characterizations of locally testable linear and affine invariant families the linear or affine invariance is the property of a function family that is closed under linear or affine transformations on the domain and closed under linear combinations of functions respectively both the linear and affine invariant families of functions are generalizations of many symmetric families for instance the low degree polynomials kaufman and sudan 2007 21 introduced the notions of constraint and characterization to characterize the locally testable affine and linear invariant families of functions over finite fields of constant size in this article it is shown that for any finite field f of size q and characteristic p and its arbitrary extension field k of size q if an affine invariant family k n f has a k local constraint then it is k locally testable for k k 2 q pq 2 q p 4 and that if a linear invariant family k n f has a k local characterization then it is k locally testable for k 2k 2 q pq 4 q p 1 consequently for any prime field f of size q any positive integer k we have that for any affine invariant family over field f the four notions of the constraint the characterization the formal characterization and the local testability are equivalent modulo a poly k q of the corresponding localities and that for any linear invariant family the notions of the characterization the formal characterization and the local testability are equivalent modulo a poly k q of the corresponding localities the results significantly improve and are in contrast to the characterizations in 21 which have locality exponential in q even if the field k is prime in the research above a missing result is a characterization of linear invariant function families by the more natural notion of constraint for this we show that a single strong local constraint is sufficient to characterize the local testability of a linear invariant boolean function family and that for any finite field f of size q greater than 2 there exists a linear invariant function family over f such that it has a strong 2 local constraint but is not q d q 1 1 locally testable the proof for this result provides an appealing approach toward more negative results in the theme of characterization of locally testable algebraic properties which is rare and of course significant

long term prediction of non linear time series using multiresolution wavelet models the long term prediction of non linear dynamical time series based on identified multiresolution wavelet models from historically observed data sets is investigated and a new direct prediction approach is introduced prediction results based on the new direct scheme are compared with those from iterative methods and it is shown that improved predictions can be obtained using the new approach

interactive proofs and the hardness of approximating cliques the contribution of this paper is two fold first a connection is established between approximating the size of the largest clique in a graph and multi prover interactive proofs second an efficient multi prover interactive proof for np languages is constructed where the verifier uses very few random bits and communication bits last the connection between cliques and efficient multi prover interaction proofs is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph of independent interest is our proof of correctness for the multilinearity test of functions

robust characterizations of polynomials withapplications to program testing the study of self testing and self correcting programs leads to the search for robust characterizations of functions here the authors make this notion precise and show such a characterization for polynomials from this characterization the authors get the following applications simple and efficient self testers for polynomial functions are constructed the characterizations provide results in the area of coding theory by giving extremely fast and efficient error detecting schemes for some well known codes this error detection scheme plays a crucial role in subsequent results on the hardness of approximating some np optimization problems

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

support vector machines for nonlinear kernel arma system identification nonlinear system identification based on support vector machines svm has been usually addressed by means of the standard svm regression svr which can be seen as an implicit nonlinear autoregressive and moving average arma model in some reproducing kernel hilbert space rkhs the proposal of this letter is twofold first the explicit consideration of an arma model in an rkhs svm arma2k is proposed we show that stating the arma equations in an rkhs leads to solving the regularized normal equations in that rkhs in terms of the autocorrelation and cross correlation of the nonlinearly transformed input and output discrete time processes second a general class of svm based system identification nonlinear models is presented based on the use of composite mercer s kernels this general class can improve model flexibility by emphasizing the input output cross information svm arma4k which leads to straightforward and natural combinations of implicit and explicit arma models svr arma2k and svr arma4k capabilities of these different svm based system identification schemes are illustrated with two benchmark problems

property testing and its connection to learning and approximation in this paper we consider the question of determining whether a function f has property p or is e far from any function with property p a property testing algorithm is given a sample of the value of f on instances drawn according to some distribution in some cases it is also allowed to query f on instances of its choice we study this question for different properties and establish some connections to problems in learning theory and approximation in particular we focus our attention on testing graph properties given access to a graph g in the form of being able to query whether an edge exists or not between a pair of vertices we devise algorithms to test whether the underlying graph has properties such as being bipartite k colorable or having a p clique clique of density p with respect to the vertex set our graph property testing algorithms are probabilistic and make assertions that are correct with high probability while making a number of queries that is independent of the size of the graph moreover the property testing algorithms can be used to efficiently i e in time linear in the number of vertices construct partitions of the graph that correspond to the property being tested if it holds for the input graph

