pinecone/core-2020-05-10-deduplication huggingface dataset:
An Efficient Algorithm for the Complex Roots Problem

AbstractGiven a univariate polynomialf(z) of degreenwith complex coefficients, whose norms are less than 2min magnitude, the root problem is to find all the roots off(z) up to specified precision 2−μ. Assuming the arithmetic model for computation, we provide an algorithm which has complexityO(nlog5nlogB), whereb= χ + μ, and χ = max{n,m}. This improves on the previous best known algorithm of Pan for the problem, which has complexityO(n2log2nlog(m+ μ)). A remarkable property of our algorithm is that it does not require any assumptions about the root separation off, which were either explicitly, or implicitly, required by previous algorithms. Moreover it also has a work-efficient parallel implementation. We also show that both the sequential and parallel implementations of the algorithm work without modification in the Boolean model of arithmetic. In this case, it follows from root perturbation estimates that we need only specify θ = ⌈n(B+ logn+ 3)⌉ bits of the binary representations of the real and imaginary parts of each of the coefficients off. We also show that by appropriate rounding of intermediate values, we can bound the number of bits required to represent all complex numbers occurring as intermediate quantities in the computation. The result is that we can restrict the numbers we use in every basic arithmetic operation to those having real and imaginary parts with at most φ bits, where[formula]and[formula]Thus, in the Boolean model, the overall work complexity of the algorithm is only increased by a multiplicative factor ofM(φ) (whereM(ψ) =O(ψ(log ψ) log log ψ) is the bit complexity for multiplication of integers of length ψ). The key result on which the algorithm is based, is a new theorem of Coppersmith and Neff relating the geometric distribution of the zeros of a polynomial to the distribution of the zeros of its high order derivatives. We also introduce several new techniques (splitting sets and “centered” points) which hinge on it. We also observe that our root finding algorithm can be efficiently parallelized to run in parallel timeO(log6nlogB) usingnprocessors
Counting Affine Roots of Polynomial Systems via Pointed Newton Polytopes

AbstractWe give a new upper bound on the number of isolated roots of a polynomial system. Unlike many previous bounds, our bound can also be restricted to different open subsets of affine space. Our methods give significantly sharper bounds than the classical Bézout theorems and further generalize the mixed volume root counts discovered in the late 1970s. We also give a complete combinatorial classification of the subsets of coefficients whose genericity guarantees that our bound is actually an exact root count in affine space. Our results hold over any algebraically closed field
On the Complexity of Sparse Elimination

AbstractSparse elimination exploits the structure of a multivariate polynomial by considering its Newton polytope instead of its total degree. We concentrate on polynomial systems that generate zero-dimensional ideals. A monomial basis for the coordinate ring is defined from a mixed subdivision of the Minkowski sum of the Newton polytopes. We offer a new simple proof relying on the construction of a sparse resultant matrix, which leads to the computation of a multiplication map and all common zeros. The size of the monomial basis equals the mixed volume and its computation is equivalent to computing the mixed volume, so the latter is a measure of intrinsic complexity. On the other hand, our algorithms have worst-case complexity proportional to the volume of the Minkowski sum. In order to derive bounds in terms of the sparsity parameters, we establish new bounds on the Minkowski sum volume as a function of mixed volume. To this end, we prove a lower bound on mixed volume in terms of Euclidean volume which is of independent interest
Complexity Estimates for the Schmüdgen Positivstellensatz

AbstractLetKbe a closed basic set inRngiven by the polynomial inequalities φ1≥ 0, ... , φm≥ 0 and let Σ be the semiring generated by the φkand the squares inR[x1, ... ,xn]. Schmüdgen has shown that ifKis compact then any polynomial function strictly positive onKbelongs to Σ. Easy consequences are (1)f≥ 0 onKif and only iff∈R++ Σ (Positivstellensatz) and (2) iff≥ 0 onKbutf∈ Σ then asdtends to 0+, in any representation off + das an element of Σ in terms of the φk, the squares and semiring operations, the integerN(d) which is the minimum over all representations of the maximum degree of the summands must become arbitrarily large. A one-dimensional example is analyzed to obtain asymptotic lower and upper bounds of the formcd−1/2≤N(d) ≤Cd−1/2log (1/d)
How to Benefit from Noise

