pinecone/core-2020-05-10-deduplication huggingface dataset:
Complexity Estimates for Representations of Schmüdgen Type

AbstractLet Φ={φ1, …, φj} and let K be a closed basic set in Rn given by the polynomial inequalities φ1⩾0, …, φj⩾0. Let Σ{Φ} be the semiring generated by the φk and the squares in R[x1, …, xn]. For example, if Φ={φ1} then Σ{Φ}=σ1+σ2φ1, where σ1, σ2 are sums of squares of polynomials. Schmüdgen has shown that if K is compact then any polynomial strictly positive on K belongs to Σ{Φ}. This paper develops a result of Schmüdgen type for functions in one dimension merely nonnegative on K. For this, it is necessary to add additional hypotheses, such as the proximity of complex zeros, to compensate for the loss of strict positivity necessary for Schmüdgen's result
Numerical Integration of Harmonic Functions with Restricted Sampling Data

AbstractBased on the idea of kriging and the radial basis function approximation, we develop in this paper a numerical scheme to integrate harmonic functions with restricted sampling data. To be more precise, the integration is performed by using the function values which are given as discrete sampling data on only part of the boundary. These problems often arise from non-destructive evaluation techniques in the engineering industry. The existence and uniqueness of the solution and the error estimation for the proposed numerical scheme are also discussed. Several numerical experimental results are presented for the verification on the accuracy and convergence of the method
Average Case Complexity of Weighted Approximation and Integration over R+

AbstractWe study weighted approximation and integration of Gaussian stochastic processes X defined over R+ whose rth derivatives satisfy a Hölder condition with exponent β in the quadratic mean. We assume that the algorithms use samples of X at a finite number of points. We study the average case (information) complexity, i.e., the minimal number of samples that are sufficient to approximate/integrate X with the expected error not exceeding ε. We provide sufficient conditions in terms of the weight and the parameters r and β for the weighted approximation and weighted integration problems to have finite complexity. For approximation, these conditions are necessary as well. We also provide sufficient conditions for these complexities to be proportional to the complexities of the corresponding problems defined over [0, 1], i.e., proportional to ε−1/α where α=r+β for the approximation and α=r+β+1/2 for the integration
Linear Complexity, k-Error Linear Complexity, and the Discrete Fourier Transform

AbstractComplexity measures for sequences of elements of a finite field play an important role in cryptology. We focus first on the linear complexity of periodic sequences. By means of the discrete Fourier transform, we determine the number of periodic sequences S with given prime period length N and linear complexity LN, 0(S)=c as well as the expected value of the linear complexity of N-periodic sequences. Cryptographically strong sequences should not only have a large linear complexity, but also the change of a few terms should not cause a significant decrease of the linear complexity. This requirement leads to the concept of the k-error linear complexity LN, k(S) of sequences S with period length N. For some k and c we determine the number of periodic sequences S with given period length N and LN, k(S)=c. For prime N we establish a lower bound on the expected value of the k-error linear complexity
Relations between Classical, Average, and Probabilistic Kolmogorov Widths

AbstractLet μ be a centered Gaussian Radon measure on a Banach space E, and let Hμ⊆E be its reproducing kernel Hilbert space with unit ball Kμ. We prove that for the μ-average widths d(a)n(E, μ) of E and the classical Kolmogorov widths dn(Kμ, E) we haved(a)n(E,  μ)≍n−α(logn)βiffdn(Kμ, E)≍n−1/2−α(logn)β for any α>0, β∈R. Moreover, order optimal subspaces for dn(Kμ, E) are order optimal for d(a)n(E, μ) as well. Furthermore, we show that for the probabilistic widths d(p)n, δ(E, μ) we have the estimate12d(a)n(E,  μ)⩽d(p)n, δ(E,  μ)⩽c 1d(a)n(E,  μ)(1+log2/δ) for some universal constant c1>0 and for all δ<δ0. These results are applied to find concrete estimates in some specific settings
Extra-Updates Criterion for the Limited Memory BFGS Algorithm for Large Scale Nonlinear Optimizatio

