following the earlier version of the kolkata restaurant problem as a generalization of the el farol bar problem and the consequent minority game problem , we introduce a very simple version of the game : the kolkata paise restaurant ( kpr ) problem , which is a repeated game with no mutual interaction .the set of agents ( restaurants ) can be macroscopically large and agents have a common preference ranking across the restaurants .we first introduce one - shot kpr games .any given one - shot kpr game is repeated each period in the kpr problem . for any one - shot kpr gamethe set of pure strategy nash equilibria coincides with the set of socially ( pareto ) efficient outcomes . by socially efficient outcomes we mean outcomes where each agent goes to a separate restaurant and each restaurant gets exactly one agent to serve .if agents are rational and is small , one can sustain any socially efficient outcome in equilibrium of the repeated game version of the kpr problem ( see ) .however , we assume that is large and hence an agent can not rely on other agents actions and therefore what matters to any agent is the past collective configuration of actions and the resulting average utilization of restaurants . assuming very ` simple ' stochastic strategies based on history and past individual experiences and assuming no mutual interaction, we observe that the kpr problems are ` intrinsically dynamic ' in the sense that , the system in general never converges to any of the states and most of the time , the system passes through socially inefficient states ( for a special example of converging dynamics , see sec . [ os ] ) .our main objective is to analyze the statistics of the fraction of restaurants utilized per day ( utilization fraction ) associated with various stochastic strategies . in particular , we investigate some of the optimally utilized states of the system , the average occupation ( utilization ) fraction of the restaurants on any day and their fluctuations .let there be restaurants in kolkata and assume that each can accommodate a fixed number of customers , normalized to unity .let there be exactly customers or agents in the city .though each of the restaurants costs the same , we assume that there is a common preference ranking of these restaurants to the agents . if , in a day , there is more than one agent ( prospective customer ) arriving in any particular restaurant , one of them is chosen randomly and the rest do not get the service . _ in kolkata , there happened to exist very cheap and fixed rate paise restaurants " ( also called `` paise hotels '' ) , popular among the daily laborers in the city . during lunch hours, the laborers used to walk down ( to save the transport costs ) to any of these restaurants and would miss the lunch if they arrived at a restaurant where their number is more than the capacity of the restaurant for such cheap lunch . walking down to the next restaurant would mean failing to report back to the job in time !paise means the smallest indian coin and there were indeed some well known rankings of these restaurants as some of them would offer more tastier items compared to the others_.  herd behavior and herd externality ( see ) can not explain the omnipresence of social inefficiency for the kpr problem since we assume that there is no ambiguity among agents over the ranking of the restaurants .the kpr problem is similar to the minority game since it punishes herd behavior and promotes diversity ( see ) .the kpr problem is different from the minority game in the following ways : ( a ) while the minority game is a simultaneous move _ two choice _ problem , the kpr problem is a simultaneous move _many choice _ problem and ( b ) for the kpr problem there is a common preference ranking of these restaurants and this sort of ranking is absent for the minority game .we assume that while deciding on the choice of the restaurant , there is no mutual interaction among the agents , and that they decide simultaneously on the basis of own past `` experiences '' and complete history .there are possible outcomes of which most are socially inefficient in the sense that there exists at least one agent not getting lunch and equivalently , there is at least one restaurant without a customer .there are exactly ( that is , a fraction exp ] ; ) socially inefficient states .hence there can be no fixed point result for the problem and the game continues dynamically forever .the system moves to an inefficient state characterized by overcrowding at higher ranked restaurants ( absence of any `` absorbing '' state ) .it is precisely this generic deviation from the efficient outcome that motivated us to investigate some of the optimally utilized states of the system and the average occupation ( utilization ) fraction of the restaurants on any day .therefore , our attempt is to analyze different versions and modifications of the kpr problem using various deterministic and stochastic strategies .how does one capture the features of the kolkata paise restaurant problem of _ any given day _ ? that is , the situation where on any given day  1 .each agent goes for lunch to any one of the restaurants , 2 .  depending on the number of agents in each restaurant only one agentis selected randomly and allowed to have lunch , and 3 .the agent derives a utility from having lunch and this utility level depends on the rank of the restaurant and if the agent does not get lunch then his utility is zero on that particular day .one can use the tools available in the game theory literature to model the kolkata paise restaurant problem _ of any given day _ as a one - shot game .we now introduce a general one - shot _ common preference restaurant game _ and identify the conditions under which it represents the _ one - shot kpr game_. let be the set of agents and let the vector represent the utility ( in terms of money ) associated with each restaurant which is common to all agents. assume without loss of generality that .these three things ( that is , , and the ranking ) are used to define the one - shot common preference restaurant game .formally , represent a one - shot common preference restaurant game where is the set of agents ( players ) , is the ( common ) strategy space of all agents where a typical strategy denotes the strategy that the -th agent s strategy is to go to the -th restaurant . in other words ,each agent on any given day selects any one of the restaurants numbered to and if any agent decides to go to restaurant then we represent it as .we denote by the expected payoff vector . in particular, is the expected payoff vector associated with any strategy combination where player s payoff is and .therefore , the strategy combination represents the selection of restaurants made by the agents and the expected payoff that an agent gets depends not only on which restaurant he has selected but also on how many other agents have selected the same restaurant . the number of agents selecting the same restaurant as that of agent under the strategy combination is given by . since we have assumed that the restaurant selects one agent randomly from the set of agents it follows that the expected payoff of agent given the strategy combination is . for each selection of a vector with the property that we associate a common preference restaurant game now introduce added restrictions on the common preference restaurant games to represent the one - shot kpr games .the first feature of the kpr problem is that agents have a common preference ranking over the restaurants . if all restaurants are identical to the agents ( that is , if ) then the problem is trivial .hence to introduce the problem in a non - trivial way it is reasonable to assume that for the one - shot kpr game .the next feature of the kpr problem is that agents prefer getting lunch to not getting lunch . to capture this featurewe assume that the payoff specification of a one - shot kpr game must be such that for all and given and this is equivalent to assuming that . intuitively , if then even the person who was at the lowest ranked restaurant , would not be willing to alter his strategy . therefore , _ if a one - shot common preference restaurant game is such that , and then it is a one - shot kpr game_.  we say that a strategy combination leads to socially efficient ( or pareto efficient ) outcome if and only if each agent goes to different restaurants that is for all , .a strategy combination is a pure strategy nash equilibrium if no agent has an incentive to deviate from his existing strategy given the strategy of the other players .for the above mentioned version of the one - shot kpr game _ the set of pure strategy nash equilibria coincides with the set of socially efficient allocations_. this is formally established in the appendix .  