global convergence analysis of a discrete time nonnegative ica algorithm when the independent sources are known to be nonnegative and well grounded which means that they have a nonzero pdf in the region of zero oja and plumbley have proposed a nonnegative principal component analysis pca algorithm to separate these positive sources generally it is very difficult to prove the convergence of a discrete time independent component analysis ica learning algorithm however by using the skew symmetry property of this discrete time nonnegative pca algorithm if the learning rate satisfies suitable condition the global convergence of this discrete time algorithm can be proven simulation results are employed to further illustrate the advantages of this theory

a hybrid technique for blind separation of non gaussian and time correlated sources using a multicomponent approach blind inversion of a linear and instantaneous mixture of source signals is a problem often encountered in many signal processing applications efficient fastica efica offers an asymptotically optimal solution to this problem when all of the sources obey a generalized gaussian distribution at most one of them is gaussian and each is independent and identically distributed i i d in time likewise weights adjusted second order blind identification wasobi is asymptotically optimal when all the sources are gaussian and can be modeled as autoregressive ar processes with distinct spectra nevertheless real life mixtures are likely to contain both gaussian ar and non gaussian i i d sources rendering wasobi and efica severely suboptimal in this paper we propose a novel scheme for combining the strengths of efica and wasobi in order to deal with such hybrid mixtures simulations show that our approach outperforms competing algorithms designed for separating similar mixtures

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

planar formulae and their uses we define the set of planar boolean formulae and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is np complete using these results we are able to provide simple and nearly uniform proofs of np completeness for planar node cover planar hamiltonian circuit and line geometric connected dominating set and of polynomial space completeness for planar generalized geography the np completeness of planar node cover and planar hamiltonian circuit and line were first proved elsewhere m r garey and d s johnson the rectilinear steiner tree is np complete siam j appl math 32 1977 pp 826 834 and m r garey d s johnson and r e tarjan the planar hamilton circuit problem is np complete siam j comp 5 1976 pp 704 714

the planar k means problem is np hard in the k means problem we are given a finite set s of points in m and integer k 1 and we want to find k points centers so as to minimize the sum of the square of the euclidean distance of each point in s to its nearest center we show that this well known problem is np hard even for instances in the plane answering an open question posed by dasgupta 2007 7

greater robustness of second order statistics than higher order statistics algorithms to distortions of the mixing matrix in blind source separation of human eeg implications for single subject and group analyses a mandatory assumption in blind source separation bss of the human electroencephalogram eeg is that the mixing matrix remains invariant i e that the sources electrodes and geometry of the head do not change during the experiment actually this is not often the case for instance it is common that some electrodes slightly move during eeg recording this issue is even more critical for group independent component analysis gica a method of growing interest in which only one mixing matrix is estimated for several subjects indeed because of interindividual anatomo functional variability this method violates the mandatory principle of invariance here using simulated experiments 1 and 2 and real experiment 3 eeg data we test how eleven current bss algorithms undergo distortions of the mixing matrix we show that this usual kind of perturbation creates non gaussian features that are virtually added to all sources impairing the estimation of real higher order statistics hos features of the actual sources by hos algorithms e g ext infomax fastica hos based methods are likely to identify more components with similar properties than actual neurological sources a problem frequently encountered by bss users in practice the quality of the recovered signal and the efficiency of subsequent source localization are substantially impaired performing dimensionality reduction before applying hos based bss does not seem to be a safe strategy to circumvent the problem second order statistics sos based bss methods belonging to the less popular sobi family class are much less sensitive to this bias

degree mixing in multilayer networks impedes the evolution of cooperation traditionally the evolution of cooperation has been studied on single isolated networks yet a player especially in human societies will typically be a member of many different networks and those networks will play different roles in the evolutionary process multilayer networks are therefore rapidly gaining on popularity as the more apt description of a networked society with this motivation we here consider two layer scale free networks with all possible combinations of degree mixing wherein one network layer is used for the accumulation of payoffs and the other is used for strategy updating we find that breaking the symmetry through assortative mixing in one layer and or disassortative mixing in the other layer as well as preserving the symmetry by means of assortative mixing in both layers impedes the evolution of cooperation we use degree dependent distributions of strategies and cluster size analysis to explain these results which highlight the importance of hubs and the preservation of symmetry between multilayer networks for the successful resolution of social dilemmas