quantum locally testable codes we initiate the study of quantum locally testable codes text qltc s classical text ltc s are very important in computational complexity these codes are defined as the linear subspace satisfying a set of local constraints with the additional requirement that their soundness r delta which is the probability that a randomly chosen constraint is violated is proportional to the proximity delta where delta n is the distance of a word from the code excellent text ltc s exist in the classical world and they are tightly related to the celebrated text pcp probabilistically checkable proof theorem in quantum complexity quantum error correcting codes provide central examples in the study of the illusive behavior of multiparticle entanglement and they have played a crucial role in many computational complexity results we provide a definition of the quantum analogue of text ltc s and motivate it by connecting its central notions in the study of both entanglement and quantum ha

a general classification rule for probability measures abstract we consider the problem of classifying an unknown probability distribution based on a sequence of random samples drawn according to this distribution specifically if a is a subset of the space of all probability measures m1 sigma over some compact polish space e we want to decide whether or not the unknown distribution belongs to a or its complement we propose an algorithm which leads a s to a correct decision for any a satisfying certain structural assumptions a refined decision procedure is also presented which given a countable collection ai c m1 sigma i 1 2 each satisfying the structural assumption will eventually determine a s the membership of the distribution in any finite number of the ai applications to density estimation and the problem of order determination of markov processes are discussed

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

semidefinite programming lower bounds and branch and bound algorithms for the quadratic minimum spanning tree problem abstract in this paper we investigate semidefinite programming sdp lower bounds for the quadratic minimum spanning tree problem qmstp two sdp lower bounding approaches are introduced here both apply lagrangian relaxation to an sdp relaxation for the problem the first one explicitly dualizes the semidefiniteness constraint attaching to it a positive semidefinite matrix of lagrangian multipliers the second relies on a semi infinite reformulation for the cone of positive semidefinite matrices and dualizes a dynamically updated finite set of inequalities that approximate the cone these lower bounding procedures are the core ingredient of two qmstp branch and bound algorithms our computational experiments indicate that the sdp bounds computed here are very strong being able to close at least 70 of the gaps of the most competitive formulation in the literature as a result their accompanying branch and bound algorithms are competitive with the best previously available qmstp exact algorithm in the literature in fact one of these new branch and bound algorithms stands out as the new best exact solution approach for the problem

nonlinear black box models in system identification mathematical foundations we discuss several aspects of the mathematical foundations of the nonlinear black box identification problem we shall see that the quality of the identification procedure is always a result of a certain trade off between the expressive power of the model we try to identify the larger the number of parameters used to describe the model the more flexible is the approximation and the stochastic error which is proportional to the number of parameters a consequence of this trade off is the simple fact that a good approximation technique can be the basis of a good identification algorithm from this point of view we consider different approximation methods and pay special attention to spatially adaptive approximants we introduce wavelet and neuron approximations and show that they are spatially adaptive then we apply the acquired approximation experience to estimation problems finally we consider some implications of these theoretical developments for the practically implemented versions of the spatially adaptive algorithms

shorter arithmetization of nondeterministic computations arithmetizing computation is a crucial component of many fundamental results in complexity theory including results that gave insight into the power of interactive proofs multi prover interactive proofs and probabilistically checkable proofs informally an arithmetization is a way to encode a machine s computation so that its correctness can be easily verified via few probabilistic algebraic checks we study the problem of arithmetizing nondeterministic computations for the purpose of constructing short probabilistically checkable proofs pcps with polylogarithmic query complexity in such a setting a pcp s proof length depends at least linearly on the length in bits of the encoded computation thus minimizing the number of bits in the encoding is crucial for minimizing pcp proof length in this paper we show how to arithmetize any t step computation on a nondeterministic turing machine by using a polynomial encoding of length o t log t 2 previously the best known length was t log t 4 for nondeterministic random access machines our length is o t log t 2 o 1 while prior work only achieved t log t 5 the polynomial encoding that we use is the reed solomon code when combined with the best pcps of proximity for this code our result yields quasilinear size pcps with polylogarithmic query complexity that are shorter by at least two logarithmic factors than in all prior work our arithmetization also enjoys additional properties first it is succinct i e the encoding of the computation can be probabilistically checked in log t o 1 time this property is necessary for constructing short pcps with a polylogarithmic time verifier furthermore our techniques extend in a certain well defined sense to the arithmetization of yet other nexp complete languages