in this sectionwe have introduced the one - shot common preference restaurant games and identified conditions under which these games also represent the one - shot kpr games .the key features that separate out a one - shot kpr game from other non - kpr one - shot common preference restaurant games are non - triviality ( ) and the restriction that ensures that it is always in the interest of any agent to choose an unoccupied restaurant .it is well known from the repeated game theory literature that if _ agents can coordinate their actions _ then any pure strategy nash equilibrium outcome of the one - shot kpr game ( which is also a socially efficient outcome ) can be sustained as an equilibrium outcome of kpr problem ( the repeated one - shot kpr game ) with appropriate punishment schemes ( see ) . however , sustaining such equilibrium in repeated play of the one - shot kpr game would require that _ set of agents is small so that they can coordinate their actions_. since expecting such coordinated actions from the agents is unrealistic for the kpr problem ( where is large ) we assume that with repetition of a one - shot kpr game agents work out their strategies without coordination .following brian arthur s tradition we assume that agents reason inductively ( see ) . by inductive reasoningwe mean that agents learn from past experiences , discard their old strategy for something new if it performs badly and continues with the old strategy if it performs well . however , our objective in this paper is not to find out strategies that survive this process of inductive reasoning but to analyze the statistics of the fraction of restaurants utilized per day associated with each such strategy .we assume that all agents take up some ( common ) stochastic strategy independent of each other . by stochastic strategies we meanthat agents choose their strategies in a probabilistic way . at any stepthis choice of probabilities can depend on the outcome of the previous period . in each casewe consider the limit and analyze the statistics of the fraction of the restaurants utilized per period .the specific one - shot kpr game that we use is the one where so that agents have a strict ranking and ._ on each day an agent selects any one restaurant with equal probability .there is no memory or learning and each day the same process is repeated . _if any restaurant is chosen by agents on a particular day , the restaurant chooses one of them randomly , and the rest agents go without any lunch that day .let there be agents ( in kpr problem ) choosing randomly from the set of restaurants .then the probability of choosing any restaurant is .the probability that any restaurant is chosen simultaneously by agents is therefore given by poisson distribution in other words , the fraction of restaurants not chosen any day is given by , giving the average fraction of restaurants occupied in the kpr problem .the distribution of the fraction utilised any day will be a gaussian around the average given above .see the simulation results in figure [ fig:1 ] for .numerical simulation results for the normalised ( by the respective half widths ) distribution of the fraction of people getting lunch on any day ( or fraction of restaurants occupied ) against where denote the average occupancies in each cases .the inset shows the bare distributions against .all the numerical simulations have been done for agents for time steps . in case nl , in case ll(1 ) and in case opr , as estimated analytically ( eqns .( [ eq : pm0 ] ) , ( [ eq : fbart ] ) and ( [ eq : ft+1_2 ] ) ) .the data points are ( ) , ( ) and ( ) for cases nl , ll(1 ) and opr respectively . , width=453 ]   we consider two types of learning strategies and show that in both the cases the utilization fraction goes down compared to the no learning case .  1 .  _if an agent gets lunch on day then the agent goes to the best restaurant on day .moreover , if on some day the agent goes to best restaurant and fails to get lunch then , on day , the agent selects a restaurant from the remaining restaurants with equal probability . _when number of agents get their lunch on any day and decide to get their next day ( day ) lunch in the best ranked restaurant , then only one of them will get into the highest ranked restaurant and others will not get their lunch .the rest agents will try independently for the remaining restaurants , following the same strategy as in the no learning case .then the recursion relation will simply be ( following equation ( [ eq : pm0 ] ) ) : giving the fixed point fraction .this again compares well with the numerical simulation results ( for ) and shows a gaussian distribution around the above value of as in figure [ fig:1 ] .  1 ._ if an agent reaches the -th ranked restaurant on day then the agent s strategy is the following : ( a ) if the agent gets lunch on day then , on day , the agent attempts to select a restaurant from the rank set with equal probability and ( b ) if the agent does not get lunch on day then , on day , the agent attempts to select a restaurant from the set with equal probability . _  for this strategy we get the fixed point fraction and it shows a gaussian distribution around . hence in this casealso there is no improvement compared to the nl case .if we can somehow introduce repetition in the strategy of the agent then we get an improvement in the utilization fraction .in fact , since the reverse choice problem , where for ( a ) the choice would be and for ( b ) would be , is identical , we get , giving .   _if an agent gets lunch on day from restaurant then the agent goes back to the same restaurant on day and competes for the best restaurant ( restaurant ) on day .if the agent fails to get lunch on day then the agent tries for a restaurant on day which was vacant on day , using the same stochastic strategy as in the nl case . _the fraction on any day of the people occupying any restaurant consists of two parts : fraction of the people already continuing in their randomly selected restaurant the earlier day and the fraction of people who have chosen today . as such , using equation ( [ eq : pm0 ] ) , .\ ] ] since the fraction chosen today is given by the fraction in equation ( [ eq : pm0 ] ) where left out agents are choosing randomly out of vacant restaurants .the next day , the fraction will therefore be given by [1-\exp(-1)].\ ] ] if we assume that and at the fixed point , then equating ( [ eq : ft_2 ] ) and ( [ eq : ft+1 ] ) (1-\exp(-1)),\ ] ] or , giving , as can be seen in the numerical simulation results in figure [ fig:1 ] .the fluctuations in the occupation density or utilization fraction is again gaussian around the most probable occupation fraction derived here .therefore , with opr strategy we have an increase in compared to the nl strategy suggesting that if the strategies are such that an agent , after getting lunch on a particular day , decides to go to the same restaurant next day we have an improvement in utilization fraction .* remark * the opr strategy is more consistent when the one - shot common preference restaurant game is such that .it may be noted that if all the restaurants are identical then the dynamics always converge to an absorbing state characterized by social efficiency ( with pavlov s * win - stay , lose - shift ( ws - ls ) * strategy ( see ) ) .the reason being the following : even if everybody starts by random choices among all the restaurants and , initially ( say , on day ) , several restaurants get more than one prospective customer and choose one of them , on day , the agents not getting lunch on day will go to a restaurant that was unoccupied on day and the agents getting lunch on day will continue to go to the same restaurant . soon( before , days ) , each agent finds one restaurant to get the lunch and the dynamics effectively stops there.[trivial ]  in the next case we think of a strategy where the queue length in a restaurant on day either acts as an inducing device or acts as a rationing device for the strategy to be adopted by an agent on day .numerical simulation results for the distribution of the fraction of people getting lunch any day ( or fraction of restaurants occupied on any day ) against , for and for case fc .all the numerical simulations have been done for agents for time steps ., width=453 ]   _ the probability of selecting a restaurant on day depends on the queue length of agents in restaurant on day . _we assume that the probability of choosing restaurant on day is given by where is the queue length ( or equivalently , the number of people who arrived at the restaurant on day ) . is a parameter that interpolates between the random ( , independent of history ) and condensation ( , strongly dependent only on history ) limits .the representative histograms are shown in the figure 2 , where the most probable fraction of occupied restaurants converge to a value 0.63 ( as obtained earlier ) as .it is also seen that the probability of a queue length of any restaurant goes as for ( exponentially decaying otherwise ) .this implies that while the utilization fraction has a gaussian distribution ( with ) the queue length distribution on any arbitrary day has a power law distribution if and it is exponentially decaying otherwise .  