family of data points on the real line such that the number of critical points of the corresponding log - likelihood function ( [ eq : loglik1 ] ) exceeds any bound .while the mle for gaussian mixtures is transcendental , this does not mean that exact methods are not available .quite to the contrary .work of yap and his collaborators in computational geometry convincingly demonstrates this . using root bounds from transcendental number theory ,they provide certified answers to geometric optimization problems whose solutions are known to be transcendental .theorem [ thm : main ] opens up the possibility of transferring these techniques to statistical inference . in our view , gaussian mixtures are an excellent domain of application for certified computation in numerical analytic geometry .transcendental number theory is a field that furnishes tools for deciding whether a given real number is a root of a nonzero polynomial in  ] . the aim is to estimate the five model parameters from a collection of data points .the _ log - likelihood function _ of the model ( [ eq : pdf - uni ] ) is this is a function of the five parameters , while are fixed constants .the principle of maximum likelihood suggests to find estimates by maximizing the function over the five - dimensional parameter space \times { \mathbb{r}}^2 \times { \mathbb{r}}^2_{>0} ] , and , and match the first mean to the first data point .the remaining function of one unknown equals  \,+\ , \text{const } \enspace .\ ] ] the lower bound tends to as .remark [ rmk : notbounded ] means that there is no global solution to the mle problem .this is remedied by restricting to a subset of the parameter space . in practice ,maximum likelihood for gaussian mixtures means computing local maxima of the function  .these are found numerically by a hill climbing method , such as the em algorithm , with particular choices of starting values .see section [ sec3 ] .this method is implemented , for instance , in the r package mclust . in order for theorem [ thm : main ] to cover such local maxima , we prove the following statement :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ there exist samples such that every non - trivial critical point of the log - likelihood function in the domain has at least one transcendental coordinate . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  here , a critical point is _ non - trivial _ if it yields an honest mixture , i.e. a distribution that is not gaussian . by the identifiability results of , this happens if and only if the estimate satisfies and .the log - likelihood function always has some algebraic critical points , for any .indeed , if we define the empirical mean and variance as then any point with and is critical .this gives a gaussian distribution with mean and variance , so it is trivial .first , we treat the univariate case .consider the partial derivative of ( [ eq : loglik1 ] ) with respect to the mixture weight :   .\end{aligned}\ ] ]  clearing the common denominator we see that if and only if \\\label{eq : partial - alpha - cleared }   & \times\prod_{j\not = i }\biggl [ \frac{\alpha}{\sigma_1 } \,\exp    \bigl(-\frac{(x_j-\mu_1)^2 } { 2\sigma_1 ^ 2 } \bigr )  \,+\ ,  \frac{1-\alpha}{\sigma_2 } \,\exp \bigl(-\frac{(x_j-\mu_2)^2    } { 2\sigma_2 ^ 2 } \bigr )   \biggr]\;=\ ; 0 \enspace .\end{aligned}\ ] ] letting and , we may rewrite the left - hand side of ( [ eq : partial - alpha - cleared ] ) as             \prod_{j\not = i } \left [ \sum_{k_j=1}^2           \frac{\alpha_{k_j}}{\sigma_{k_j } } \,\exp     \bigl(-\frac{(x_j-\mu_{k_j})^2 } { 2\sigma_{k_j}^2 } \bigr )  \right ] .   \end{aligned}\ ] ]  we expand the products , collect terms , and set . with this, the partial derivative is zero if and only if the following vanishes :  \\    & \ ! \ ! = \!\ !\sum_{k\in\{1,2\}^n } \!\!\!\ !   \exp \! \left ( \ ! -\sum_{j=1}^n \frac{(x_j-\mu_{k_j})^2    } { 2\sigma_{k_j}^2 } \right ) \!\ ! \left(\prod_{j=1}^n   \frac{1}{\sigma_{k_j } }  \right )    \alpha^{n_1(k)-1}(1-\alpha)^{n - n_1(k)-1}(n_1(k)-n\alpha ) \enspace .\,\,\ ,  \end{aligned}\ ] ]  let be a non - trivial isolated critical point of the likelihood function .this means that and .this point depends continuously on the choice of the data . by moving the vector with these coordinates along a general line in , the mixture parameter moves continuously in the critical equation above . by the implicit function theorem, it takes on all values in some open interval of , and we can thus choose our data points general enough so that is not an integer multiple of . we can further ensure that the last sum above is a -linear combination of exponentials with nonzero coefficients .suppose that is algebraic .the lindemann - weierstrass theorem implies that the arguments of are all the same .then the numbers are all identical . however , for , and for general choice of data as above , this can only happen if .this contradicts our hypothesis that the critical point is non - trivial .we conclude that all non - trivial critical points of the log - likelihood function ( [ eq : loglik1 ] ) are transcendental .  in the multivariate case ,the model parameters comprise the mixture weight  ] . if is large , then any global maximum of is in the interior of \times \mathbb{r} ] . running the em algorithm ( as explained in the proof of theorem [ thm : manyhills ] below ) yields the non - trivial critical points reported in table [ tab : k7 ] .their coordinates are seen to be close to the cluster midpoints for all .the observed symmetry under reversing the order of the rows also holds for all larger .our proof of theorem [ thm : manyhills ] will be based on the _ em algorithm_. we first recall this algorithm .let be the mixture density from ( [ eq : pdf - uni ] ) , and let be the two gaussian component densities .define which can be interpreted as the conditional probability that data point belongs to the first mixture component .further , define and , which are expected cluster sizes . following ( * ? ? ?