AbstractWe compare sequential and non-sequential designs for estimating linear functionals in the statistical setting, where experimental observations are contaminated by random noise. It is known that sequential designs are no better in the worst case setting for convex and symmetric classes, as well as in the average case setting with Gaussian distributions.In the statistical setting the opposite is true. That is, sequential designs can be significantly better. Moreover, by using sequential designs one can obtain much better estimators for noisy data than for exact data. In this way, problems that are computationally intractable for exact data may become tractable for noisy data. These results hold because adaptive observations and noise make it possible to simulate Monte Carlo
Computing Discrepancies of Smolyak Quadrature Rules

AbstractIn recent years, Smolyak quadrature rules (also called quadratures on hyperbolic cross points or sparse grids) have gained interest as possible competition to number theoretic quadratures for high dimensional problems. A standard way of comparing the quality of multivariate quadrature formulas consists in computing theirL2-discrepancy. Especially for larger dimensions, such computations are a highly complex task. In this paper we develop a fast recursive algorithm for computing theL2-discrepancy (and related quality measures) of general Smolyak quadratures. We carry out numerical comparisons between the discrepancies of certain Smolyak rules and Hammersley and Monte Carlo sequences
Topological Complexity with Continuous Operations

AbstractThe topological complexity of algorithms is studied in a general context in the first part and for zero-finding in the second part. In the first part thelevel of discontinuityof a functionfis introduced and it is proved that it is a lower bound for the total number of comparisons plus 1 in any algorithm computingfthat uses only continuous operations and comparisons. This lower bound is proved to be sharp if arbitrary continuous operations are allowed. Then there exists even a balanced optimal computation tree forf. In the second part we use these results in order to determine the topological complexity of zero-finding for continuous functionsfon the unit interval withf(0) ·f(1) < 0. It is proved that roughly log2log2ϵ−1comparisons are optimal during a computation in order to approximate a zero up to ϵ. This is true regardless of whether one allows arbitrary continuous operations or just function evaluations, the arithmetic operations {+, −, *, /}, and the absolute value. It is true also for the subclass of nondecreasing functions. But for the subclass of increasing functions the topological complexity drops to zero even for the smaller class of operations
On the Expected Number of Shadow Vertices of the Convex Hull of Random Points

AbstractLeta1, ... ,ambe independent random points in Rnthat are independent and identically distributed spherically symmetrical in Rn. Moreover, letXbe the random polytope generated as the convex hull ofa1, ... ,amand letLkbe an arbitraryk-dimensional subspace of Rnwith 2 ≤k≤n− 1. LetXkbe the orthogonal projection image ofXinLk. We call those vertices ofXwhose projection images inLkare vertices ofXkshadow vertices ofXwith respect to the subspaceLk. We derive a distribution independent sharp upper bound for the expected number of shadow vertices ofXinLk
Numerical Integration of Peak Functions

AbstractWe study the logarithmic error of numerical methods for the integration of monotone or unimodal positive functions. We compare adaptive and nonadaptive methods in the worst case setting. It turns out that adaption significantly helps for the class of unimodal functions, but it does not help for the class of monotone functions. We do not assume any smoothness properties of the integrands and obtain numerical methods that are reliable even for discontinuous integrands. Numerical examples show that our method is very competitive in the case of nonsmooth and/or peak functions
Topological Complexity of Zero-Finding

AbstractThe topological complexity of zero-finding is studied using a BSS machine over the reals with an information node. The topological complexity depends on the class of functions, the class of arithmetic operations, and on the error criterion. For the root error criterion the following results are established. If only Hölder operations are permitted as arithmetic operations then the topological complexity is roughly −log2ϵ and bisection is optimal. This holds even for the small class of linear functions. On the other hand, for the class of all increasing functions, if we allow the sign function or division, together with log and exp, then the topological complexity drops to zero. For the residual error criterion, results can be totally different than for the root error criterion. For example, the topological complexity can be zero for the residual error criterion, and roughly −log2ϵ for the root error criterion
Worst Case Complexity of Problems with Random Information Noise