if the agents follow * avoid the crowd ( ac ) * strategy that is if ( people choosing to avoid the last day s crowded restaurants ) , the most probable restaurant utilization fraction is independent of the value of  . here, no power law for the queue length was observed for any value of .the observations of this subsection are similar to the hospital waiting list size distribution ( see , ) though the hospital problem is significantly different from the kpr problem ( for example , the absence of possibilities of iterative learning and the presence of heterogeneity of choices in the hospital waiting list problem ) .till now we had assumed that on any day one can have potentially all the agents turning up in one particular restaurant . if the arrival of the agents were sequential then all agents turning up in one restaurant tantamount to saying that queues of all sizes are allowed to form in any restaurant and the restaurant then picks up any one agent to serve .all other agents have to go back without having lunch on day . herewe are assuming that formation of queue of all sizes is allowed .what if all restaurants have limited capacity to accommodate agents in a queue and that an agent gets the option to move to another restaurant ( in the same day ) if the agent arrives at a restaurant that has reached its queue limit ? before concluding we consider this particular modification of the kpr problem since it has important implications about the robustness of our occupancy statistics ._ a modified kpr problem : _ on each day the restaurants are filled up sequentially .we also assume that _ each restaurant allows for a queue length , so that the person arriving at the restaurant has to search for another restaurant with a queue length less than . when all the customers have arrived , one of the customers gets lunch at the restaurant , while the number of prospective customers do not get any lunch that day . in terms of the dynamics ,the most probable restaurant utilization fraction occurs for .we find that for , the fraction _ drastically falls _ to . also for , as discussed and derived earlier .figure 3 shows the result for different values of .note that approaches the value for the random occupancy case as becomes large .thus , except for the special case where , for all other cases we get conclusions that are closer to the kpr problem derived in the previous section .numerical simulation results for the distribution of the fraction of people getting lunch any day ( or fraction of restaurants occupied on any day ) against , for for the modified kpr problem .all the numerical simulations have been done for agents for time steps ., width=453 ]for the general kpr problem we have a perpetual dynamics characterized by large fluctuations in the measure of daily misuse having a gaussian distribution of utilization fraction around the average where differs for different strategies .we have argued this point by showing that the average utilization fraction with no learning and other simple stochastic strategies could be analytically calculated and the distributions are seen to be gaussian . for the `` follow / avoid the crowd '' strategies we observe the statistics of the number of people _ going without lunch _ to have power law fluctuations ( see also , , , for similar behavior ) .our main findings in terms of utilization fraction comparisons across different strategies are :  1 .if we have a one - shot kpr game then any pure strategy nash equilibrium leads to a socially efficient outcome , as shown in sec .[ os ] , and hence we have full capacity utilization . however , if we have a kpr problem ( which is a repeated game where any given one - shot kpr game is repeated ) and agents can not coordinate their action then we do not have full capacity utilization for many ( common ) stochastic strategies .in such a situation utilization fraction under different stochastic strategies is of prime importance .2 .  the utilization fraction is higher on an average with no learning strategy in comparison to some limited learning strategies ( sec .[ sec : stochgm ] ) .3 .  the statistics for utilization fraction can improve with a strategy where each agent ,after getting lunch from a restaurant on day , goes back to the same restaurant on day ( ignoring the ranking of the restaurants ) ( sec .[ ss : opr ] ) .4 .  following ( avoiding )the crowd strategy can not improve on the statistics of utilization fraction in comparison to the no learning case ( sec . [ ss : fc ] ) .our findings about the statistics of utilization fraction are robust since we have shown that by adding different queue limits to the krp problem the conclusions are similar to our findings except for the very special case where the queue limit is unity ( sec .[ sec : lql ] ) .* acknowledgment : * we are grateful to satya ranjan chakravarty , pradip maiti , pradeep kumar mohanty , abhirup sarkar and arunava sen for some useful discussions and suggestions .ac thanks camcs , saha institute of nuclear physics for supporting a visit from ictp , during which the work started .ac was supported by the complexmarkets e.u. strep project 516446 under fp6 - 2003-nest - path-1 .a strategy combination for the one - shot common preference restaurant game leads to _ socially ( pareto ) efficient _ outcome if and only if for all . in other words ,given any one - shot common preference restaurant game a strategy combination leads to a socially efficient outcome if and only if each agent goes to a different restaurant so that the sum of utilities of the agents is .let denote the strategy combination of all but player .a strategy combination is a _ pure strategy nash equilibrium _ of the one - shot common preference restaurant game if for all , for all . for a typical one - shot common preference restaurant game , let denote the set of all strategy combinations that lead to socially efficient allocations and let denote the set of all pure strategy nash equilibria .let represent any one - shot kpr game we introduced in sec .[ os ] , that is any one - shot common preference restaurant game with the property that .we have the following result .* proof : * consider any one - shot kpr game and a strategy combination .take any agent and any deviation .since the number of agents is equal to the number of restaurants and since for a strategy combination , for all , it follows that .therefore , since for the one - shot kpr game for all and hence agent has no incentive to deviate .since the selection of was arbitrary it follows that implying .  to prove consider any strategy combination for all but player .given , the best response of agent is to select a best ranked restaurant from the set  of unoccupied restaurants .the reasons being ( 1 ) given , for all and all integers implying that given the strategy of all other agents , agent strictly prefers to occupy a best unoccupied restaurant as opposed to either occupying a different unoccupied restaurant ( which is not a best restaurant ) or crowding in an occupied restaurant and ( 2 ) given the number of agents is equal to the number of restaurants , for any implying that given any strategy of all other agents , agent can always find an unoccupied restaurant and hence agent can also find a best unoccupied restaurant since .therefore , it is in the interest of each agent to occupy an unoccupied restaurant and given the fact that no matter what others do an agent can always find an unoccupied restaurant it follows that any pure strategy nash equilibrium for the one - shot kpr game necessarily leads to a socially efficient allocation , that is .chakrabarti , b. k. , 2007 , kolkata restaurant problem as a generalised el farol bar problem , in econophysics of markets and business networks , pages 239 - 246 , eds .a. chatterjee and b. k. chakrabarti , new economic windows series , springer , milan ( 2007 ) ; http://www.arxiv.org/0705.2098

Abstract: we study the dynamics of the `` kolkata paise restaurant problem '' . the problem is the following : in each period , agents have to choose between restaurants . agents have a common ranking of the restaurants . restaurants can only serve one customer . when more than one customer arrives at the same restaurant , one customer is chosen at random and is served ; the others do not get the service . we first introduce the one - shot versions of the kolkata paise restaurant problem which we call one - shot kpr games . we then study the dynamics of the kolkata paise restaurant problem ( which is a repeated game version of any given one shot kpr game ) for large . for statistical analysis , we explore the long time steady state behavior . in many such models with myopic agents we get under - utilization of resources , that is , we get a lower aggregate payoff compared to the social optimum . we study a number of myopic strategies , focusing on the average occupation fraction of restaurants .