blind separation of instantaneous mixtures of nonstationary sources most source separation algorithms are based on a model of stationary sources however it is a simple matter to take advantage of possible nonstationarities of the sources to achieve separation this paper develops novel approaches in this direction based on the principles of maximum likelihood and minimum mutual information these principles are exploited by efficient algorithms in both the off line case via a new joint diagonalization procedure and in the on line case via a newton like procedure some experiments showing the good performance of our algorithms and evidencing an interesting feature of our methods are presented their ability to achieve a kind of super efficiency the paper concludes with a discussion contrasting separating methods for non gaussian and nonstationary models and emphasizing that as a matter of fact what makes the algorithms work is strictly speaking not the nonstationarity itself but rather the property that each realization of the source signals has a time varying envelope

new families of skewed higher order kernel estimators to solve the bss ica problem for multimodal sources mixtures this letter suggests two new types of asymmetrical higher order kernels hok that are generated using the orthogonal polynomials laguerre positive or right skew and bessel negative or left skew these skewed hok are implemented in the blind source separation independent component analysis bss ica algorithm the tests for these proposed hok are accomplished using three scenarios to simulate a real environment using actual sound sources an environment of mixtures of multimodal fast changing probability density function pdf sources that represent a challenge to the symmetrical hok and an environment of an adverse case near gaussian the separation is performed by minimizing the mutual information mi among the mixed sources the performance of the skewed kernels is compared to the performance of the standard kernels such as epanechnikov bisquare trisquare and gaussian and the performance of the symmetrical hok generated using the polynomials chebyshev1 chebyshev2 gegenbauer jacobi and legendre to the tenth order the gaussian hok are generated using the hermite polynomial and the wand and schucany procedure the comparison among the 96 kernels is based on the average intersymbol interference ratio aisir and the time needed to complete the separation in terms of aisir the skewed kernels performance is better than that of the standard kernels and rivals most of the symmetrical kernels performance the importance of these new skewed hok is manifested in the environment of the multimodal pdf mixtures in such an environment the skewed hok come in first place compared with the symmetrical hok these new families can substitute for symmetrical hoks in such applications

performance analysis of the fastica algorithm and crame spl acute r rao bounds for linear independent component analysis the fastica or fixed point algorithm is one of the most successful algorithms for linear independent component analysis ica in terms of accuracy and computational complexity two versions of the algorithm are available in literature and software a one unit deflation algorithm and a symmetric algorithm the main result of this paper are analytic closed form expressions that characterize the separating ability of both versions of the algorithm in a local sense assuming a good initialization of the algorithms and long data records based on the analysis it is possible to combine the advantages of the symmetric and one unit version algorithms and predict their performance to validate the analysis a simple check of saddle points of the cost function is proposed that allows to find a global minimum of the cost function in almost 100 simulation runs second the crame spl acute r rao lower bound for linear ica is derived as an algorithm independent limit of the achievable separation quality the fastica algorithm is shown to approach this limit in certain scenarios extensive computer simulations supporting the theoretical findings are included

a bad instance for k means k means is a seeding technique for the k means method with an expected approximation ratio of o logk where k denotes the number of clusters examples are known on which the expected approximation ratio of k means is w logk showing that the upper bound is asymptotically tight however it remained open whether k means yields a constant approximation with probability 1 poly k or even with constant probability we settle this question and present instances on which k means achieves an approximation ratio no better than 2 email protected logk with probability exponentially close to 1

clustering under approximation stability a common approach to clustering data is to view data objects as points in a metric space and then to optimize a natural distance based objective such as the k median k means or min sum score for applications such as clustering proteins by function or clustering images by subject the implicit hope in taking this approach is that the optimal solution for the chosen objective will closely match the desired target clustering e g a correct clustering of proteins by function or of images by who is in them however most distance based objectives including those mentioned here are np hard to optimize so this assumption by itself is not sufficient assuming p np to achieve clusterings of low error via polynomial time algorithms in this article we show that we can bypass this barrier if we slightly extend this assumption to ask that for some small constant c not only the optimal solution but also all c approximations to the optimal solution differ from the target on at most some e fraction of points we call this c e approximation stability we show that under this condition it is possible to efficiently obtain low error clusterings even if the property holds only for values c for which the objective is known to be np hard to approximate specifically for any constant c 1 c e approximation stability of k median or k means objectives can be used to efficiently produce a clustering of error o e with respect to the target clustering as can stability of the min sum objective if the target clusters are sufficiently large thus we can perform nearly as well in terms of agreement with the target clustering as if we could approximate these objectives to this np hard value