AbstractWe study the worst case complexity of solving problems for which information is partial and contaminated by random noise. It is well known that if information is exact then adaption does not help for solving linear problems, i.e., for approximating linear operators over convex and symmetric sets. On the other hand, randomization can sometimes help significantly. It turns out that for noisy information, adaption may lead to much better approximations than nonadaption, even for linear problems. This holds because, in the presence of noise, adaption is equivalent to randomization. We present sharp bounds on the worst case complexity of problems with random noise in terms of the randomized complexity with exact information. The results obtained are applied to thed-variate integration andL∞-approximation of functions belonging to Hölder and Sobolev classes. Information is given by function evaluations with Gaussian noise of variance σ2. For exact information, the two problems are intractable since the complexity is proportional to (1/ϵ)qwhereqgrows linearly withd. For noisy information the situation is different. For integration, the ϵ-complexity is of order σ2/ϵ2as ϵ goes to zero. Hence the curse of dimensionality is broken due to random noise. for approximation, the complexity is of order σ2(1/ϵ)q+2ln(1/ϵ), and the problem is intractable also with random noise
The Complexity of Definite Elliptic Problems with Noisy Data

AbstractWe study the complexity of 2mth order definite elliptic problemsLu=f(with homogeneous Dirichlet boundary conditions) over ad-dimensional domain Ω, error being measured in theHm(Ω)-norm. The problem elementsfbelong to the unit ball ofWr,p(Ω), wherep∈ [2, ∞] andr>d/p. Information consists of (possibly adaptive) noisy evaluations offor the coefficients ofL. The absolute error in each noisy evaluation is at most δ. We find that thenth minimal radius for this problem is proportional ton−r/d+ δ, and that a noisy finite element method with quadrature (FEMQ), which uses only function values, and not derivatives, is a minimal error algorithm. This noisy FEMQ can be efficiently implemented using multigrid techniques. Using these results, we find tight bounds on the ϵ-complexity (minimal cost of calculating an ϵ-approximation) for this problem, said bounds depending on the costc(δ) of calculating a δ-noisy information value. As an example, if the cost of a δ-noisy evaluation isc(δ) = δ−s(fors> 0), then the complexity is proportional to (1/ϵ)d/r+s
How Partial Knowledge Helps to Solve Linear Programs

AbstractWe present results on how partial knowledge helps to solve linear programs. In particular, if a linear system,Ax=bandx≥0, has an interior feasible point, then we show that finding a feasible point to this system can be done inO(n2.5c(A)) iterations by the layered interior-point method, and each iteration solves a least-squares problem, wherenis the dimension of vectorxandc(A) is the condition number of matrixAdefined by Vavasis and Ye. This complexity bound is reduced by a factornfrom that when this property does not exists. We also present a result for solving the problem using a little strong knowledge
Graeffe's, Chebyshev-like, and Cardinal's Processes for Splitting a Polynomial into Factors

AbstractNumerical splitting of a real or complex univariate polynomial into factors is the basic step of the divide-and-conquer algorithms for approximating complex polynomial zeros. Such algorithms are optimal (up to polylogarithmic factors) and are quite promising for practical computations. In this paper, we develop some new techniques, which enable us to improve numerical analysis, performance, and computational cost bounds of the known splitting algorithms. In particular, we study a Chebyshev-like modification of Graeffe's lifting iteration (which is a basic block of the splitting algorithms, as well as of several other known algorithms for approximating polynomial zeros), analyze its numerical performance, compare it with Graeffe's, prove some results on numerical stability of both lifting processes (that is, Graeffe's and Chebyshev-like), study their incorporation into polynomial root-finding algorithms, and propose some improvements of Cardinal's recent effective technique for numerical splitting of a polynomial into factors. Our improvement relies, in particular, on a modification of the matrix sign iteration, based on the analysis of some conformal mappings of the complex plane and of techniques of recursive lifting/recursive descending. The latter analysis reveals some otherwise hidden correlations among Graeffe's, Chebyshev-like, and Cardinal's iterative processes, and we exploit these correlations in order to arrive at our improvement of Cardinal's algorithm. Our work may also be of some independent interest for the study of applications of conformal maps of the complex plane to polynomial root-finding and of numerical properties of the fundamental techniques for polynomial root-finding such as Graeffe's and Chebyshev-like iterations
An Algebraic Proof of Barlet's Join Theorem