universal quantum computers promise to substantially outperform classical computers  , however , they have been experiencing challenges to build in practice , due to stringent requirement of high - fidelity quantum gate and scalability . for example, shor s algorithm  , solving integer factorization problem , is one of the most attractive quantum algorithms because of its potential to crack current mainstream rsa cryptosystem .an universal quantum computer requires 1025 qubits to factor a 1024-bit number ( )  , a key size thought to be crackable on classical computers  .however , the largest number factored using shor s algorithm is 21 untill now  .this motivates the research of specific quantum computation with quantum speedup and more favorable experimental conditions .boson sampling  is a specific quantum computation thought to be an outstanding candidate for beating the most powerful classical computer in the near term .it samples the distribution of bosons coming from a complex interference network . unlike universal quantum computation, quantum boson sampling seems to be more straightforward , since it only requires identical bosons , linear evolution and measurement . as for classical computers ,the distribution can be obtained by computing permanents of matrices derived from the unitary transformation matrix of the network  . however , computing the permanent has been proved as a # p - hard task with classical computers .the best - known classical algorithm scales exponentially with the size of a matrix .these motivate a lot of advances in building larger quantum boson sampling machine to outperform classical computers , including principle - of - proof experiments  , simplified models easy to implement  , implementation techniques  , robustness of boson sampling  , validation of a large scale boson sampling  , varied models for other quantum states  , etc .    however , what is the upper - bound of computing the permanent on classical computer ? solving boson sampling problem with bosons on classical computers depends on computing permanents of matrices .therefore , the upper - bound is strongly related to the condition on which a quantum boson sampling machine will surpass classical computers .  herewe present the test results of computing permanents on tianhe-2  ( fig .[ fig : tianhe2 ] ) , the supercomputer which has retained its position as the world s no.1 in top500 supercomputer lists since june 2013 .the performance of tianhe-2 is comparable with that of several millions of typical personal computers working simultaneously .we developed , optimized and tested parallel programs of two best - known algorithms , ryser s algorithm and balasubramanian - bax / franklin - glynn s ( bb / fg s ) algorithm . in this paper , we report the speed performance of ryser s algorithm and bb / fg s algorithm in real tests on tianhe-2 supercomputer .we then show the predicted performance of computing the permanent of a matrix on the full system of tianhe-2 supercomputer .in addition , we show that ryser s algorithm has an unacceptable accumulating rounding error from the limited word length of classical computers , compared to bb / fg s algorithm .this implies the result of ryser s algorithm for large - size matrices may be untrustable .ryser s algorithm is shown in equation  and bb / fg s algorithm in equation . in the equations, denotes the permanent of the matrix . is the element of in the row and column . is the matrix size . , where and for .      to test the upper - bound speed of computing permanents classically , we tried exploiting tianhe-2 s computing resources as many as possible ( see section [ sec : methods ] and appendix [ app : methodology ] ) .tianhe-2 has 16,000 compute nodes , and each comprises two intel ivy bridge xeon e5 processors ( cpus ) and three xeon phi coprocessors ( denoted as mics , because they adopt intel many integrated core architecture ) .ryser s algorithm was tested with only cpus , while bb / fg s algrithm was tested under two types of configurations : with only cpus or with cpus and mics heterogeneously .we use real tests on small - scale subsystems ( with compute nodes ) to evaluate and optimize our programs .note that even a small - scale subsystem with 256 nodes still has a predicted linpack performance of 541.8 teraflops and a theoretical peak performance of 848.4 teraflops that may be ranked in the top 160 in top500 s list in nov .2015 .the key of gaining speedup from a parallel computer system is to implement the scalable parallel program , which can get more and more performance improvement with the increase of the computing resources . to evaluate the scalability of our programs, we tested them with different combinations of problem scale and parallel scale . according to the complexity ( , is the matrix size ) , the number of operations almost grows double with the increase of .execution time grows by 1.95 times in fig .[ fig : executiontime](a ) , meaning that the additional cost of larger matrix is very small with the increase of problem scale. meanwhile , execution time decreases with nearly linear speedup ( slopes of 0.54 and 0.59 in fig .[ fig : executiontime](b ) ) when the parallel scale increases .ideal slope is , meaning no additional cost .these suggest our programs have execellent scalability with only small and fixed cost from their serial parts .rccc + & & + & & * lower bound * & * upper bound * + & & & + * b : * & & & +  + & 284.2 & + * r_square : * & 0.994 & + * adjusted r_square : * & 0.9938 & + * rmse : * & 2.665 & +

Abstract: boson sampling , a specific quantum computation problem , is widely regarded to be one of the most achievable fields in which quantum machine will outperform the most powerful classical computer in the near term , although up to now no upper - bound of how fast the classical computers can compute matrix permanents , core problem of boson sampling , has been reported . here we test the computing of the matrix permanent on tianhe-2 , a supercomputer retaining its position as the world s no . 1 system for six times since june 2013 . we arrived at the time ( about minutes ) to compute the permanent of a matrix in an acceptable precision . in addition , we have found that ryser s algorithm will produce an unacceptable error with the increase of problem scale , compared to balasubramanian - bax / franklin - glynn s algorithm in the same complexity . the precision issue suggests carefully check in future research of boson sampling , and comprehensive comparison between quantum computer and classical computer .


the problem of variable selection has received a huge amount of attention over the last 15 years , motivated by the desire to understand structure in massive data sets that are now routinely encountered across many scientific disciplines .it is now very common , e.g. in biological applications , image analysis and portfolio allocation problems as well as many others , for the number of variables ( or predictors ) that are measured to exceed the number of observations . in such circumstances , variable selection is essential for model interpretation .  in a notable recent contribution to the now vast literature on this topic , proposed stability selection as a very general technique designed to improve the performance of a variable selection algorithm .the basic idea is that instead of applying one s favourite algorithm to the whole data set to determine the selected set of variables , one instead applies it several times to random subsamples of the data of size , and chooses those variables that are selected most frequently on the subsamples .stability selection is therefore intimately connected with bagging and subagging .a particularly attractive feature of stability selection is the error control provided by an upper bound on the expected number of falsely selected variables ( * ? ? ?* theorem 1 ) .such control is typically unavailable when applying the original selection procedure to the whole data set , and allows the practitioner to select the threshold for the proportion of subsamples for which a variable must be selected in order for it to be declared significant .  however , the bound does have a couple of drawbacks .firstly , it applies to the ` population version ' of the subsampling process , i.e. to the version of the procedure that aggregates results over the non - random choice of all subsamples . even for as small as 15 , it is unrealistic to expect this version to be used in practice , and in fact choosing around random subsamples is probably typical .more seriously , the bound is derived under a very strong exchangeability assumption on the selection of noise variables ( as well as a weak one on the quality of the original selection procedure , namely that it is not worse than random guessing ) .  in this paper, we develop the methodology and conceptual understanding of stability selection in several respects .we introduce a variant of stability selection , where the subsamples are drawn as complementary pairs from .thus the subsampling procedure outputs index sets , where each is a subset of of size , and .we call this variant complementary pairs stability selection ( cpss ) .  