AbstractThis paper studies recent modifications of the limited memory BFGS (L-BFGS) method for solving large scale unconstrained optimization problems. Each modification technique attempts to improve the quality of the L-BFGS Hessian by employing (extra) updates in a certain sense. Because at some iterations these updates might be redundant or worsen the quality of this Hessian, this paper proposes an updates criterion to measure this quality. Hence, extra updates are employed only to improve the poor approximation of the L-BFGS Hessian. The presented numerical results illustrate the usefulness of this criterion and show that extra updates improve the performance of the L-BFGS method substantially
Computation of Edit Probabilities and Edit Distances for the A5-Type Keystream Generator

AbstractSeveral edit probabilities and edit distances incorporating the stop/go clocking in the well-known A5-type keystream generator are proposed. Recursive algorithms for their efficient computation are derived. It is shown how the edit probabilities can be used to mount statistically optimal correlation attacks on any two of the three stop/go clocked shift registers. By using a method for estimating the underlying false alarm probability, it is argued that the minimum output sequence length required to be known for a successful attack is linear in the total length of the respective shift registers
A Two-Dimensional Bisection Envelope Algorithm for Fixed Points

AbstractIn this paper we present a new algorithm for the two-dimensional fixed point problem f(x)=x on the domain [0, 1]×[0, 1], where f is a Lipschitz continuous function with respect to the infinity norm, with constant 1. The computed approximation x satisfies ‖f(x)−x‖∞⩽ε for a specified tolerance ε<0.5. The upper bound on the number of required function evaluations is given by 2⌈log2(1/ε)⌉+1. Similar bounds were derived for the case of the 2-norm by Z. Huang et al. (1999, J. Complexity15, 200–213), our bound is the first for the infinity norm case
Tractability of Integration in Non-periodic and Periodic Weighted Tensor Product Hilbert Spaces

AbstractWe study strong tractability and tractability of multivariate integration in the worst case setting. This problem is considered in weighted tensor product reproducing kernel Hilbert spaces. We analyze three variants of the classical Sobolev space of non-periodic and periodic functions whose first mixed derivatives are square integrable. We obtain necessary and sufficient conditions on strong tractability and tractability in terms of the weights of the spaces. For the three Sobolev spaces periodicity has no significant effect on strong tractability and tractability. In contrast, for general reproducing kernel Hilbert spaces anything can happen: we may have strong tractability or tractability for the non-periodic case and intractability for the periodic one, or vice versa
Quantum Summation with an Application to Integration

AbstractWe study summation of sequences and integration in the quantum model of computation. We develop quantum algorithms for computing the mean of sequences that satisfy a p-summability condition and for integration of functions from Lebesgue spaces Lp([0, 1]d), and analyze their convergence rates. We also prove lower bounds showing that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of G. Brassard et al. (2000, “Quantum Amplitude Amplification and Estimation,” Technical Report, http://arXiv.org/abs/quant-ph/0005055) on computing the mean for bounded sequences and complements results of E. Novak (2001, J. Complexity17, 2–16) on integration of functions from Hölder classes. The analysis requires an appropriate model of quantum computation, capable of covering the typical features of numerical problems such as dealing with real numbers and real-valued functions and with vector and function spaces. We develop and study such a model, which can be viewed as a quantum setting for information-based complexity theory
The Discrepancy and Gain Coefficients of Scrambled Digital Nets

AbstractDigital sequences and nets are among the most popular kinds of low discrepancy sequences and sets and are often used for quasi-Monte Carlo quadrature rules. Several years ago Owen proposed a method of scrambling digital sequences and recently Faure and Tezuka have proposed another method. This article considers the discrepancy of digital nets under these scramblings. The first main result of this article is a formula for the discrepancy of a scrambled digital (λ, t, m, s)-net in base b with n=λbm points that requires only O(n) operations to evaluate. The second main result is exact formulas for the gain coefficients of a digital (t, m, s)-net in terms of its generator matrices. The gain coefficients, as defined by Owen, determine both the worst-case and random-case analyses of quadrature error
The Brownian Bridge Does Not Offer a Consistent Advantage in Quasi-Monte Carlo Integration