AbstractIn this article we give an algebraic proof of Barlet's join theorem, using an improved version of an algorithm for computing the join that was presented in (J. Dalbec and B. Sturmfels,in“Invariant Methods in Discrete and Computational Geometry, Curaçao, 1994,” pp. 37–58, Kluwer Academic, Dordrecht, 1995). The improvements in the algorithm consist of proving the correctness of the algorithm in large prime characteristics and giving explicit formulas for the functionsp(q,L) that were left unspecified in the original. The first three sections are largely expository; the fourth presents the modified algorithm and the proof of the main theorem. The last section gives an example of the algorithm; unfortunately, the complexity of the process makes it difficult to compute anything without resorting to tricky shortcuts, as we do in the example
Metropolis, Simulated Annealing, and Iterated Energy Transformation Algorithms: Theory and Experiments

AbstractIn this paper, we compare from the theoretical and experimental points of view three stochastic optimization algorithms: the Metropolis, simulated annealing, and iterated energy transformation algorithms. We give the optimal exponents for the concentration of the marginal distribution of the final state of these algorithms around the global minima of the virtual energy function. Experiments are performed on an N.P. complete benchmark which tries to retain the main aspects of scheduling problems. They lead to the same qualitative ranking of algorithms as the theory does
Semi-algebraic Complexity—Additive Complexity of Matrix Computational Tasks

AbstractThis paper is devoted to the study of lower bounds on the inherent number of additions and subtractions necessary to solve some natural matrix computational tasks such as computing the nullspace, some band transformation, and some triangulation of a givenm×mmatrix. The additive complexities of such tasks are shown to grow asymptotically like that of them×mmatrix multiplication. The paper is a continuation of an earlier paper by the authors, and also of4where multiplicative complexity has been considered. We also propose a formalization of semialgebraic computational tasks
Newton's Iteration for Inversion of Cauchy-Like and Other Structured Matrices

AbstractWe specify some initial assumptions that guarantee rapid refinement of a rough initial approximation to the inverse of a Cauchy-like matrix, by means of our new modification of Newton's iteration, where the input, output, and all the auxiliary matrices are represented with their short generators defined by the associated scaling operators. The computations are performed fast since they are confined to operations with short generators of the given and computed matrices. Because of the known correlations among various structured matrices, the algorithm is immediately extended to rapid refinement of rough initial approximations to the inverses of Vandermonde-like, Chebyshev–Vandermonde-like, and Toeplitz-like matrices, where again the computations are confined to operations with short generators of the involved matrices
Polar Varieties, Real Equation Solving, and Data Structures: The Hypersurface Case

AbstractIn this paper we apply for the first time a new method for multivariate equation solving which was developed for complex root determination to therealcase. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm yields a new method for symbolically solving zero-dimensional polynomial equation systems over the complex numbers. One feature of central importance of this algorithm is the use of a problem-adapted data type represented by the data structures arithmetic network and straight-line program (arithmetic circuit). The algorithm finds the complex solutions of any affine zero-dimensional equation system in nonuniform sequential time that ispolynomialin the length of the input (given in straight-line program representation) and an adequately definedgeometric degree of the equation system.Replacing the notion of geometric degree of the given polynomial equation system by a suitably definedreal (or complex) degreeof certain polar varieties associated to the input equation of the real hypersurface under consideration, we are able to find for each connected component of the hypersurface a representative point (this point will be given in a suitable encoding). The input equation is supposed to be given by a straight-line program and the (sequential time) complexity of the algorithm is polynomial in the input length and the degree of the polar varieties mentioned above
Elimination of Constants from Machines over Algebraically Closed Fields

