pinecone/core-2020-05-10-deduplication huggingface dataset:
Intractability Results for Integration and Discrepancy

AbstractWe mainly study multivariate (uniform or Gaussian) integration defined for integrand spaces Fd such as weighted Sobolev spaces of functions of d variables with smooth mixed derivatives. The weight γj moderates the behavior of functions with respect to the jth variable. For γj≡1, we obtain the classical Sobolev spaces whereas for decreasing γj's the weighted Sobolev spaces consist of functions with diminishing dependence on the jth variables. We study the minimal errors of quadratures that use n function values for the unit ball of the space Fd. It is known that if the smoothness parameter of the Sobolev space is one, then the minimal error is the same as the discrepancy. Let n(ε, Fd) be the smallest n for which we reduce the initial error, i.e., the error with n=0, by a factor ε. The main problem studied in this paper is to determine whether the problem is intractable, i.e., whether n(ε, Fd) grows faster than polynomially in ε−1 or d. In particular, we prove intractability of integration (and discrepancy) if limsupd∑dj=1γj/lnd=∞. Previously, such results were known only for restricted classes of quadratures. For γj≡1, the, following bounds hold for discrepancy •with boundary conditions1.0628d(1+o(1))⩽n(ε, Fd)⩽1.5dε−2,asd→∞,•without boundary conditions1.0463d(1+o(1))⩽n(ε, Fd)⩽1.125dε−2,asd→∞. These results are obtained by analyzing arbitrary linear tensor product functionals Id defined over weighted tensor product reproducing kernel Hilbert spaces Fd of functions of d variables. We introduce the notion of a decomposable kernel. For reproducing kernels that have a decomposable part we prove intractability of all Id with non-zero subproblems with respect to the decomposable part of the kernel, as long as the weights satisfy the condition mentioned above
A One-Dimensional Optimization Algorithm and Its Convergence Rate under the Wiener Measure

AbstractIn this paper we describe an adaptive algorithm for approximating the global minimum of a continuous function on the unit interval, motivated by viewing the function as a sample path of a Wiener process. It operates by choosing the next observation point to maximize the probability that the objective function has a value at that point lower than an adaptively chosen threshold. The error converges to zero for any continuous function. Under the Wiener measure, the error converges to zero at rate e−nδn, where {δn} (a parameter of the algorithm) is a positive sequence converging to zero at an arbitrarily slow rate
Complexity of Neural Network Approximation with Limited Information: A Worst Case Approach

AbstractIn neural network theory the complexity of constructing networks to approximate input-output functions is of interest. We study this in the more general context of approximating elements f of a normed space F using partial information about f. We assume information about f and the size of the network are limited, as is typical in radial basis function networks. We show complexity can be essentially split into two independent parts, information ε-complexity and neural ε-complexity. We use a worst case setting, and integrate elements of information-based complexity and nonlinear approximation. We consider deterministic and/or randomized approximations using information possibly corrupted by noise. The results are illustrated by examples including approximation by piecewise polynomial neural networks
Nonlinear Lebesgue and Itô Integration Problems of High Complexity

AbstractWe analyze the complexity of nonlinear Lebesgue integration problems in the average case setting for continuous functions with the Wiener measure and the complexity of approximating the Itô stochastic integral. G. W. Wasilkowski and H. Woźniakowski (2001, Math. Comp., 685–698) studied these problems, observed that their complexities are closely related, and showed that for certain classes of smooth functions with boundedness conditions on derivatives the complexity is proportional to ε−1. Here ε>0 is the desired precision with which the integral is to be approximated. They showed also that for certain natural function classes with weaker smoothness conditions the complexity is at most of order ε−2 and conjectured that this bound is sharp. We show that this conjecture is true
What Is the Complexity of Surface Integration?