at first glanceit would seem that cpss would be expected to yield very similar results to the original version of stability selection .however , we show that cpss in fact has the following properties :  a.  the meinshausen bhlmann bound holds for cpss regardless of the number of complementary pairs chosen even with . b.  there is a corresponding bound for the number of important variables excluded by cpss . c.  our results have no conditions on the original selection procedure , and in particulardo not require the strong exchangeability assumption on the selection of noise variables .indeed , we argue that even a precise definition of ` signal ' and ` noise ' variables is not helpful in trying to understand the properties of cpss , and we instead state the bounds in terms of the expected number of variables chosen by cpss that have low selection probability under the base selection procedure , and the expected number of high selection probability variables that are excluded by cpss .see section [ sec : cpss ] for further discussion .d.  the bound on the number of low selection probability variables chosen by cpss can be significantly sharpened under mild shape restrictions ( e.g. unimodality or -concavity ) on the distribution of the proportion of times a variable is selected in both and .we discuss these conditions in detail in sections [ sec : improvedtail ] and [ sec : r_concave ] respectively , and compare both the original and new bounds to demonstrate the marked improvement .our improved bounds are based on new versions of markov s inequality that hold for random variables whose distributions are unimodal or -concave .however , it is important to note at this point that the results are not just a theoretical contribution ; they allow the practitioner to reduce ( and therefore select more variables ) for the same control of the number of low selection probability variables chosen by cpss . in section [ sec :practice ] , we give recommendations on how a practitioner can make use of the bounds in applying cpss .  in section [ sec :simstudy ] , we present the results of an extensive simulation study designed to illustrate the appropriateness of our shape restrictions , and to compare stability selection and cpss with their base selection procedures.  a review of some of the extensive literature on variable selection can be found in  .work related more specifically to stability selection includes , who studied the bolasso ( short for bootstrapped enhanced lasso ) .this involves applying the lasso to bootstrap ( with replacement ) samples from the original data , rather than subsampling without replacement .a final estimate is obtained by applying the lasso to the intersection of the set of variables selected across the bootstrap samples .various authors , particularly in the machine learning literature , have considered the _ stability _ of a feature selection algorithm , i.e. the insensitivity of the output of the algorithm to variations in the training set ; such studies include , , , and . consider obtaining a final feature ranking by aggregating the rankings across bootstrap samples .in order to keep our discussion rather general , we only assume that we have vector - valued data which we take to be a realisation of independent and identically distributed random elements .informally , we think of some of the components of as being ` signal variables ' , and others as being ` noise variables ' , though for our purposes it is not necessary to define these notions precisely . formally , we let and , thought of as the index sets of the signal and noise variables respectively .a _ variable selection procedure _ is a statistic taking values in the set of all subsets of , and we think of as an estimator of . as a typical example , we may often write with the covariate and the response , and our ( pseudo ) log - likelihood might be of the form for some . in this context , we regard as the signal indices , as noise indices .examples from graphical modelling can also be cast within our framework .note however that we do not require a ( pseudo ) log - likelihood of the form ( [ eq : loglike ] ) .we define the selection probability of a variable index under as we take the view that for understanding the properties of stability selection , the selection probabilities are the fundamental quantities of interest .since an application of stability selection is contingent on a choice of base selection procedure , all we can hope is that it selects variables having high selection probability under the base procedure , and avoids selecting those variables with low selection probability .indeed this turns out to be the case ; see theorem [ thm : hi_low ] below .  of course, has a bernoulli distribution with parameter , so we may view as an unbiased estimator of ( though is not a model parameter in the conventional sense ) .the key idea of stability selection is to improve on this simple estimator of through subsampling .  for a subset with ,we shall write   let be randomly chosen independent pairs of subsets of of size such that . for  ] ,the simultaneous selection version of is , where   for our purposes , simultaneous selection is a tool for understanding the properties of cpss .however , the special case of of simultaneous selection was studied by , and a variant involving all possible disjoint pairs of subsets was considered in .in theorem [ thm : hi_low ] below , we show that the expected number of low selection probability variables chosen by cpss is controlled in terms of the expected number chosen by the original selection procedure , with a corresponding result for the expected number of high selection probability variables not chosen by cpss .the appealing feature of these results is their generality : they require no assumptions on the underlying model or on the quality of the original selection procedure , and they apply regardless of the number of complementary pairs of subsets chosen .  for  ] , then b.  let and .if , then   in many applications , and for a good base selection procedure , we imagine that the set of selection probabilities is positively skewed in  } \\  \dfrac{4(1- \tau + 1/2b)}{1 + 1/b } & \mbox{if }. \end{array } \right.\ ] ]  the proof of theorem [ thm : unimodal ] is based on a new version of markov s inequality ( theorem [ thm : unimarkov ] in the appendix ) for random variables with unimodal distributions supported on a finite lattice .there is also an explicit expression for when , which follows from theorem [ thm : unimarkov ] in the same way , but we do not present it here because it is a little more complicated , and because we anticipate the bound when is ( much ) smaller than being of most use in practice . see section [ sec : practice ] for further discussion .figure [ fig : bounds_compare ] compares the bounds provided by theorems [ thm : hi_low ] and theorem [ thm : unimodal ] as a function of , for the illustration discussed after the statement of theorem [ thm : hi_low ] .[ ] [ ] [ _ bounds_compare.ps]0.4 [ ] [ ] [ _ bounds_compare.ps]0.5 [ ] [ ] [ _ bounds_compare.ps]0.6 [ ] [ ] [ _ bounds_compare.ps]0.7 [ ] [ ] [ _ bounds_compare.ps]0.8 [ ] [ ] [ _ bounds_compare.ps]0.9 [ ] [ ] [ _ bounds_compare.ps]1.0 [ ] [ ] [ _ bounds_compare.ps]0 [ ] [ ] [ _ bounds_compare.ps]5 [ ] [ ] [ _ bounds_compare.ps]10 [ ] [ ] [ _ bounds_compare.ps]15 [ ] [ ] [ 0.9]bound on [ ] [ ] [ 0.9]   the unimodal assumption allows for a significant improvement in the bounds attainable from a naive application of markov s inequality . however , figure [ fig : compare_dist ] suggests that further gains may be realised by placing tighter constraints on the family of distributions for that we consider , in order to match better the empirical distributions that we see in practice .a very natural constraint to impose on the distribution of is log - concavity . by this , we mean that , if denotes the probability mass function of , then the linear interpolant to is a log - concave function on  ] as an interpolation between log - concavity and unimodality .[ prop : log_iff._r ] a function is log - concave if and only if it is -concave for every .[ prop : uni_implies_r ] let be a unimodal probability mass function supported on and suppose both that and that , for some . then is -concave for some .  in proposition [ prop : rbound ] in the appendix, we present a result that characterises those -concave distributions that attain equality in a version of markov s inequality for random variables with -concave distributions on .if we assume that is -concave for all , using , for these variables we can obtain a bound of the form where denotes the maximum of over all -concave random variables supported on with .although does not appear to have a closed form , it is straightforward to compute numerically , as we describe in section [ sec : alg ] .the lack of a simple form means a direct analogue theorem [ thm : unimodal ] is not available .we can nevertheless obtain the following bound on the expected number of low selection probability variables chosen by cpss : our simulation studies suggest that is a sensible choice to use for the bound . in other words , if denotes the probability mass function of , then the linear interpolant to is typically well approximated by a convex function .this is illustrated in the bottom left panel of figure [ fig : compare_dist ] ( note that the right - hand tail in this plot corresponds to tiny probabilities ) .the bounds obtained thus far have used the relationship to convert a markov bound for into a corresponding one for the statistic of interest , .the advantage of this approach is that is much smaller than for variables with low selection probability , so the markov bound is quite tight .however , for close to , the inequality starts to become weak , and bounds can only be obtained for in any case .  