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Abstract: This is a follow-up paper of “Liberating the dimension for function approximation”, where we studied approximation of infinitely variate functions by algorithms that use linear information consisting of finitely many linear functionals. In this paper, we study similar approximation problems, however, now the algorithms can only use standard information consisting of finitely many function values. We assume that the cost of one function value depends on the number of active variables. We focus on polynomial tractability, and occasionally also study weak tractability. We present non-constructive and constructive results. Non-constructive results are based on known relations between linear and standard information for finitely variate functions, whereas constructive results are based on Smolyak’s construction generalized to the case of infinitely variate functions. Surprisingly, for many cases, the results for standard information are roughly the same as for linear information. [Copyright &y& Elsevier]

In this paper, we present algorithms for the approximation of multivariate periodic functions by trigonometric polynomials. The approximation is based on sampling of multivariate functions on rank-1 lattices. To this end, we study the approximation of periodic functions of a certain smoothness. Our considerations include functions from periodic Sobolev spaces of generalized mixed smoothness. Recently an algorithm for the trigonometric interpolation on generalized sparse grids for this class of functions was investigated by Griebel and Hamaekers (2014). The main advantage of our method is that the algorithm is based mainly on a single one-dimensional fast Fourier transform, and that the arithmetic complexity of the algorithm depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. Therefore, we investigate trigonometric polynomials with frequencies supported on hyperbolic crosses and energy norm based hyperbolic crosses in more detail. Furthermore, we present an algorithm for sampling multivariate functions on perturbed rank-1 lattices and show the numerical stability of the suggested method. Numerical results are presented up to dimension d = 10 , which confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

The efficient evaluation of multivariate polynomials at many points is an important operation for polynomial system solving. Kedlaya and Umans have recently devised a theoretically efficient algorithm for this task when the coefficients are integers or when they lie in a finite field. In this paper, we assume that the set of points where we need to evaluate is fixed and "sufficiently generic". Under these restrictions, we present a quasi-optimal algorithm for multi-point evaluation over general fields. We also present a quasi-optimal algorithm for the opposite interpolation task. [ABSTRACT FROM AUTHOR]

This paper describes an algorithm which computes the characteristic polynomial of a matrix over a field within the same asymptotic complexity, up to constant factors, as the multiplication of two square matrices. Previously, this was only achieved by resorting to genericity assumptions or randomization techniques, while the best known complexity bound with a general deterministic algorithm was obtained by Keller-Gehrig in 1985 and involves logarithmic factors. Our algorithm computes more generally the determinant of a univariate polynomial matrix in reduced form, and relies on new subroutines for transforming shifted reduced matrices into shifted weak Popov matrices, and shifted weak Popov matrices into shifted Popov matrices. [ABSTRACT FROM AUTHOR]

In this paper, we study tractability of L 2 -approximation of one-periodic functions from weighted Korobov spaces in the worst-case setting. The considered weights are of product form. For the algorithms we allow information from the class Λ all consisting of all continuous linear functionals and from the class Λ std , which only consists of function evaluations. We provide necessary and sufficient conditions on the weights of the function space for quasi-polynomial tractability, uniform weak tractability, weak tractability and (σ , τ) -weak tractability. Together with the already known results for strong polynomial and polynomial tractability, our findings provide a complete picture of the weight conditions for all current standard notions of tractability. [ABSTRACT FROM AUTHOR]

Abstract: The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of homogeneous equations in variables is satisfiable (the resultant is a polynomial in the system’s coefficients which vanishes if and only if the system is satisfiable). In this paper, we investigate the complexity of computing the multivariate resultant. First, we study the complexity of testing the multivariate resultant for zero. Our main result is that this problem is -hard under deterministic reductions in any characteristic, for systems of low-degree polynomials with coefficients in the ground field (rather than in an extension). In null characteristic, we observe that this problem is in the Arthur–Merlin class if the generalized Riemann hypothesis holds true, while the best known upper bound in positive characteristic remains . Second, we study the classical algorithms to compute the resultant. They usually rely on the computation of the determinant of an exponential-size matrix, known as Macaulay matrix. We show that this matrix belongs to a class of succinctly representable matrices, for which testing the determinant for zero is proved -complete. This means that improving Canny’s upper bound requires either to look at the fine structure of the Macaulay matrix to find an ad hoc algorithm for computing its determinant, or to use altogether different techniques. [Copyright &y& Elsevier]

Abstract: Many methods to compute the winding number of plane curves have been proposed, often with the aim of counting the number of roots of polynomials (or, more generally, zeros of analytic functions) inside some region by using the principle of argument. In this paper we propose another method, which are not based on numerical integration, but on discrete geometry. We give conditions that ensure its correct behavior, and a complexity bound based on the distance of the curve to singular cases. Besides, we provide a modification of the algorithm that detects the proximity to a singular case of the curve. If this proximity is such that the number of operations required grows over certain threshold, set by the user, the modified algorithm returns without winding number computation, but with information about the distance to singular case. When the method is applied to polynomials, this information refers to the localization of the roots placed near the curve. [Copyright &y& Elsevier]