AbstractWe study the worst case complexity of computing ε-approximations of surface integrals. This problem has two sources of partial information: the integrand f and the function g defining the surface. The problem is nonlinear in its dependence on g. Here, f is an r times continuously differentiable scalar function of l variables, and g is an s times continuously differentiable injective function of d variables with l components. We must have d⩽l and s⩾1 for surface integration to be well-defined. Surface integration is related to the classical integration problem for functions of d variables that are min{r, s−1} times continuously differentiable. This might suggest that the complexity of surface integration should be Θ((1/ε)d/min{r, s−1}). Indeed, this holds when d<l and s=1, in which case the surface integration problem has infinite complexity. However, if d⩽l and s⩾2, we prove that the complexity of surface integration is O((1/ε)d/min{r, s}). Furthermore, this bound is sharp whenever d<l
Non-linear Approximations Using Sets of Finite Cardinality or Finite Pseudo-dimension

AbstractWe investigate optimal non-linear approximations of multivariate periodic functions with mixed smoothness. In particular, we study optimal approximation using sets of finite cardinality (as measured by the classical entropy number), as well as sets of finite pseudo-dimension (as measured by the non-linear widths introduced by Ratsaby and Maiorov). Approximation error is measured in the Lq(Td)-sense, where Td is the d-dimensional torus. The functions to be approximated are in the unit ball SBrp, θ of the mixed smoothness Besov space or in the unit ball SWrp of the mixed smoothness Sobolev space. For 1<p, q<∞, 0<θ⩽∞ and r>0 satisfying some restrictions, we establish asymptotic orders of these quantities, as well as construct asymptotically optimal approximation algorithms. We particularly prove that for either r>1/p and θ⩾p or r>(1/p−1/q)+ and θ⩾min{q, 2}, the asymptotic orders of these quantities for the Besov class SBrp, θ are both n−r(logn)(d−1)(r+1/2−1/θ)
Uniform Distribution, Discrepancy, and Reproducing Kernel Hilbert Spaces

AbstractIn this paper we define a notion of uniform distribution and discrepancy of sequences in an abstract set E through reproducing kernel Hilbert spaces of functions on E. In the case of the finite-dimensional unit cube these discrepancies are very closely related to the worst case error obtained for numerical integration of functions in a reproducing kernel Hilbert space. In the compact case we show that the discrepancy tends to zero if and only if the sequence is uniformly distributed in our sense. Next we prove an existence theorem for such uniformly distributed sequences and investigate the relation to the classical notion of uniform distribution. Some examples conclude this paper
A Theory of Complexity for Continuous Time Systems

AbstractWe present a model of computation with ordinary differential equations (ODEs) which converge to attractors that are interpreted as the output of a computation. We introduce a measure of complexity for exponentially convergent ODEs, enabling an algorithmic analysis of continuous time flows and their comparison with discrete algorithms. We define polynomial and logarithmic continuous time complexity classes and show that an ODE which solves the maximum network flow problem has polynomial time complexity. We also analyze a simple flow that solves the Maximum problem in logarithmic time. We conjecture that a subclass of the continuous P is equivalent to the classical P
Kantorovich's Theorem on Newton's Method in Riemannian Manifolds

AbstractNewton's method for finding a zero of a vectorial function is a powerful theoretical and practical tool. One of the drawbacks of the classical convergence proof is that closeness to a non-singular zero must be supposed a priori. Kantorovich's theorem on Newton's method has the advantage of proving existence of a solution and convergence to it under very mild conditions. This theorem holds in Banach spaces. Newton's method has been extended to the problem of finding a singularity of a vectorial field in Riemannian manifold. We extend Kantorovich's theorem on Newton's method to Riemannian manifolds
A Probabilistic Analysis of Linear Operator Testing