testing reed muller codes a code is locally testable if there is a way to indicate with high probability that a vector is far enough from any codeword by accessing only a very small number of the vector s bits we show that the reed muller codes of constant order are locally testable specifically we describe an efficient randomized algorithm to test if a given vector of length n 2 sup m is a word in the rth order reed muller code r r m of length n 2 sup m for a given integer r spl ges 1 and real spl epsi 0 the algorithm queries the input vector spl upsi at o 1 spl epsi r2 sup 2r positions on the one hand if spl upsi is at distance at least spl epsi n from the closest codeword then the algorithm discovers it with probability at least 2 3 on the other hand if spl upsi is a codeword then it always passes the test our result is almost tight any algorithm for testing r r m must perform spl omega 1 spl epsi 2 sup r queries

multiscale support vector learning with projection operator wavelet kernel for nonlinear dynamical system identification a giant leap has been made in the past couple of decades with the introduction of kernel based learning as a mainstay for designing effective nonlinear computational learning algorithms in view of the geometric interpretation of conditional expectation and the ubiquity of multiscale characteristics in highly complex nonlinear dynamic systems 1 3 this paper presents a new orthogonal projection operator wavelet kernel aiming at developing an efficient computational learning approach for nonlinear dynamical system identification in the framework of multiresolution analysis the proposed projection operator wavelet kernel can fulfill the multiscale multidimensional learning to estimate complex dependencies the special advantage of the projection operator wavelet kernel developed in this paper lies in the fact that it has a closed form expression which greatly facilitates its application in kernel learning to the best of our knowledge it is the first closed form orthogonal projection wavelet kernel reported in the literature it provides a link between grid based wavelets and mesh free kernel based methods simulation studies for identifying the parallel models of two benchmark nonlinear dynamical systems confirm its superiority in model accuracy and sparsity

a swarm intelligence approach to the quadratic minimum spanning tree problem the quadratic minimum spanning tree problem q mst is an extension of the minimum spanning tree problem mst in q mst in addition to edge costs costs are also associated with ordered pairs of distinct edges and one has to find a spanning tree that minimizes the sumtotal of the costs of individual edges present in the spanning tree and the costs of the ordered pairs containing only edges present in the spanning tree though mst can be solved in polynomial time q mst is np hard in this paper we present an artificial bee colony abc algorithm to solve q mst the abc algorithm is a new swarm intelligence approach inspired by intelligent foraging behavior of honey bees computational results show the effectiveness of our approach

linearity testing in characteristic two let dist f g pr sub u f u spl ne g u denote the relative distance between functions f g mapping from a group g to a group h and let dist f denote the minimum over all linear functions homomorphisms g of dist f g given a function f g spl rarr h we let err f pr sub u spl upsi f u f spl upsi spl ne f u spl upsi denote the rejection probability of the blum luby rubinfeld 1993 linearity test linearity testing is the study of the relationship between err f and dist f and in particular lower bounds on err f in terms of dist f we discuss when the underlying groups are g gf 2 sup n and h gf 2 in this case the collection of linear functions describe a hadamard code of block length 2 sup n and for an arbitrary function f mapping gf 2 sup n to gf 2 the distance dist l measures its distance to a hadamard code err f is a parameter that is easy to measure and linearity testing studies the relationship of this parameter to the distance of f the code and corresponding test are used in the construction of efficient probabilistically checkable proofs and thence in the derivation of hardness of approximation improved analyses translate into better nonapproximability results we present a description of the relationship between err f and dist f which is nearly complete in all its aspects and entirely complete in some we present functions l u 0 1 spl rarr 0 1 such that for all x spl isin 0 1 we have l x spl les err f spl les u x whenever dist f x with the upper bound being tight on the whole range and the lower bound tight on a large part of the range and close on the rest part of our strengthening is obtained by showing a new connection between linearity testing and fourier analysis