AbstractThe Brownian bridge has been suggested as an effective method for reducing the quasi-Monte Carlo error for problems in finance. We give an example of a digital option where the Brownian bridge performs worse than the standard discretization. Hence, the Brownian bridge does not offer a consistent advantage in quasi-Monte Carlo integration. We consider integrals of functions of d variables with Gaussian weights such as the ones encountered in the valuation of financial derivatives and in risk management. Under weak assumptions on the class of functions, we study quasi-Monte Carlo methods that are based on different covariance matrix decompositions. We show that different covariance matrix decompositions lead to the same worst case quasi-Monte Carlo error and are, therefore, equivalent
Worst Case Complexity of Weighted Approximation and Integration over Rd

AbstractWe study the worst case complexity of weighted approximation and integration for functions defined over Rd. We assume that the functions have all partial derivatives of order up to r uniformly bounded in a weighted Lp-norm for a given weight function ψ. The integration and the error for approximation are defined in a weighted sense for another given weight ϱ. We present a necessary and sufficient condition on weight functions ϱ and ψ for the complexity of the problem to be finite. Under additional conditions, we show that the complexity of the weighted problem is proportional to the complexity of the corresponding classical problem defined over a unit cube and with ϱ=ψ=1. Similar results have been obtained recently for scalar functions (d=1) and for multivariate functions under restriction that ψ=1 and p=∞
The New k-Windows Algorithm for Improving thek -Means Clustering Algorithm

AbstractThe process of partitioning a large set of patterns into disjoint and homogeneous clusters is fundamental in knowledge acquisition. It is called Clustering in the literature and it is applied in various fields including data mining, statistical data analysis, compression and vector quantization. The k-means is a very popular algorithm and one of the best for implementing the clustering process. The k-means has a time complexity that is dominated by the product of the number of patterns, the number of clusters, and the number of iterations. Also, it often converges to a local minimum. In this paper, we present an improvement of the k-means clustering algorithm, aiming at a better time complexity and partitioning accuracy. Our approach reduces the number of patterns that need to be examined for similarity, in each iteration, using a windowing technique. The latter is based on well known spatial data structures, namely the range tree, that allows fast range searches
Verifying Topological Indices for Higher-Order Rank Deficiencies

AbstractIt has been known how to use computational fixed point theorems to verify existence and uniqueness of a true solution to a nonlinear system of equations within a small region about an approximate solution. This can, be done in O(n3) operations, where n is the number of equations and unknowns. However, these standard techniques are only valid if the Jacobi matrix for the system is nonsingular at the solution. In previous work and a dissertation (of Dian), we have shown, both theoretically and practically, that existence and multiplicity can be verified in a complex setting, and in the real setting for odd multiplicity, when the rank defect of the Jacobi matrix at an isolated solution is 1. Here, after reviewing work to date, we discuss the case of higher rank defect. In particular, it appears that p-dimensional searches are required if the rank defect is p, and that the work increases exponentially in p
On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree

AbstractIn this contribution the isolation of real roots and the computation of the topological degree in two dimensions are considered and their complexity is analyzed. In particular, we apply Stenger's degree computational method by splitting properly the boundary of the given region to obtain a sequence of subintervals along the boundary that forms a sufficient refinement. To this end, we properly approximate the function using univariate polynomials. Then we isolate each one of the zeros of these polynomials on the boundary of the given region in various subintervals so that these subintervals form a sufficiently refined boundary
On the Representation of Band Limited Functions Using Finitely Many Bits