AbstractWe test for β-conformance of an implementation linear operator A to a specification linear operator S where the operator domain and range are separable Hilbert spaces and the domain space F is equipped with a Gaussian measure μ. Given an error bound ε>0 and a tolerance parameter β∈(0, 1), we want to determine either that there is an element f in a ball Bq of radius q in domain F such that ‖Sf−Af‖>ε or that Aβ-conforms to S on a set of measure at least 1−β in the ball Bq; i.e., μq(f:‖Sf−Af‖⩽ε)⩾1−β where μq is the truncated Gaussian measure to Bq. We present a deterministic algorithm that solves this problem and uses almost a minimal number of tests where each test is an evaluation of operators S and A at an element of F. We prove that optimal tests are conducted on the eigenvectors of the covariance operator of μ. They are universal; they are independent of the operators under consideration and other problem parameters. We show that finite testing is conclusive in this probabilistic setting. In contrast, finite testing is inconclusive in the worst and average case settings; see [5, 7]. We discuss the upper and lower bounds on the minimal number of tests. For q=∞ we derive the exact bounds on the minimal number of tests, which depend on β very weakly. On the other hand, for a finite q, the bounds on the minimal number of tests depend on β more significantly. We explain our approach by an example with the Wiener measure
Learnability in Hilbert Spaces with Reproducing Kernels

AbstractWe explore the question of the learnability of classes of functions contained in a Hilbert space which has a reproducing kernel. We show that if the evaluation functionals are uniformly bounded and if the class is norm bounded then it is learnable. We formulate a learning procedure related to the well known support vector machine (SVM), which requires solving a system of linear equations, rather than the quadratic programming needed for the SVM. As a part of our discussion, we estimate the fat-shattering dimension of the unit ball of the dual of a Banach space when considered as a set of functions on the unit ball of the space itself. Our estimate is based on a geometric property of the Banach space called the type
Wrap-Around L2-Discrepancy of Random Sampling, Latin Hypercube and Uniform Designs

AbstractFor comparing random designs and Latin hypercube designs, this paper con- siders a wrap-around version of the L2-discrepancy (WD). The theoretical expectation and variance of this discrepancy are derived for these two designs. The expectation and variance of Latin hypercube designs are significantly lower than those of the corresponding random designs. We also study construction of the uniform design under the WD and show that one-dimensional uniform design under this discrepancy can be any set of equidistant points. For high dimensional uniform designs we apply the threshold accepting heuristic for finding low discrepancy designs. We also show that the conjecture proposed by K. T. Fang, D. K. J. Lin, P. Winker, and Y. Zhang (2000, Technometrics) is true under the WD when the design is complete
Variations on (0, s)-Sequences

AbstractIn the first part of this paper, after a survey on the general theory of (t, s)-sequences, we review the main constructions of (0, s)-sequences and point out some relations between them. From these comparisons and from examples we have met in another context, we have found new (0, s)-sequences in prime base b to which the second part is devoted; the proofs are based on the general framework of formal Laurent series introduced by Niederreiter; the construction can be randomized and should have numerical applications
Tractability of Multivariate Integration for Periodic Functions

AbstractWe study multivariate integration in the worst case setting for weighted Korobov spaces of smooth periodic functions of d variables. We wish to reduce the initial error by a factor ε for functions from the unit ball of the weighted Korobov space. Tractability means that the minimal number of function samples needed to solve the problem is polynomial in ε−1 and d. Strong tractability means that we have only a polynomial dependence in ε−1. This problem has been recently studied for quasi-Monte Carlo quadrature rules and for quadrature rules with non-negative coefficients. In this paper we study arbitrary quadrature rules. We show that tractability and strong tractability in the worst case setting hold under the same assumptions on the weights of the Korobov space as for the restricted classes of quadrature rules. More precisely, let γj moderate the behavior of functions with respect to the jth variable in the weighted Korobov space. Then strong tractability holds iff ∑∞j=1γj<∞, whereas tractability holds iff limsupd→∞∑dj=1γj/lnd<∞. We obtain necessary conditions on tractability and strong tractability by showing that multivariate integration for the weighted Korobov space is no easier than multivariate integration for the corresponding weighted Sobolev space of smooth functions with boundary conditions. For the weighted Sobolev space we apply general results from E. Novak and H. Woźniakowski (J. Complexity17 (2001), 388–441) concerning decomposable kernels
The Price of Pessimism for Multidimensional Quadrature