universality considerations in vlsi circuits the problem of embedding the interconnection pattern of a circuit into a two dimensional surface of minimal area is discussed since even for some natural patterns graphs containing m connections may require m2 area in order to achieve compact embeddings restricted classes of graphs have to be considered for example arbitrary trees of bounded degree can be embedded in linear area without edges crossing over planar graphs can be embedded efficiently only if crossovers are allowed in the embedding

blind source separation by entropy rate minimization by assuming latent sources are statistically independent independent component analysis separates underlying sources from a given linear mixture since in many applications latent sources are both non gaussian and have sample dependence it is desirable to exploit both properties jointly in this paper we use mutual information rate to construct a general framework for analysis and derivation of algorithms that take both properties into account we discuss two types of source models for entropy rate estimation a markovian and an invertible filter model and give the general independent component analysis cost function update rule and performance analysis based on these we also introduce four algorithms based on these two models and show that their performance can approach the cramer rao lower bound in addition we demonstrate that the algorithms with flexible models exhibit very desirable performance for natural data

fast and robust fixed point algorithms for independent component analysis independent component analysis ica is a statistical method for transforming an observed multidimensional random vector into components that are statistically as independent from each other as possible we use a combination of two different approaches for linear ica comon s information theoretic approach and the projection pursuit approach using maximum entropy approximations of differential entropy we introduce a family of new contrast functions for ica these contrast functions enable both the estimation of the whole decomposition by minimizing mutual information and estimation of individual independent components as projection pursuit directions the statistical properties of the estimators based on such contrast functions are analyzed under the assumption of the linear mixture model and it is shown how to choose contrast functions that are robust and or of minimum variance finally we introduce simple fixed point algorithms for practical optimization of the contrast functions

a minimum range approach to blind extraction of bounded sources in spite of the numerous approaches that have been derived for solving the independent component analysis ica problem it is still interesting to develop new methods when among other reasons specific a priori knowledge may help to further improve the separation performances in this paper the minimum range approach to blind extraction of bounded source is investigated the relationship with other existing well known criteria is established it is proved that the minimum range approach is a contrast and that the criterion is discriminant in the sense that it is free of spurious maxima the practical issues are also discussed and a range measure estimation is proposed based on the order statistics an algorithm for contrast maximization over the group of special orthogonal matrices is proposed simulation results illustrate the performances of the algorithm when using the proposed range estimation criterion

complexity pursuit separating interesting components from time series a generalization of projection pursuit for time series that is signals with time structure is introduced the goal is to find projections of time series that have interesting structure defined using criteria related to kolmogoroff complexity or coding length interesting signals are those that can be coded with a short code length we derive a simple approximation of coding length that takes into account both the nongaussianity and the autocorrelations of the time series also we derive a simple algorithm for its approximative optimization the resulting method is closely related to blind separation of nongaussian time dependent source signals

fast approximate joint diagonalization incorporating weight matrices we propose a new low complexity approximate joint diagonalization ajd algorithm which incorporates nontrivial block diagonal weight matrices into a weighted least squares wls ajd criterion often in blind source separation bss when the sources are nearly separated the optimal weight matrix for wls based ajd takes a nearly block diagonal form based on this observation we show how the new algorithm can be utilized in an iteratively reweighted separation scheme thereby giving rise to fast implementation of asymptotically optimal bss algorithms in various scenarios in particular we consider three specific yet common scenarios involving stationary or block stationary gaussian sources for which the optimal weight matrices can be readily estimated from the sample covariance matrices which are also the target matrices for the ajd comparative simulation results demonstrate the advantages in both speed and accuracy as well as compliance with the theoretically predicted asymptotic optimality of the resulting bss algorithms based on the weighted ajd both on large scale problems with matrices of the size 100times100