to solve this problem , we can apply our versions of markov s inequality directly to .we have found , through our simulations , that for variables with low selection probability , the distribution of can be modelled very well as a -concave distribution ( see figure [ fig : minus_quarter_concave ] ) . that the distribution of is closer to log - concavity than that of is intuitive becausealthough the summands in are not independent , terms involving subsamples which have little overlap will be close to independent .if we assume that is -concave and that is -concave for all , we can obtain our best bound which is valid for all  ] with the remaining equally spaced in  ] . as explained in section [ sec : practice ] , we expect our bound to weaken in this situation .however , even when the correlation is as high as 0.9 , we recover a sizeable proportion of the signal we would select had we used the optimal .   herewe illustrate our cpss methodology on the widely studied colon data set of , freely available at ` http://microarray.princeton.edu/oncology/affydata/index.html ` .the data consist of 2000 gene expression levels from 40 colon tumour samples and 22 normal colon tissue samples , measured using affymetrix oligonucleotide arrays .our goal is to identify a small subset of genes which we are confident are linked with the development of colon cancer .such a task is important for improving scientific understanding of the disease and for selecting genes as potential drug targets .the data were first preprocessed by averaging over the expression levels for repeated genes ( which had been tiled more than once on each array ) , log - transforming each gene expression level , standardising each row to have mean zero and unit variance , and finally removing the columns corresponding to control genes , so that genes remained .the transformation and standardisation are very common preprocessing steps to reduce skewness in the data and help eliminate the effects of systematic variations between different microarrays ( see for example and ) .we applied cpss with ( lasso ) penalised logistic regression as the base procedure , with , and choosing both using the -concave bound of section [ sec : practice ] , and the original bound of .we estimated the expected classification error in the two cases by averaging over 128 repetitions of stratified random subsampling validation , taking 8 cancerous and 4 normal observations in each test set .thus when applying cpss , we had .we looked at and , and set to control with and .  rather than subsampling completely at random when using cpss , we also stratified these subsamples to include the same proportion of cancerous to normal samples as in the training data supplied to the procedure . without this step , some of the subsamples may not include any samples from one of the classes , and applying to such a subsample would give misleading results .using stratified random subsampling is still compatible with our theory , provided that is interpreted as an expectation over random data which contain the same class proportions as observed in the original data . in general , this approach of stratified random subsampling is useful when the response is categorical .the results in table [ tab01 ] show that , as expected , the new error bounds allow one to select more variables than the conservative bounds of for the same level of error control , and as a consequence , the expected prediction error is reduced .figure [ fig : robustness ] demonstrates the robustness of the selected set to the different values of .finally , we also applied cpss on the entire dataset with and and using the -concave bound of section [ sec : practice ] to choose to control ( cf .figure [ fig : genemap ] ) .we see that with just 5 genes out of 1908 , we manage to separate the two classes quite well .   for ( left ) and ( right ) ,we have plotted the proportion of times a gene was selected by our -concave cpss procedure for all genes which were selected at least 5% of the time among the 128 repetitions .solid black means the gene was selected in every repetition , and white means it was never selected .thus dark vertical lines indicate that the choice of has little effect on the end result of cpss . ]\(i ) let be randomly chosen independent pairs of subsets of of size such that .then now because and are independent conditional on .it follows using that where we have used markov s inequality in the final step .\(ii ) define and by replacing with in the definitions of and respectively . then , using the bound corresponding to and markov s inequality again ,     the proof of theorem [ thm : unimodal ] requires several preliminary results , and we use the following notation .let denote the finite lattice  ] . moreover, the hyperplane is closed . also , is a continuous function on , so ) ] , we obtain all the relevant probability mass functions supported on via the tail probability of , when the threshold is , is and we may maximise this over $ ] to obtain an optimal for each .this is easily accomplished using a general purpose optimiser such as ` optimize ` in ` r ` . to summarise , we have the following simple procedure for computing .   then .when we wish to evaluate for a range of values of , the process is simplified by the observation that is increasing in , and thus in step 2 we need only consider those which are at least .using the algorithm described above , we have computed over a grid of and values ( cf .tables [ tab : r_conc1 ] and [ tab : r_conc2 ] ) .an ` r ` implementation of the algorithm is available from both authors websites .  75 alon , u. , barkai , n. , notterman , d.a . , gish , k. , ybarra , s. , mack , d. and levine , a.j .broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays .usa , * 96 * , 6745-6750 .lange , t. , braun , m. , roth , v. and buhmann , j. ( 2003 ) stability - based model selection . in s. becker , s. thrun , and k. obermayer ( eds . ) , _ advances in neural information processing systems _ ,* 15 * ( pp . 617624 ) .cambridge , ma : mit press .loscalzo , s. , yu , l. and ding , c. ( 2009 ) consensus group based stable feature selection . in _ proc .15th acm sigkdd international conference on knowledge discovery and data mining ( kdd-09 ) _ , pp .

Abstract: stability selection was recently introduced by as a very general technique designed to improve the performance of a variable selection algorithm . it is based on aggregating the results of applying a selection procedure to subsamples of the data . we introduce a variant , called complementary pairs stability selection ( cpss ) , and derive bounds both on the expected number of variables included by cpss that have low selection probability under the original procedure , and on the expected number of high selection probability variables that are excluded . these results require no ( e.g. exchangeability ) assumptions on the underlying model or on the quality of the original selection procedure . under reasonable shape restrictions , the bounds can be further tightened , yielding improved error control , and therefore increasing the applicability of the methodology .   key words : complementary pairs stability selection , -concavity , subagging , subsampling , variable selection  _ plots_9_40.7 _ bounds_compare.ps0.8 _ comp_plots_9_2_20.8


a spherical gravitational wave antenna ideally has equal sensitivity to gravitational waves from all directions and polarizations and is able to determine the directional information and tensorial character of a gravitational wave .the solution for the inverse problem for a noiseless antenna has been known for some time , and an analytic solution for an noisy antenna was recently found .these solutions are quite elegant as they only require linear algebra to estimate the wave direction and polarization from the detector outputs .  by monitoring the five quadrupole modes of an elastic sphere, one has a direct measurement of the effective force of a gravitational wave on the sphere .the standard technique for doing so on resonant detectors is to position resonant transducers on the surface of the sphere that strongly couple to the quadrupole modes .a number of proposals have been made for the type and positions of the transducers .what all of these proposals have in common is that the outputs of the transducers are combined into five `` mode channels '' that are constructed to have a one - to - one correspondence with the quadrupole modes of the sphere and thus the spherical amplitudes of the gravitational wave  .the mode channels can be collected to form a `` detector response '' matrix   ,   \label{eqn : cartesian_strain_tensor}\ ] ]  which , in the absence of noise in the detector , is equal to the gw strain tensor , expressed in lab frame coordinates .the latter tensor has the canonical form   ,   \label{eqn : wave_strain}\ ] ]  in the wave frame , and is related to by an orthogonal transformation a rotation . clearly has an eigenvector , , say , with zero eigenvalue which corresponds to the wave propagation direction .the same therefore applies to , and this enables the determination of the wave direction by a straightforward algebraic procedure directly from detector data : it is the eigenvector of with null eigenvalue .things change when the detector is noisy : noise gets added to the signal in the mode channels , destroying the equivalence between the data matrix and the signal matrix .however , it has been shown that under ideal conditions of the noise a modified version of the above procedure can be used : the eigenvector of the noisy with eigenvalue _ closest to zero _ is the _best _ approximation to the actual incidence direction of the gravitational wave .  