AbstractIn this paper, we consider the question of representing an entire function of finite order and type in terms of finitely many bits, and reconstructing the function from these. Instead of making any further assumptions about the function, we measure the error in reconstruction in a suitably weighted Lp norm. The optimal number of bits in order to obtain a given accuracy is given by the Kolmogorov entropy. We determine this entropy in the case of certain compact subsets of these weighted Lp spaces and obtain constructive algorithms to determine the asymptotically optimal bit representation from finitely many samples of the function. Our theory includes both equidistant and non-uniform sampling. The reconstructions are polynomials, having several other optimality properties
The covering number in learning theory

AbstractThe covering number of a ball of a reproducing kernel Hilbert space as a subset of the continuous function space plays an important role in Learning Theory. We give estimates for this covering number by means of the regularity of the Mercer kernel K. For convolution type kernels K(x,t)=k(x−t) on [0,1]n, we provide estimates depending on the decay of k̂, the Fourier transform of k. In particular, when k̂ decays exponentially, our estimate for this covering number is better than all the previous results and covers many important Mercer kernels. A counter example is presented to show that the eigenfunctions of the Hilbert–Schmidt operator LmK associated with a Mercer kernel K may not be uniformly bounded. Hence some previous methods used for estimating the covering number in Learning Theory are not valid. We also provide an example of a Mercer kernel to show that LK1/2 may not be generated by a Mercer kernel
Interior-Point Methods: Worst Case and Average Case Analysis of a Phase-I Algorithm and a Termination Procedure

AbstractWe are interested in the average behavior of interior-point methods (IPMs) for linear programming problems (LPs). We use the rotation-symmetry-model as the probabilistic model for the average case analysis. This model had been used by Borgwardt in his average case analysis of the simplex-method. IPMs solve LPs in three phases. First, one has to find an appropriate starting point, then a sequence of interior points is generated, which converges to the optimal face. Finally, the optimum has to be calculated, as it is not an interior point. We present upper bounds on the average number of iterations in the first and the third phase by looking at random figures of the underlying polyhedron. These bounds show, that IPMs solve LPs in strongly polynomial time in the average case, so only the dimension parameters and not the encoding length of the problem determine the average behavior of IPMs
A Constructive Approach to Strong Tractability Using Quasi-Monte Carlo Algorithms

AbstractWe prove in a constructive way that multivariate integration in appropriate weighted Sobolev classes is strongly tractable and the ε-exponent of strong tractability is 1 (which is the best-possible value) under a stronger assumption than Sloan and Woźniakowski's assumption. We show that quasi-Monte Carlo algorithms based on the Sobol sequence and Halton sequence achieve the convergence order O(n−1+δ) for any δ>0 independent of the dimension with a worst-case deterministic guarantee (where n is the number of function evaluations). This implies that quasi-Monte Carlo algorithms based on the Sobol and Halton sequences converge faster and therefore are superior to Monte Carlo methods independent of the dimension for integrands in suitable weighted Sobolev classes
Support Vector Machines are Universally Consistent

AbstractWe show that support vector machines of the 1-norm soft margin type are universally consistent provided that the regularization parameter is chosen in a distinct manner and the kernel belongs to a specific class—the so-called universal kernels—which has recently been considered by the author. In particular it is shown that the 1-norm soft margin classifier with Gaussian RBF kernel on a compact subset X of Rd and regularization parameter cn=nβ−1 is universally consistent, if n is the training set size and 0<β<1/d
Complexity of Nonlinear Two-Point Boundary-Value Problems

AbstractWe study upper and lower bounds on the worst-case ε-complexity of nonlinear two-point boundary-value problems. We deal with general systems of equations with general nonlinear boundary conditions, as well as with second-order scalar problems. Two types of information are considered: standard information defined by the values or partial derivatives of the right-hand-side function, and linear information defined by arbitrary linear functionals. The complexity depends significantly on the problem being solved and on the type of information allowed. We define algorithms based on standard or linear information, using perturbed Newton's iteration, which provide upper bounds on the ε-complexity. The upper and lower bounds obtained differ by a factor of log log 1/ε. Neglecting this factor, for general problems the ε-complexity for the right-hand-side functions having r(r⩾2) continuous bounded partial derivatives turns out to be of order (1/ε)1/r for standard information, and (1/ε)1/(r+1) for linear information. For second-order scalar problems, linear information is even more powerful. The ε-complexity in this case is shown to be of order (1/ε)1/(r+2), while for standard information it remains at the same level as in the general case
Applicability of Smolyak's Algorithms to Certain Banach Spaces of Multivariate Functions