AbstractMultidimensional quadrature error for Hilbert spaces of integrands is studied in three settings: worst-case, random-case, and average-case. Explicit formulae are derived for the expected errors in each case. These formulae show the relative, pessimism of the three approaches. The first is the trace of a hermitian and nonnegative definite matrix ΛμQ, the second is the spectral radius of the same matrix ΛμQ, and the third is the trace of the matrix ΣΛμQ for a hermitian and nonnegative matrix Σ with trace (Σ)=1. Several examples are studied, including Monte Carlo quadrature and shifted lattice rules. Some of the results for Hilbert spaces of integrands can be extended to Banach spaces of integrands
Pricing Algorithms of Multivariate Path Dependent Options

AbstractFinancial derivatives which are multivariate in nature are abundant in the financial markets. The underlying state variables may be the stock prices, interest rates, exchange rates, stochastic volatility, average of stock prices, extremum values of stock priors, etc. Option contracts whose life and payoff depend on the stochastic movement of the underlying asset prices are termed path dependent options. In this paper, we examine the pricing methods of several prototype path dependent options. These include options with sequential barriers, options with an external barrier and two-asset lookback options. The governing equations for the option prices are seen to resemble the diffusion type equations but with cross derivative terms, a feature which differs from the usual diffusion equations in engineering. Various techniques to reduce the complexity of the multivariate nature of these prototype option pricing models are discussed. It is illustrated that the dimensionality of a path dependent option model may be reduced by some ingenious choices of similarity variables. We also examine the design of pricing algorithms of these multivariate options, in particular, with regard to the treatment of discrete monitoring feature and the prescription of numerical boundary conditions. The possible generalizations of the numerical techniques presented in this paper to other models with more complicated path dependent payoff structures are also discussed
Quasi-random Simulation of Linear Kinetic Equations

AbstractWe study the improvement achieved by using quasi-random sequences in place of pseudo-random numbers for solving linear spatially homogeneous kinetic equations. Particles are sampled from the initial distribution. Time is discretized and quasi-random numbers are used to move the particles in the velocity space. Quasi-random points are not blindly used in place of pseudo-random numbers: at each time step, the number order of the particles is scrambled according to their velocities. Convergence of the method is proved. Numerical results are presented for a sample problem in dimensions 1, 2 and 3. We show that by using quasi-random sequences in place of pseudo-random points, we are able to obtain reduced errors for the same number of particles
Calculation of the Quality Parameter of Digital Nets and Application to Their Construction

AbstractIn quasi-Monte Carlo methods, point sets of low discrepancy are crucial for accurate results. A class of point sets with low theoretic upper bounds of discrepancy are the digital point sets known as digital (t, m, s)-nets which can be implemented very efficiently. The parameter t is indicative of the quality; i.e., small values of t lead to small upper bounds of the discrepancy. We introduce an effective way to establish this quality parameter t for digital nets constructed over arbitrary finite fields and give an application to the construction of digital nets of high quality
An Algorithm to Compute Bounds for the Star Discrepancy

AbstractWe propose an algorithm to compute upper and lower bounds for the star discrepancy of an arbitrary sequence of points in the s-dimensional unit cube. The method is based on a particular partition of the unit cube into subintervals and on a specialized procedure for orthogonal range counting. The cardinality of the partition depends on the dimension and on an accuracy parameter that has to be specified. We have implemented this method and here we present results of some computational experiments obtained with this implementation
Complexity of Weighted Approximation over Rd

AbstractWe study approximation of multivariate functions defined over Rd. We assume that all rth order partial derivatives of the functions considered are continuous and uniformly bounded. Approximation algorithms U(f) only use the values of f or its partial derivatives up to order r. We want to recover the function f with small error measured in a weighted Lq norm with a weight function ρ. We study the worst case (information) complexity which is equal to the minimal number of function and derivative evaluations needed to obtain error ε. We provide necessary and sufficient conditions in terms of the weight ρ and the parameters q and r for the weighted approximation problem to have finite complexity. We also provide conditions guaranteeing that the complexity is of the same order as the complexity of the classical approximation problem over a finite domain. Since the complexity of the weighted integration problem is equivalent to the complexity of the weighted approximation problem with q=1, the results of this paper also hold for weighted integration. This paper is a continuation of [7], where weighted approximation over R was studied
Integration and Approximation Based on Scramble Sampling in Arbitrary Dimensions