in this paper we shall be taking an _ analytic _approach to the diagonalization problem , whereby errors in the estimated gw parameters can be assessed to any desired degree of accuracy .inherent in this approach is the unambiguous definition of the incidence direction estimate , as well as a cartesian coordinate convention for it , which rids us of the ambiguities intrinsically associated to the euler angle characterization for incidence directions near the poles .errors in these quantities will be shown to be incidence direction independent .in addition , we shall also address the problem of estimating the gw amplitudes and , and their errors .previous authors found it impossible to give isotropic estimates of these quantities , a very strange result for a spherical detector .we explain why these results come about and we show how the problem can be solved by properly including all the necessary information .the paper is organized as follows .we begin in section [ sec : a_ev ] by deriving analytic expressions for the eigenvalues of .we then use these expressions to find the first order statistical errors in the eigenvalues in section [ sec : uncertainties_ev ] , followed by the direction estimation error in section [ sec : dir_error ] .higher order corrections to these errors are presented in section [ sec : corrections ] . in section [ sec : polarization ] we discuss the errors on the polarization amplitude estimates .we explain why past solutions have direction dependent errors on these quantities and we describe a maximum likelihood algorithm , based a hypothesis on the physical nature of the source , that fulfills the natural property of source location independence .the detector response matrix is symmetric and traceless and has the eigenvalue equation    where the eigenvectors are normalized in the usual way    expanding equation we find    where we have defined   %   d \equiv \det(\bi{a}).\end{aligned}\ ] ] solving this cubic equation we find the eigenvalues of to be    where    the eigenvalue is identically zero in the absence of noise , so it will generally be the one closest to zero in the presence of noise .random fluctuations may eventually change this ( more likely for low snr ) , but we shall always take the corresponding eigenvector as the best approximation to the direction of the source. the amplitude of the wave can be calculated in many ways from the mode channels ( for example , is an estimate for the amplitude ) , but the semi - difference of the other two eigenvalues will give the best estimate , assume that the mode channels have uncorrelated noise with zero mean and variance .the lowest order statistical errors in the eigenvalues are easily calculated by where the derivatives of the eigenvalues are given by and the derivatives of the determinant are   if the variances on the mode channels are equal then it is easily seen that equations and lead to    where is the variance on any one mode channel .note that from equation all three eigenvalues have equal variance to first order .cross correlations between these eigenvalues are also easily calculated , and for equal mode channel variances they are equal for all pairs :    shown in figs .[ fig : ev_error_1]-[fig : ev_error_3 ] is this variance as a function of the . also shown is the results of a monte carlo type simulation of the errors which take into account the higher order perturbations at low values of snr .as shown , the analytic expressions match the simulated errors for high snr. for low snr discrepancies arise between the analytic and simulated values .this is within expectation since equation is only accurate for large snr .higher order correction will be considered below .we assume that the eigenvector points in the propagation direction of the gravitational wave .we want to estimate the fluctuations in the determination of this direction caused by the presence of noisy fluctuations in the mode channels .we represent with a difference between a given quantity and its ideal value if there were no noise ( i.e. is the difference between the position calculated from noisy data and its real position in the sky ) .we now take equation for and consider fluctuations in it .if these are not too large ( high snr ) we can retain only first order terms \delta\bi{v}_3    =   -\left[\delta\bi{a}-\delta\lambda_3\right]\bi{v}_3 .\label{pert}\ ] ] this is an equation for , but the matrix $ ] is not invertible. the only consequence of this is that we can not determine the component of which is parallel to itself .the orthogonal components ( those parallel to and ) can easily be found by multiplying equation [ pert ] on the left by and   an appropriate assessment of the error on a direction measurement is the solid angle error . since , this erroris given by where is the quadratic error in the determination of .to find it we need to calculate the expectation of the squared modulus of the above fluctuations ,    first order calculations only require us to take expectations in , while leaving the rest untouched ,    where we have defined    explicitly ,   , %   \\bi{a}_2    =   \left[\begin{array}{ccc }    0 & 1 & 0 \\   1 & 0 & 0 \\   0 & 0 & 0   \end{array}\right ] , %   \\bi{a}_3    =   \left[\begin{array}{ccc }   0 & 0 & 0 \\   0 & 0 & 1 \\   0 & 1 & 0   \end{array}\right ] , \\[1 em ] %   \bi{a}_4   =    \left[\begin{array}{ccc }   0 & 0 & 1 \\   0 & 0 & 0 \\   1 & 0 & 0    \end{array}\right ] , %   \\bi{a}_5    =   \left[\begin{array}{ccc }    -\frac{1}{\sqrt{3 } } & 0 & 0 \\   0 & -\frac{1}{\sqrt{3 } } & 0 \\   0 & 0 & \frac{2}{\sqrt{3 } }   \end{array}\right].\end{aligned}\ ] ] we thus have  \,\frac{\sigma_{g_m}^2}{g^2}.   \label{eq : deltav}\ ] ]  again , setting the variances on the mode channels equal to , the sum in equation can easily be done , giving   from equation we see that the error in the incidence direction is independent of this direction , as expected of an omnidirectional antenna . substituting this into we find  this expression is in perfect agreement with the solid angle estimation error found by zhou and michelson who used a maximum likelihood technique to estimate the wave direction . this is not surprising as the two methods of estimating the wave direction have been shown to be equivalent ( though the assumptions behind each are quite different )the advantage of our approach is that , by using unit vectors ( cartesian components ) , we are all the time free from the anomalously high errors and correlations intrinsically associated to the euler angle parametrization .  shown in fig .[ fig : solid_angle_error ] is the solid angle estimation error as a function of the snr .also shown is the results of a monte carlo type simulation of the errors . as shown, the analytic expressions match the simulated errors for high snr .deviations however appear for lower values of snr .this again is due to the insufficiency of the first order analytical estimates of the errors . in the next section we presentimproved theoretical estimates of the errors by going one order beyond the first in the calculations of variances .in order to improve our theoretical understanding of the error behaviours of the monte carlo simulations displayed in figs . [fig : ev_error_1]-[fig : solid_angle_error ] we need to go one step beyond the linear error terms of equations and .this requires calculations of higher order derivatives for the added terms , and then the resulting general expressions become quite complicated .they somewhat simplify for equal mode channel variances , but are still rather cumbersome .for example , the next order correction to the eigenvalues , assuming the mode channel noises are zero - mean independent gaussian processes , is given by ( see appendix )    after rather long algebra it is found that    it turns out that the series for converges _ very slowly _ for low snr , so that equation only constitutes an improvement on ( [ eq : ev_error ] ) for a rather limited range of snr see figs .[ fig : ev_error_1]-[fig : ev_error_3 ] .equation shows that the errors in and split from the error in for low values of snr , and reproduces the observed behaviour that falls below and .it is a reasonable approximation to for snr between 30 and 6 , but it is not quite as good as regards and for those values of snr .higher order terms would be required for an improvement , but these imply still much longer calculations of derivatives of the eigenvalues up to the fifth order , as can be seen in equation ( [ eq : sigma_f ] ) of the appendix .similar corrections can be applied to the incidence direction error estimate of equation .they appear to be given by    for equal mode channel variances , .solid angle errors can be directly inferred from here :    as shown in fig .[ fig : solid_angle_error ] , the above theoretical prediction is a better approach to the behaviour observed in the numerical simulations than is equation . for snr less than 2 ,equation is also insufficient .it is important to remind ourselves that for very low snr the uncertainties on the direction estimation are so large that the measurement is almost meaningless .zhou and michelson decided that a minimum snr of 10 in energy was necessary for a direction measurement .looking at fig .[ fig : solid_angle_error ] this lower limit seems reasonable , thus it is only necessary to have accurate analytical expressions down to that level .we now come to the discussion of errors in the polarization amplitudes and .it is easily shown that the uncertainty on the measurement of the polarization amplitudes is direction independent if the source position is known ahead of time , but it has been claimed that in the unknown direction case there is a strong direction dependency .this is disturbing given that a spherical detector is equally sensitive to waves of all polarization and direction .we argue in this section that the difficulties to find isotropic estimates of and are ultimately due to the use of unsuitable criteria to set up those estimates .we discuss a more natural procedure , based on data set processing , that leads to a solution of this problem .  