AbstractWe investigate the applicability of Smolyak's algorithm to tensor product problems on certain Banach spaces of multivariate functions. First, we show that the algorithm can be efficiently used for the integration problem on these function classes. For approximation problems on the Sobolev space W1,γr1,…,rd, we prove that the algorithm is applicable as well; the range spaces can be any Banach spaces of functions, provided that the tensor product of these spaces is natural. On the other hand, if the range spaces are the univariate smooth function classes Cγkrk, the same conclusion can be drawn for approximation problems on any natural tensor products of Banach spaces of functions. Applications are illustrated for the integration problem on Wpr1,…,rd([0,1]d)
Piecewise Linear Approximation of Smooth Compact Fibers

AbstractLet H:RN→Rm be a twice differentiable map. Let N≔n+m. Assume that the fiber F≔H−1(y0) of H over a point y0∈Rm is a compact n-dimensional differentiable manifold and that ∂H is of maximal rank in a neighborhood of F. Works of Allgower and Gnutzmann, and of Allgower and Schmidt give an algorithm to find a global piecewise linear approximation of F based on a triangulation of RN. Several authors have given algorithms for obtaining a piecewise linear approximation of F in a way that does not depend on the triangulation in RN. This effects a reduction of the combinatorial complexity from O(N!) to O(n!), but the approximations are not global. In this paper, a probability one algorithm is given, which, given H, F, N, m as above, uses homotopy continuation to construct a differentiable map H:R2n+1→Rn+1 with a compact differentiable manifold F≔H−1(0), which is diffeomorphic to F by an explicit diffeomorphism, and with ∂H of maximal rank in a neighborhood of F. This reduces the problem when N>2n+1 to the known piecewise linear algorithm in R2n+1 at the simpler expense of carrying out a well-behaved homotopy continuation starting at vertices of the simplices of the triangulation of a neighborhood of π(F)⊂R2n+1, where π:RN→R2n+1 is a generic linear projection. Consequently, a global algorithm is obtained with a combinatorial complexity of O((2n+1)!)
Topological Complexity of Zero Finding with Algebraic Operations

AbstractHow many tests does one have to perform in order to compute an ε-approximation of a zero of a function f out of a given class of continuous functions on the unit interval, for a given ε>0? We study this question in the context of the real number oracle machine model which uses the four standard arithmetic operations, comparisons with zero, and function values at adaptively chosen points. Let 0<ε<12. We show that for the class of all continuous functions f on the unit interval with f(0)<0 and f(1)>0 one needs exactly ⌈log2(1/(2ε))⌉ tests in the worst case during a computation. For the subclass of all functions which are additionally nondecreasing one needs roughly log2 log2ε−1 tests, and for the subclass of all functions which are additionally increasing one needs exactly 1 test
Component-by-Component Construction of Good Lattice Rules with a Composite Number of Points

AbstractWe develop algorithms to construct rank-1 lattice rules in weighted Korobov spaces of periodic functions and shifted rank-1 lattice rules in weighted Sobolev spaces of non-periodic functions. Analyses are given which show that the rules so constructed achieve strong QMC tractability error bounds. Unlike earlier analyses, there is no assumption that n, the number of quadrature points, be a prime number. However, we do assume that there is an upper bound on the number of distinct prime factors of n. The generating vectors and shifts characterizing the rules are constructed ‘component-by-component,’ that is, the (d+1)th components of the generating vectors and shifts are obtained using one-dimensional searches, with the previous d components kept unchanged