the complexity of colouring problems on dense graphs abstract we consider the problem of deciding whether or not a graph has a vertex k colouring restricted to the class of graphs in which each vertex has degree at least n where is a fixed positive constant and n is the number of vertices of the graph we show that there is a deterministic polynomial time algorithm for this problem when k 3 k 2 and that this is best possible we also consider the corresponding enumeration problems

algebraic testing and weight distributions of codes we study the testing problem that is the problem of determining maybe probabilistically if a function to which one has oracle access satisfies a given property we propose a framework in which to formulate and carry out the analysis of several known tests this framework establishes a connection between testing and the theory of weight distributions of codes we illustrate this connection by giving a coding theoretic interpretation of several tests that fall under the label of low degree tests we also show how the connection naturally suggests a new way of testing for linearity over finite fields we derive from the macwilliams theorems a general result the duality testing lemma and use it to analyze the simpler tests that fall into our framework in contrast to other analyses of tests the ones we present elicit the fact that a test s probability of rejecting a fuction depends on how far away the function is from every function that satisfies the property of interest

small bias probability spaces efficient constructions and applications it is shown how to efficiently construct a small probability space on n binary random variables such that for every subset its parity is either zero or one with almost equal probability they are called epsilon biased random variables the number of random bits needed to generate the random variables is o log n log frac 1 epsilon thus if epsilon is polynomially small then the size of the sample space is also polynomial random variables that are epsilon biased can be used to construct almost k wise independent random variables where epsilon is a function of k these probability spaces have various applications l derandomization of algorithms many randomized algorithms that require only k wise independence of their random bits where k is bounded by o log n can be derandomized by using epsilon biased random variables 2 reducing the number of random bits required by certain randomized algorithms e g verification of matrix multiplication 3 exhaustive tes

constant round interactive proofs for delegating computation the celebrated sf ip sf pspace theorem lund fortnow karloff and nisan j acm 39 1992 pp 859 868 shamir j acm 39 1992 pp 869 877 allows an all powerful but untrusted pro

solving the quadratic minimum spanning tree problem abstract given an undirected graph with costs associated to its edges and pairs of edges the quadratic minimum spanning tree problem qmstp requires to determine a spanning tree of minimum total cost this is a proper model for network problems in which both routing and interference costs need to be considered it is np hard in the strong sense and not approximable unless p np this paper describes a tabu search algorithm with two independent and adaptively tuned tabu lists and a variable neighbourhood search algorithm both metaheuristics are based on the same neighbourhood but the tabu search proves more effective and robust than the variable neighbourhood search to assess the quality of these results we provide a comparison with the heuristic algorithms proposed in the recent literature and we reimplement with minor improvements an exact algorithm drawn from the literature which confirms the optimality of the results obtained on small instances

constant rate pcps for circuit sat with sublinear query complexity the pcp theorem arora et al 1998 arora and safra 1998 says that every np proof can be encoded to another proof namely a probabilistically checkable proof pcp which can be tested by a verifier that queries only a small part of the pcp a natural question is how large is the blow up incurred by this encoding that is how long is the pcp compared to the original np proof the state of the art work of ben sasson and sudan 2008 and dinur 2007 shows that one can encode proofs of length n by pcps of length n poly log n that can be verified using a constant number of queries in this work we show that if the query complexity is relaxed to ne then one can construct pcps of length o n for circuit sat and pcps of length o tlog t for any language in ntime t more specifically for any e 0 we present nonuniform probabilistically checkable proofs pcps of length 2o 1 e n that can be checked using ne queries for circuit sat instances of size n our pcps have perfect completeness and constant soundness this is the first constant rate pcp construction that achieves constant soundness with nontrivial query complexity o n our proof replaces the low degree polynomials in algebraic pcp constructions with tensors of transitive algebraic geometry ag codes we show that the automorphisms of an ag code can be used to simulate the role of affine transformations that are crucial in earlier high rate algebraic pcp constructions using this observation we conclude that any asymptotically good family of transitive ag codes over a constant sized alphabet leads to a family of constant rate pcps with polynomially small query complexity such codes are constructed in the appendix to this article for the first time for every message length building on an earlier construction for infinitely many message lengths by stichtenoth 2006