back to the future radial basis function network revisited radial basis function rbf networks are a classical family of algorithms for supervised learning the most popular approach for training rbf networks has relied on kernel methods using regularization based on a norm in a reproducing kernel hilbert space rkhs which is a principled and empirically successful framework in this paper we aim to revisit some of the older approaches to training the rbf networks from a more modern perspective specifically we analyze two common regularization procedures one based on the square norm of the coefficients in the network and another one using centers obtained by k k means clustering we show that both of these rbf methods can be recast as certain data dependent kernels we provide a theoretical analysis of these methods as well as a number of experimental results pointing out very competitive experimental performance as well as certain advantages over the standard kernel methods in terms of both flexibility incorporating of unlabeled data and computational complexity finally our results shed light on some impressive recent successes of using soft k k means features for image recognition and other tasks

consistent algorithms for clustering time series the problem of clustering is considered for the case where every point is a time series the time series are either given in one batch offline setting or they are allowed to grow with time and new time series can be added along the way online setting we propose a natural notion of consistency for this problem and show that there are simple computationally efficient algorithms that are asymptotically consistent under extremely weak assumptions on the distributions that generate the data the notion of consistency is as follows a clustering algorithm is called consistent if it places two time series into the same cluster if and only if the distribution that generates them is the same in the considered framework the time series are allowed to be highly dependent and the dependence can have arbitrary form if the number of clusters is known the only assumption we make is that the marginal distribution of each time series is stationary ergodic no parametric memory or mixing assumptions are made when the number of clusters is unknown stronger assumptions are provably necessary but it is still possible to devise nonparametric algorithms that are consistent under very general conditions the theoretical findings of this work are illustrated with experiments on both synthetic and real data

tight lower bound instances for k means in two dimensions the k means seeding algorithm is one of the most popular algorithms that is used for finding the initial k centers when using the lloyd s algorithm for the k means problem it was conjectured by brunsch and roglin 9 that k means behaves well for datasets with small dimension more specifically they conjectured that the k means seeding algorithm gives o log d approximation with high probability for any d dimensional dataset in this work we refute this conjecture by giving two dimensional datasets on which the k means seeding algorithm achieves an o log k approximation ratio with probability exponentially small in k this solves open problems posed by mahajan et al 12 and by brunsch and roglin 9

the fastica algorithm revisited convergence analysis the fast independent component analysis fastica algorithm is one of the most popular methods to solve problems in ica and blind source separation it has been shown experimentally that it outperforms most of the commonly used ica algorithms in convergence speed a rigorous local convergence analysis has been presented only for the so called one unit case in which just one of the rows of the separating matrix is considered however in the fastica algorithm there is also an explicit normalization step and it may be questioned whether the extra rotation caused by the normalization will affect the convergence speed the purpose of this paper is to show that this is not the case and the good convergence properties of the one unit case are also shared by the full algorithm with symmetrical normalization a local convergence analysis is given for the general case and the global behavior is illustrated numerically for two sources and two mixtures in several typical cases

a blind source separation technique using second order statistics separation of sources consists of recovering a set of signals of which only instantaneous linear mixtures are observed in many situations no a priori information on the mixing matrix is available the linear mixture should be blindly processed this typically occurs in narrowband array processing applications when the array manifold is unknown or distorted this paper introduces a new source separation technique exploiting the time coherence of the source signals in contrast with other previously reported techniques the proposed approach relies only on stationary second order statistics that are based on a joint diagonalization of a set of covariance matrices asymptotic performance analysis of this method is carried out some numerical simulations are provided to illustrate the effectiveness of the proposed method

beyond independent components trees and clusters we present a generalization of independent component analysis ica where instead of looking for a linear transform that makes the data components independent we look for a transform that makes the data components well fit by a tree structured graphical model this tree dependent component analysis tca provides a tractable and flexible approach to weakening the assumption of independence in ica in particular tca allows the underlying graph to have multiple connected components and thus the method is able to find clusters of components such that components are dependent within a cluster and independent between clusters finally we make use of a notion of graphical models for time series due to brillinger 1996 to extend these ideas to the temporal setting in particular we are able to fit models that incorporate tree structured dependencies among multiple time series