AbstractThis paper considers integration in the worst case setting and approximation in the average case setting based on the scramble sampling scheme proposed by A. B. Owen (1995, Lecture Notes in Statistics, Vol. 106, pp. 299–317, Springer-Verlag, New York.) The tractability and strong tractability exponents are found for function spaces with reproducing/covariance kernels that are scramble-invariant. Integration and approximation for a space with a non-scramble-invariant kernel are no harder than the corresponding problems with the associated scramble-invariant kernel. This enables us to derive tractability results for weighted Sobolev spaces
What Is the Complexity of Volume Calculation?

AbstractWe study the worst case complexity of computing ε-approximations of volumes of d-dimensional regions g([0, 1]d), by sampling the function g. Here, g is an s times continuously differentiable injection from [0, 1]d to Rd, where we assume that s⩾1. Since the problem can be solved exactly when d=1, we concentrate our attention on the case d⩾2. This problem is a special case of the surface integration problem we studied earlier (J. Complexity17, 442–446). Let c be the cost of one function evaluation. The earlier results (cited above) might suggest that the ε-complexity of volume calculation should be proportional to c(1/ε)d/s when s⩾2. However, using integration by parts to reduce the dimension, we show that if s⩾2, then the complexity is proportional to c(1/ε)(d−1)/s. Next, we consider the case s=1, which is the minimal smoothness for which our volume problem is well-defined. We show that when s=1, an ε-approximation can be computed with cost proportional to at most c(1/ε)(d−1)d/2. Since a lower bound proportional to c(1/ε)d−1 holds when s=1, it follows that the complexity in the minimal smoothness case is proportional to c(1/ε) when d=2, and that there is a gap between the lower and upper bounds when d⩾3
On Asymptotically Optimal Methods of Prediction and Adaptive Coding for Markov Sources

AbstractThe problem of predicting a sequence x1, x2, … generated by a discrete source with unknown statistics is considered. Each letter xt+1 is predicted using information on the word x1x2…xt only. In fact, this problem is a classical problem which has received much attention. Its history can be traced back to Laplace. To estimate the efficiency of a method of prediction, three quantities are considered: the precision as given by the Kullback–Leibler divergence, the memory size of the program needed to implement the method on a computer, and the time required, measured by the number of binary operations needed at each time instant. A method is presented for which the memory size and the average time are close to the minimum. The results can readily be translated to results about adaptive coding
Direct Estimation of Linear Functionals from Indirect Noisy Observations

AbstractThe authors study the efficiency of the linear-functional strategy, as introduced by Anderssen in 1986, for inverse problems with observations blurred by Gaussian white noise with known intensity δ. The optimal accuracy is presented and it is shown how this can be achieved by a linear-functional strategy based on the noisy observations. This optimal linear-functional strategy is obtained from Tikhonov regularization of some dual problem. Next, the situation is treated when only a finite number of noisy observations, given beforehand, is available. Under appropriate smoothness assumptions best possible accuracy still can be attained if the number of observations corresponds to the noise intensity in a proper way. It is also shown that, at least asymptotically, this number of observations cannot be reduced
Complexity Theory for Lie-Group Solvers

AbstractCommencing with a brief survey of Lie-group theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Lie-group structure: Runge–Kutta–Munthe-Kaas schemes, Fer and Magnus expansions. This is followed by derivation of the computational cost of Fer and Magnus expansions, whose conclusion is that for order four, six, and eight an appropriately discretized Magnus method is always cheaper than a Fer method of the same order. Each Lie-group method of the kind surveyed in this paper requires the computation of a matrix exponential. Classical methods, e.g., Krylov-subspace and rational approximants, may fail to map elements in a Lie algebra to a Lie group. Therefore we survey a number of approximants based on the splitting approach and demonstrate that their cost is compatible (and often superior) to classical methods