locally testable codes and pcps of almost linear length we initiate a systematic study of locally testable codes that is error correcting codes that admit very efficient membership tests specifically these are codes accompanied with tests that make a constant number of random queries into any given word and reject non codewords with probability proportional to their distance from the code locally testable codes are believed to be the combinatorial core of pcps however the relation is less immediate than commonly believed nevertheless we show that certain pcp systems can be modified to yield locally testable codes on the other hand we adapt techniques that we develop for the construction of the latter to yield new pcps our main results are locally testable codes and pcps of almost linear length specifically we prove the existence of the following constructs locally testable binary linear codes in which k information bits are encoded by a codeword of length k exp o log k this improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet pcp systems of almost linear length for sat the length of the proof is n exp o log n and verification in performed by a constant number i e 19 of queries as opposed to previous results that used proof length n 1 p o 1 q for verification by q queries the novel techniques in use include a random projection of certain codewords and pcp oracles that preserves local testability an adaptation of pcp constructions to obtain linear pcp oracles for proving conjunctions of linear conditions and design of pcps with some new soundness properties a direct construction of locally testable linear codes of subexponential length

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

fuzzy quadratic minimum spanning tree problem in this paper a fuzzy quadratic minimum spanning tree problem is formulated as expected value model chance constrained programming and dependent chance programming according to different decision criteria then the crisp equivalents are derived when the fuzzy costs are characterized by trapezoidal fuzzy numbers furthermore a simulation based genetic algorithm using prufer number representation is designed for solving the proposed fuzzy programming models as well as their crisp equivalents finally a numerical example is provided for illustrating the effectiveness of the genetic algorithm

a polynomial approximation scheme for scheduling on uniform processors using the dual approximation approach in this paper we present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors more specifically the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible we give a family of polynomial time algorithms a such that a delivers a solution that is within a relative error of e of the optimum the technique employed is the dual approximation approach where infeasible but superoptimal solutions for a related dual problem are converted to the desired feasible but possibly suboptimal solution

high rate locally correctable and locally testable codes with sub polynomial query complexity locally correctable codes lccs and locally testable codes ltcs are error correcting codes that admit local algorithms for correction and detection of errors those algorithms are local in the sense that they only query a small number of entries of the corrupted codeword the fundamental question about lccs and ltcs is to determine the optimal tradeoff among their rate distance and query complexity in this work we construct the first lccs and ltcs with constant rate constant relative distance and sub polynomial query complexity specifically we show that there exist lccs and ltcs with block length n constant rate which can even be taken arbitrarily close to 1 and constant relative distance whose query complexity is exp o slog n for lccs and log n o log log n for ltcs in addition to having small query complexity our codes also achieve better tradeoffs between the rate and the relative distance than were previously known to be achievable by lccs or ltcs specifically over large but constant size alphabet our codes approach the singleton bound that is they have almost the best possible relationship between their rate and distance over the binary alphabet our codes meet the zyablov bound such tradeoffs between the rate and the relative distance were previously not known for any o n query complexity our results on lccs also immediately give locally decodable codes with the same parameters

the complexity of promise problems with applications to public key cryptography a promise problem is a formulation of partial decision problem complexity issues about promise problems arise from considerations about cracking problems for public key cryptosystems using a notion of turing reducibility between promise problems this paper disproves a conjecture made by even and yacobi 1980 that would imply nonexistence of public key cryptosystems with np hard cracking problems in its place a new conjecture is raised having the same consequence in addition the new conjecture implies that np complete sets cannot be accepted by turing machines that have at most one accepting computation for each input word