* section 9.2.2 ) , the likelihood equations for our model can be written in the following fixed - point form : in the present context , the em algorithm amounts to solving these equations iteratively .more precisely , consider any starting point .then the e - step ( `` expectation '' ) computes the estimated frequencies via ( [ probs ] ) . in the subsequent m - step ( `` maximization '' ) , one obtains a new parameter vector by evaluating the right - hand sides of the equations ( [ firstlik])-([lastlik ] ) .the two steps are repeated until a fixed point is reached , up to the desired numerical accuracy .the updates never decrease the log - likelihood . for our problemit can be shown that the algorithm will converge to a critical point ; see e.g.  .fix .we choose starting parameter values to suggest that the pair belongs to the first mixture component , while the rest of the sample belongs to the second .explicitly , we set we shall argue that , when running the em algorithm , the parameters will always stay close to these starting values .specifically , we claim that throughout all em iterations , the parameter values satisfy the inequalities  the starting values proposed above obviously satisfy the inequalities in ( [ eq : box1 ] ) , and it is not difficult to check that ( [ eq : box2 ] ) and ( [ eq : box3 ] ) are satisfied as well .to prove the theorem , it remains to show that ( [ eq : box1])-([eq : box3 ] ) continue to hold after an em update .  in the remainder , we assume that . for smaller values of the claim of the theorem can be checked by running the em algorithm . in particular , for , the second standard deviation satisfies the simpler bounds key property is that the quantity , computed in the e - step , is always very close to zero for . to see why , rewrite ( [ probs ] ) as since , we have . on the other hand , . using that , their product is thus bounded below by 0.3209 .turning to the exponential term , the second inequality in ( [ eq : box1 ] ) implies that for or , which index the data points closest to the pair .using ( [ eq : box4 ] ) , we obtain from , we deduce that .the exponential term becomes only smaller as the considered data point move away from the pair . as increases , decreases and can be bounded above by a geometric progression starting at and with ratio .this makes with negligible .indeed , from the limit of geometric series , we have and similarly , satisfies the two sums and are relevant for the m - step .  the probabilities and give the main contribution to the averages that are evaluated in the m - step .they satisfy .moreover , we may show that the values of and are similar , namely : which we prove by writing and using to bound   bringing it all together , we have using and ( [ series ] ) , as well as the lower bound in ( [ eq : bound : gamma : ratio ] ) , we find using the upper bound in ( [ eq : bound : gamma : ratio ] ) , we also have .hence , the second inequality in ( [ eq : box1 ] ) holds .the inequalities for the other parameters are verified similarly .for instance , holds for .therefore , the first inequality in ( [ eq : box1 ] ) continues to be true .we conclude that running the em algorithm from the chosen starting values yields a sequence of parameter vectors that satisfy the inequalities ( [ eq : box1])-([eq : box4 ] ) .the sequence has at least one limit point , which must be a non - trivial critical point of the log - likelihood function .therefore , for every , the log - likelihood function has a non - trivial critical point with .we showed that the maximum likelihood estimator ( mle ) in gaussian mixture models is not an algebraic function of the data , and that the log - likelihood function may have arbitrarily many critical points .hence , in contrast to the models studied so far in algebraic statistics , there is no notion of an ml degree for gaussian mixtures .however , certified likelihood inference may still be possible , via transcendental root separation bounds , as in .  the _ cauchy - location model_ , treated in , is an example where the ml estimation is algebraic but the ml degree , and also the maximum number of local maxima , depends on the sample size and increases beyond any bound .  the ml estimation problem admits a population / infinite - sample version . herethe maximization of the likelihood function is replaced by _minimization of the kullback - leibler divergence _ between a given data - generating distribution and the distributions in the model .the question of whether this population problem is subject to local but not global maxima was raised in  in the context of gaussian mixtures with known and equal variances .it is known that the kullback - leibler divergence for such gaussian mixtures is not an analytic function .readers of japanese should be able to find details in .as previously mentioned , theorem [ thm : main ] shows that likelihood inference is in a fundamental way more complicated than the classical _ method of moments _ .the latter involves only the solution of polynomial equation systems .this was recognized also in the computer science literature on learning gaussian mixtures , where most of the recent progress is based on variants of the method of moments rather than likelihood inference .we refer to for a study of the method of moments from an algebraic perspective .section 3 in that paper illustrates the behavior of pearson s method for the sample used in theorem [ thm : manyhills ] .m. buot , s. hoten , and d. richards , _ counting and locating the solutions of polynomial systems of maximum likelihood equations .ii . the behrens - fisher problem _ , statistica sinica * 17 * ( 2007 ) , no . 4 , 13431354 .   s.w .choi , s. pae , h. park and c. yap : _ decidability of collision between a helical motion and an algebraic motion _, in g. hanrot and p. zimmermann ( eds ) : 7th conf . on real numbers and computers , loria , nancy , france , 2006 , pp. 6982 .

Abstract: gaussian mixture models are central to classical statistics , widely used in the information sciences , and have a rich mathematical structure . we examine their maximum likelihood estimates through the lens of algebraic statistics . the mle is not an algebraic function of the data , so there is no notion of ml degree for these models . the critical points of the likelihood function are transcendental , and there is no bound on their number , even for mixtures of two univariate gaussians .