AbstractLet K be an algebraically closed field of characteristic 0. We show that constants can be removed efficiently from any machine over K solving a problem which is definable without constants. This gives a new proof of the Blum–Cucker–Shub–Smale transfer theorem for the problem P=?NP. We have similar results in positive characteristic for non-uniform complexity classes. We also construct explicit and correct test sequences (in the sense of Heintz and Schnorr) for the class of polynomials which are easy to compute
On Computing a Set of Points Meeting Every Cell Defined by a Family of Polynomials on a Variety

AbstractWe consider a family ofspolynomials, P = {P1, …,Ps}, inkvariables with coefficients in a real closed fieldR, each of degree at mostd, and an algebraic varietyVof real dimensionk′ which is defined as the zero set of a polynomialQof degree at mostd. The number of semi-algebraically connected components of all non-empty sign conditions on P overVis bounded bysk′(O(d))k. In this paper we present a new algorithm to compute a set of points meeting every semi-algebraically connected component of each non-empty sign condition of P overV. Its complexity issk′ + 1dO(k). This interpolates a sequence of results between the Ben-Or–Kozen–Reif algorithm which is the casek′ = 0, in one variable, and the Basu–Pollack–Roy algorithm which is the casek′ =k. It improves the results where the same problem was solved in timesk′ + 1dO(k′k)
Nearly Sharp Complexity Bounds for Multiprocessor Algebraic Computations

AbstractThe complexity lower bound of Ω(logN) was obtained [12,14] for recognizing a semialgebraic set withNconnected components by some parallel computational models, like the accepting network or the algebraic PRAM. It is unknown whether a parallel computation has an advantage versus its sequential counterpart for this sort of problem, i.e., whether it could recognize a semialgebraic set faster than within complexityO(logN) (the complexity lower bound for sequential algebraic computation trees [1,15]). We introduce a computational model called the multiprocessor algebraic computation which extends the notions of accepting network and algebraic PRAM. For this model an Ω(logN) complexity lower bound still holds. We design a multiprocessor algebraic computation which recognizes a linear complex in Rn(i.e., a set given by a Boolean combination ofNlinear inequalities) within the complexityO(logNloglogN) for smalln
Decoding of Reed Solomon Codes beyond the Error-Correction Bound

AbstractWe present a randomized algorithm which takes as inputndistinct points {(xi,yi)}i= 1nfromF×F(whereFis a field) and integer parameterstanddand returns a list of all univariate polynomialsfoverFin the variablexof degree at mostdwhich agree with the given set of points in at leasttplaces (i.e.,yi=f(xi) for at leasttvalues ofi), providedt= Ω(nd). The running time is bounded by a polynomial inn. This immediately provides a maximum likelihood decoding algorithm for Reed Solomon Codes, which works in a setting with a larger number of errors than any previously known algorithm. To the best of our knowledge, this is the first efficient (i.e., polynomial time bounded) algorithm which provides error recovery capability beyond the error-correction bound of a code for any efficient (i.e., constant or even polynomial rate) code
The Fast Generalized Parker–Traub Algorithm for Inversion of Vandermonde and Related Matrices

AbstractIn this paper we compare the numerical properties of the well-knownfastO(n2) Traub and Björck–Pereyra algorithms, which both use the special structure of a Vandermonde matrix to rapidly compute the entries of its inverse. The results of numerical experiments suggest that the Parker variant of what we shall call the Parker–Traub algorithm allows one not onlyfastO(n2) inversion of a Vandermonde matrix, but it also gives moreaccuracy. We also show that the Parker–Traub algorithm is connected to the well-known concept ofdisplacement rank,introduced by Kailath, Kung, and Morf about two decades ago, and therefore this algorithm can be generalized to invert the more general class ofVandermonde-likematrices, naturally suggested by the idea of displacement