exact algorithms for size constrained 2 clustering in the plane we study the problem of determining an optimal bipartition a b of a set x of n points in r 2 under the size constraints a k and b n k that minimizes the dispersion of points around their centroid in a and b both in the cases of euclidean and manhattan norms under the euclidean norm we show that the problem can be solved in o n k 3 log 2 n time by using known properties on k sets and convex hulls moreover the solutions for all k 1 2 n 2 can be computed in o n 2 log n time in the case of manhattan norm we present an algorithm working in o n 2 log n time which uses an extended version of red black trees to maintain a bipartition of a planar point set also in this case we provide a full version of the algorithm yielding the solutions for all size constraints k all these procedures work in o n space and rely on separation results of the clusters of optimal solutions

hybrid manifold embedding in this brief we present a novel supervised manifold learning framework dubbed hybrid manifold embedding hyme unlike most of the existing supervised manifold learning algorithms that give linear explicit mapping functions the hyme aims to provide a more general nonlinear explicit mapping function by performing a two layer learning procedure in the first layer a new clustering strategy called geodesic clustering is proposed to divide the original data set into several subsets with minimum nonlinearity in the second layer a supervised dimensionality reduction scheme called locally conjugate discriminant projection is performed on each subset for maximizing the discriminant information and minimizing the dimension redundancy simultaneously in the reduced low dimensional space by integrating these two layers in a unified mapping function a supervised manifold embedding framework is established to describe both global and local manifold structure as well as to preserve the discriminative ability in the learned subspace experiments on various data sets validate the effectiveness of the proposed method

k means under approximation stability one of the most popular algorithms for finding centers for initializing lloyd s heuristic is the k means seeding algorithm the algorithm is a simple sampling procedure that can be described as follows the algorithm picks the first center randomly from among the given points and then for i 2 3 k picks a point to be the ith center with probability proportional to the squared euclidean distance of this point to the nearest center out of the i 1 previously chosen centers the k means seeding algorithm is known to exhibit nice properties it has been noticed that this seeding algorithm tends to perform well when the optimal clusters are separated in some sense intuitively this is because the algorithm gives preference to further away points when picking centers one separation condition that has been studied in the past was due to ostrovsky et al 9 jaiswal and garg 8 showed that if any dataset satisfies the separation condition of 9 then this sampling algorithm gives a constant approximation with probability 1 k on this dataset another separation condition that is strictly weaker than 9 is the approximation stability condition studied by balcan et al 5 in this work we show that the sampling algorithm gives a constant approximation with probability 1 k on any dataset that satisfies the separation condition of 5 and the optimal k clusters are not too small we give a negative result for datasets that have small optimal clusters

predicting peak of participants in collective action extending existing models of human decisions on collective action utilizing related attributions and factors of collective action exploring how to predict peaks of participants in collective action in terms of the number of participants almost each collective action has a life cycle where the number grows from zero to its peak where its maximum potential power or influence is acquired then it decreases to zero eventually therefore we concentrate on modeling simulating and predicting the peaks the model is constructed based on previous models and the data is collected from simulations preliminarily it suggests that there exists a peak for collective action when its jointness of supply is less than one under complete homogeneity the ideal peak is calculated and the ideal peaks function ipf is obtained then heterogeneity is introduced into to the model and the form of real peaks function rpf can be obtained based on simulations and statistical methods for those who intend to organize a collective action and increase the peak of participants should take measures such as ideology leadership and propagation to enhance homogeneity or try to reduce heterogeneity

an lp based k means algorithm for balancing weighted point sets the classical k means algorithm for partitioning n points in rd into k clusters is one of the most popular and widely spread clustering methods the need to respect prescribed lower bounds on the cluster sizes has been observed in many scientific and business applications in this paper we present and analyze a generalization of k means that is capable of handling weighted point sets and prescribed lower and upper bounds on the cluster sizes we call it weight balanced kmeans the key difference to existing models lies in the ability to handle the combination of weighted point sets with prescribed bounds on the cluster sizes this imposes the need to perform partial membership clustering and leads to significant differences for example while finite termination of all k means variants for unweighted point sets is a simple consequence of the existence of only finitely many partitions of a given set of points the situation is more involved for weighted point sets as there are infinitely many partial membership clusterings using polyhedral theory we show that the number of iterations of weight balanced k means is bounded above by no dk so in particular it is polynomial for fixed k and d this is similar to the known worst case upper bound for classical k means for unweighted point sets and unrestricted cluster sizes despite the much more general framework we conclude with the discussion of some additional favorable properties of our method