free bits pcps and nonapproximability towards tight results this paper continues the investigation of the connection between probabilistically checkable proofs pcps and the approximability of np optimization problems the emphasis is on proving tight nonapproximability results via consideration of measures such as the free bit complexity and the amortized free bit complexity of proof systems the first part of the paper presents a collection of new proof systems based on a new error correcting code called the long code we provide a proof system that has amortized free bit complexity of 2 epsilon implying that approximating maxclique within n frac13 e and approximating the chromatic number within n frac15 e are hard assuming np neq corp for any e 0 we also derive the first explicit and reasonable constant hardness factors for min vertex cover msat 2 and max cut and we improve the hardness factor for msat 3 we note that our nonapproximability factors for maxsnp problems are appreciably close to the values known to be achievable by polynomial time algorithms finally we note a general approach to the derivation of strong nonapproximability results under which the problem reduces to the construction of certain gadgets the increasing strength of nonapproximability results obtained via the pcp connection motivates us to ask how far this can go and whether pcps are inherent in any way the second part of the paper addresses this the main result is a reversal of the connection due to feige et al fglss connection j acm 43 1996 pp 268 292 where the latter had shown how to translate proof systems for np into np hardness of approximation results for maxclique we show how any np hardness of approximation result for maxclique yields a proof system for np roughly our result says that for any constant f if maxclique is np hard to approximate within n 1 1 f then np subseteq overline fpcp log f the latter being the class of languages possessing proofs of logarithmic randomness and amortized free bit complexity f this suggests that pcps are inherent to obtaining nonapproximability results furthermore the tight relation suggests that reducing the amortized free bit complexity is necessary for improving the nonapproximability results for maxclique the third part of our paper initiates a systematic investigation of the properties of pcp and fpcp free pcp as a function of the following various parameters randomness query complexity free bit complexity amortized free bit complexity proof size etc we are particularly interested in triviality results which indicate which classes are not powerful enough to capture np we also distill the role of randomized reductions in this area and provide a variety of useful transformations between proof checking complexity classes

orthonormal wavelets with simple closed form expressions two classes of orthonormal wavelets that have simple closed form expressions are derived from pulses with the raised cosine spectrum these wavelets which are bandlimited and polynomial decaying in time are found to be particular examples of the lemarie meyer 1992 1993 wavelets the derivation reveals interesting connections between wavelet construction and intersymbol interference isi free signaling for digital communications

towards long term prediction abstract this paper describes a simple method of obtaining longer term predictions from a nonlinear time series assuming one already has a reasonably good short term predictor the usefulness of the technique is that it eliminates to some extent the systematic errors of the iterated short term predictor the technique we describe also provides an indication of the prediction horizon we consider systems with both observational and dynamic noise and analyse a number of artificial and experimental systems obtaining consistent results we also compare this method of longer term prediction with ensemble prediction

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection

two query pcp with subconstant error we show that the np complete language 3sat has a pcp verifier that makes two queries to a proof of almost linear size and achieves subconstant probability of error eeo 1 the verifier performs only projection tests meaning that the answer to the first query determines at most one accepting answer to the second query the number of bits representing a symbol in the proof depends only on the error e previously by the parallel repetition theorem there were pcp theorems with two query projection tests but only arbitrarily small constant error and polynomial size there were also pcp theorems with subconstant error and almost linear size but a constant number of queries that is larger than 2 as a corollary we obtain a host of new results in particular our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem a partial list includes the following 1 3sat cannot be efficiently approximated to within a factor of 7 8po 1 unless penp this holds even under almost linear reductions previously the best known np hardness factor was 7 8pe for any constant e 0 under polynomial reductions hastad 2 3lin cannot be efficiently approximated to within a factor of 1 2po 1 unless penp this holds even under almost linear reductions previously the best known np hardness factor was 1 2pe for any constant e 0 under polynomial reductions hastad 3 a pcp theorem with amortized query complexity 1 p o 1 and amortized free bit complexity o 1 previously the best known amortized query complexity and free bit complexity were 1pe and e respectively for any constant e 0 samorodnitsky and trevisan one of the new ideas that we use is a new technique for doing the composition step in the classical proof of the pcp theorem without increasing the number of queries to the proof we formalize this as a composition of new objects that we call locally decode reject codes ldrc the notion of ldrc was implicit in several previous works and we make it explicit in this work we believe that the formulation of ldrcs and their construction are of independent interest