to understand why a simple estimate of the errors leads to direction dependencies ,let us look at the basics of the solution to the inverse problem .we are given 3 eigenvectors and 3 eigenvalues , but these are not independent .the eigenvectors are orthogonal , so we actually only get 2 pieces of information from them .we use that information to determine the direction of the wave .next , the strain tensor is traceless so the third eigenvalue can be determined from the other two , therefore , we only get 2 pieces of information from them .we use one of them ( actually a combination of 2 ) to get the wave amplitude .past reasoning suggested that we can use the last piece of information to determine the polarization angle .this is wrong .the last eigenvalue does not tell us the polarization , but rather is related to the scalar component of the gw ( or lack thereof ) .we assume this to be zero for gr , so this can be interpreted as a measurement of the non - zeroness of this component .the fact that we decide that this should be zero _ a priori _ does not give us additional information about the polarization , only the level of noise in our system . without any additional information we have an under - determined system which will lead to direction dependent errors as seen in reference  .the additional piece of information needed is the polarization angle : the angle between the gw s axes and the eigenvectors and perpendicular to the incidence direction . to this end, we submit our diagonal form of the matrix to a rotation of angle about to obtain best estimates of and by the formulas   where the pure and uncorrelated noise term in was dropped out .the gw amplitude has been shown to have a best estimate given by equation . can be determined with isotropic sensitivity since that is the case with and , as we have just seen . for a _ fixed _ , equations and give us an estimate of and in the presence of noise in the detector .we can use the results of section [ sec : uncertainties_ev ] to see that   and these errors are indeed isotropic , for they only depend on .  in the absence of further information on the specific _ physical _ nature of the source , _any _ polarization angle is valid , for the canonical form of the tensor ( [ eqn : wave_strain ] ) is invariant to rotations about the third axis .a particular choice of is thus a matter of taste in this case , and equations give the correct error estimates .a common way to resolve the arbitrariness in  is to set the first euler angle in the rotation relating the lab frame to the wave frame equal to zero .however , this is a very much observer dependent criterion , for detectors at different locations would claim different values for and , even if they agreed to be seeing the _ same _ source .errors in and based on such criterion have been shown e.g. by to be strongly direction dependent , which is certainly not surprising .it is however paradoxical that a spherical detector should prefer certain directions to others to detect a gw signal , therefore this must be reassessed .we now propose a more consistent solution .it is clear from the above discussion that any criterion to resolve the arbitrariness in , therefore to estimate and , should be established _relatively to the gw source _ , be it known ahead of time or based on a hypothesis to be checked _a posteriori_.  let us , for concreteness , consider a _coalescing binary system _ as the gw source .the signal generated by such a system is given by somewhat complicated functions of the space - time variables and a number of system parameters ; it will not be necessary for our purposes to consider in detail the explicit form of such functions ( see for example ) , it will suffice to use formal expressions indicating the signal dependencies :    here is the source position , and is the time . stands for the _ set of characteristic source parameters _ , which in this case include the masses of the stars , the inclination of the orbital plane , the semimajor axis , the eccentricity of the orbit , the periastron position , etc .note that these amplitudes are referred to a set of _ source _ axes , so they are independent of the detector s location .  the usual way to estimate the parameters is to resort to classical statistics , as has been done for example in for interferometric detectors or in for spherical detectors .the fundamental quantity required by such method is the _ likelihood function _ , , which is a functional of the ( unknown ) signal parameters _ and _ the detector data .we then proceed as follows : we construct the likelihood function associated to the hypothesis that equations and be a fit to equations and for suitable values of the parameters .it will thus have the generic form    standard manipulations of yield both best estimates of the signal parameters _ and _ of the polarization angle  , as well as errors and cross correlations between any pair of these it is recalled that such are identified as the coefficients of the _ covariance matrix _ , which is the inverse of the matrix of second derivatives of  .we shall not attempt to give a detailed discussion of this process here .the important point to stress is that in the approach just proposed , we have managed to have as the only combination of actual data entering the likelihood function .errors and cross correlations between parameter estimates will thus ultimately be functions only of the errors and cross correlations between the eigenvalue estimates , and , which we have proved in section [ sec : uncertainties_ev ] to be direction independent .so not only but also the source parameters can be determined with isotropic sensitivity by means of a spherical gw detector .the same therefore applies to the gw amplitudes and , as indeed expected .the _ quantitative _ estimation of the errors in and can not however be given explicitly until the full parameter estimation problem has been completely solved , as interactions between all those estimates will strongly affect one another .with analytic expressions for the uncertainties on the eigenvectors and the eigenvalues of the mode channel matrix ( [ eqn : cartesian_strain_tensor ] ) we can turn our attention to the physical interpretation of these values .an unambiguous selection of can be made on the basis that it is the _ third root _ in equation .this will usually be the one closest to zero .then the other two represent the amplitude measurement .as proved in , the best estimate of the gw amplitude is the semi - difference of these two , .the third eigenvalue ideally should be zero if general relativity is correct .once noise is introduced this is no longer the case , but the variance on this eigenvalue gives us a level of the `` non - zeroness '' .one can imagine setting a threshold on this eigenvalue that is a function of ( a function of the snr ) to veto any candidate events that have an excessive .many non - gw sources are likely to produce a non - zero , therefore becoming easily identified and discarded.  the errors in both eigenvalues and eigenvectors are direction independent .the last step in the analysis is the splitting of the gw amplitude into the usual and components .we have shown that this can be accomplished by making suitable reference to the source properties , whereby isotropic sensitivity to these quantities obtains .this solves the paradox of the anisotropies in the determination of and , and stresses the fact that the most fundamental magnitudes to estimate from the detector data are the eigenvalues and eigenvectors of the mode channel matrix : these are _ source independent _ , and any further progress in signal deconvolution explicitly requires reference to the source properties , be them known ahead of time or be them stated in the form of a hypothesis to test .let be a set of independent gaussian random variables , with mean and variances .let be a regular function of its arguments . because is a random variableso is , although it will of course be generally non - gaussian .we want to find the mean and variance of , and to this end we taylor expand it around the mean : where and the usual convention of summation over repeated indices is adopted in ( [ eq : taylor ] ) .the mean of is its expectation value , , while its variance is the difference ^ 2   \label{eq : ms}\ ] ]  expectation values are to be taken on the basis of the expansion ( [ eq : taylor ] ) . given that is a set of independent gaussian variables , the expectation of the product of an _ odd _ number of s is zero , while etc . , where no summation over repeated indices is exceptionally assumed in these expressions .if the assumption is made that all the s have equal variances , , then one easily finds where is a shorthand for .likewise ,   ^ 2  =  f^2 + ff_{ii}\;\sigma^2 +    \frac{1}{4}\,\left(ff_{iijj}+f_{ij}f_{ij}\right)\,\sigma^4 \nonumber\\    + \frac{1}{24}\,\left(ff_{iijjkk}+3\,f_{ij}f_{ijkk}\right)\,\sigma^6    + \ldots,\end{aligned}\ ] ]  and   , finally ,

Abstract: a single spherical antenna is capable of measuring the direction and polarization of a gravitational wave . it is possible to solve the inverse problem using only linear algebra even in the presence of noise . the simplicity of this solution enables one to explore the error on the solution using standard techniques . in this paper we derive the error on the direction and polarization measurements of a gravitational wave . we show that the solid angle error and the uncertainty on the wave amplitude are direction independent . we also discuss the possibility of determining the polarization amplitudes with isotropic sensitivity for any given gravitational wave source .