efficient variant of algorithm fastica for independent component analysis attaining the cramer rao lower bound fastica is one of the most popular algorithms for independent component analysis ica demixing a set of statistically independent sources that have been mixed linearly a key question is how accurate the method is for finite data samples we propose an improved version of the fastica algorithm which is asymptotically efficient i e its accuracy given by the residual error variance attains the cramer rao lower bound crb the error is thus as small as possible this result is rigorously proven under the assumption that the probability distribution of the independent signal components belongs to the class of generalized gaussian gg distributions with parameter alpha denoted gg alpha for alpha 2 we name the algorithm efficient fastica efica computational complexity of a matlab implementation of the algorithm is shown to be only slightly about three times higher than that of the standard symmetric fastica simulations corroborate these claims and show superior performance of the algorithm compared with algorithm jade of cardoso and souloumiac and nonparametric ica of boscolo on separating sources with distribution gg alpha with arbitrary alpha as well as on sources with bimodal distribution and a good performance in separating linearly mixed speech signals

efficient certifiably optimal clustering with applications to latent variable graphical models motivated by the task of clustering either d variables or d points into k groups we investigate efficient algorithms to solve the peng wei p w k means semi definite programming sdp relaxation the p w sdp has been shown in the literature to have good statistical properties in a variety of settings but remains intractable to solve in practice to this end we propose force a new algorithm to solve this sdp relaxation compared to off the shelf interior point solvers our method reduces the computational complexity of solving the sdp from widetilde mathcal o d 7 log epsilon 1 to widetilde mathcal o d 6 k 2 epsilon 1 arithmetic operations for an epsilon optimal solution our method combines a primal first order method with a dual optimality certificate search which when successful allows for early termination of the primal method we show for certain variable clustering problems that with high probability force is guaranteed to find the optimal solution to the sdp relaxation and provide a certificate of exact optimality as verified by our numerical experiments this allows force to solve the p w sdp with dimensions in the hundreds in only tens of seconds for a variation of the p w sdp where k is not known a priori a slight modification of force reduces the computational complexity of solving this problem as well from widetilde mathcal o d 7 log epsilon 1 using a standard sdp solver to widetilde mathcal o d 4 epsilon 1

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

structural effects of participation propensity in online collective actions based on big data and delphi methods abstract the probability or propensity of participation or internet users individuals shapes the evolutionary dynamics and final outcomes of online collective actions this paper investigates the structural factors and the aggregate kernel that may influence or determine the propensity of participation for cyber collective actions for structural factors such as interest i rule r and moral m a structural model is built to explain the participate propensity and macro processes of online cases the variable energy is the aggregation of structural factors energy interest rule moral we have 310 online collective actions with the numbers of participation and witness based on which their participation propensity or percentage can be obtained for each online collective action three researchers pj ns and wz are invited to score the structural factors independently therefore separate from pj ns and wz and averaged interest i rule r and moral m scores can be obtained for each online collective action hence the linkage between structural factors and observed propensity is built the statistical analysis shows that interest rule and moral have positive and significant effects on the propensity their joint explanatory power is much higher adjusted r2 around 90 for propensity and even closed to 100 for its logarithm adjusted r2 around 95 energy aggregates the structural factors and avoids possible collinearities between interest rule and moral scores the single variable of energy explains the most part of propensity adjusted r2 around 90 and even more part of it adjusted r2 round 95 which is why energy is deemed as the kernel of online collective actions that determines the participation propensity

on the complexity of clustering with relaxed size constraints in fixed dimension abstract we study the computational complexity of the problem of computing an optimal clustering a 1 a 2 a k of a set of points assuming that every cluster size a i belongs to a given set m of positive integers we present a polynomial time algorithm for solving the problem in dimension 1 i e when the points are simply rational values for an arbitrary set m of size constraints which extends to the l 1 norm an analogous procedure known for the euclidean norm moreover we prove that in dimension 2 assuming euclidean norm the problem is strongly np hard with size constraints m 2 4 this result is extended also to the size constraints m 2 3 both in the case of euclidean and l 1 norm