Abstract: In this paper we study planar polynomial differential systems of this form: where and , , and. A lot of properties of planar polynomial differential systems are related to irreducible Darboux polynomials of the corresponding derivation: . Darboux polynomials are usually computed with the method of undetermined coefficients. With this method we have to solve a polynomial system. We show that this approach can give rise to the computation of an exponential number of reducible Darboux polynomials. Here we show that the Lagutinskii–Pereira algorithm computes irreducible Darboux polynomials with degree smaller than , with a polynomial number, relatively to , and , binary operations. We also give a polynomial-time method to compute, if it exists, a rational first integral with bounded degree. [Copyright &y& Elsevier]

Abstract: Regularized classifiers (a leading example is support vector machine) are known to be a kind of kernel-based classification methods generated from Tikhonov regularization schemes, and the polynomial kernels are the original and also probably the most important kernels used in them. In this paper, we provide an error analysis for the regularized classifiers using multivariate polynomial kernels. We introduce Bernstein–Durrmeyer polynomials, whose reproducing kernel Hilbert space norms and approximation properties in space play a key role in the analysis of regularization error. We also introduce the standard estimation of sample error, and derive explicit learning rates for these algorithms. [Copyright &y& Elsevier]

Abstract: Valiant introduced 20 years ago an algebraic complexity theory to study the complexity of polynomial families. The basic computation model used is the arithmetic circuit, which makes these classes very easy to define and open to combinatorial techniques. In this paper we gather known results and new techniques under a unifying theme, namely the restrictions imposed upon the gates of the circuit, building a hierarchy from formulas to circuits. As a consequence we get simpler proofs for known results such as the equality of the classes VNP and or the completeness of the Determinant for VQP, and new results such as a characterization of the classes VQP and VP (which we can also apply to the Boolean class LOGCFL) or a full answer to a conjecture in Bürgisser''s book [Completeness and reduction in algebraic complexity theory, Algorithms and Computation in Mathematics, vol. 7, Springer, Berlin, 2000]. We also show that for circuits of polynomial depth and unbounded size these models all have the same expressive power and can be used to characterize a uniform version of VNP. [Copyright &y& Elsevier]

Abstract: In this paper we prove the existence of digitally shifted polynomial lattice rules which achieve strong tractability results for Sobolev spaces of arbitrary high smoothness. The convergence rate is shown to be the best possible up to a given degree of smoothness of the integrand. Indeed we even show the existence of polynomial lattice rules which automatically adjust themselves to the smoothness of the integrand up to a certain given degree. Further we show that strong tractability under certain conditions on the weights can be obtained and that polynomial lattice rules exist for which the worst-case error can be bounded independently of the dimension. These results hold independent of the smoothness. [Copyright &y& Elsevier]

Abstract: Let be multivariate polynomials (with complex coefficients) in the variables . The common zero locus of these polynomials, for , determines an algebraic set. This algebraic set decomposes into a union of simpler, irreducible components. The set of polynomials imposes on each component a positive integer known as the multiplicity of the component. Multiplicity plays an important role in many practical applications. It determines roughly “how many times the component should be counted in a computation”. Unfortunately, many numerical methods have difficulty in problems where the multiplicity of a component is greater than one. The main goal of this paper is to present an algorithm for determining the multiplicity of a component of an algebraic set. The method sidesteps the numerical stability issues which have obstructed other approaches by incorporating a combined numerical-symbolic technique. [Copyright &y& Elsevier]

Multivariate integration of high dimension s occurs in many applications. In many such applications, for example in finance, integrands can be well approximated by sums of functions of just a few variables. In this situation the superposition (or effective) dimension is small, and we can model the problem with finite-order weights, where the weights describe the relative importance of each distinct group of variables up to a given order (where the order is the number of variables in a group), and ignore all groups of variables of higher order.In this paper we consider multivariate integration for the anchored and unanchored (non-periodic) Sobolev spaces equipped with finite-order weights. Our main interest is tractability and strong tractability of QMC algorithms in the worst-case setting. That is, we want to find how the minimal number of function values needed to reduce the initial error by a factor ϵ depends on s and ϵ−1. If there is no dependence on s, and only polynomial dependence on ϵ−1, we have strong tractability, whereas with polynomial dependence on both s and ϵ−1 we have tractability.We show that for the anchored Sobolev space we have strong tractability for arbitrary finite-order weights, whereas for the unanchored Sobolev space we have tractability for all bounded finite-order weights. In both cases, the dependence on ϵ−1 is quadratic. We can improve the dependence on ϵ−1 at the expense of polynomial dependence on s. For finite-order weights, we may achieve almost linear dependence on ϵ−1 with a polynomial dependence on s whose degree is proportional to the order of the weights.We show that these tractability bounds can be achieved by shifted lattice rules with generators computed by the component-by-component (CBC) algorithm. The computed lattice rules depend on the weights. Similar bounds can also be achieved by well-known low discrepancy sequences such as Halton, Sobol and Niederreiter sequences which do not depend on the weights. We prove that these classical low discrepancy sequences lead to error bounds with almost linear dependence on n−1 and polynomial dependence on d. We present explicit worst-case error bounds for shifted lattice rules and for the Niederreiter sequence. Better tractability and error bounds are possible for finite-order weights, and even for general weights if they satisfy certain conditions. We present conditions on general weights that guarantee tractability and strong tractability of multivariate integration. [Copyright &y& Elsevier]