The lower estimate for the linear combinations of Bernstein–Kantorovich operators

AbstractIn this paper we obtain a new strong type of Steckin inequality for the linear combinations of Bernstein–Kantorovich operators, which gives the optimal approximation rate. On the basis of this inequality, we further obtain the lower estimate for these operators

Learning Arbitrary Statistical Mixtures of Discrete Distributions

We study the problem of learning from unlabeled samples very general statistical mixture models on large finite sets. Specifically, the model to be learned, $\vartheta$, is a probability distribution over probability distributions $p$, where each such $p$ is a probability distribution over $[n] = \{1,2,\dots,n\}$. When we sample from $\vartheta$, we do not observe $p$ directly, but only indirectly and in very noisy fashion, by sampling from $[n]$ repeatedly, independently $K$ times from the distribution $p$. The problem is to infer $\vartheta$ to high accuracy in transportation (earthmover) distance. We give the first efficient algorithms for learning this mixture model without making any restricting assumptions on the structure of the distribution $\vartheta$. We bound the quality of the solution as a function of the size of the samples $K$ and the number of samples used. Our model and results have applications to a variety of unsupervised learning scenarios, including learning topic models and collaborative filtering.Comment: 23 pages. Preliminary version in the Proceeding of the 47th ACM Symposium on the Theory of Computing (STOC15

Analysis of Approximation by Linear Operators on Variable L

This paper is concerned with approximation on variable Lρp(·) spaces associated with a general exponent function p and a general bounded Borel measure ρ on an open subset Ω of Rd. We mainly consider approximation by Bernstein type linear operators. Under an assumption of log-Hölder continuity of the exponent function p, we verify a conjecture raised previously about the uniform boundedness of Bernstein-Durrmeyer and Bernstein-Kantorovich operators on the Lρp(·) space. Quantitative estimates for the approximation are provided for high orders of approximation by linear combinations of such positive linear operators. Motivating connections to classification and quantile regression problems in learning theory are also described

Some numerical applications of generalized Bernstein operators

Inthispaper,somerecentapplicationsoftheso-calledGeneralizedBernsteinpolynomialsarecollected. This polynomial sequence is constructed by means of the samples of a continuous function f on equispaced points of [0, 1] and depends on an additional parameter which can be suitable chosen in order to improve the rate of convergence to the function f , as the smoothness of f increases, overcoming the well-known low degree of approximation achieved by the classical Bernstein polynomials or by the piecewise polynomial approximation. The applications considered here deal with the numerical integration and the simultaneous approximation. Quadrature rules on equidistant nodes of [0, 1] are studied for the numerical computation of ordinary integrals in one or two dimensions, and usefully em- ployed in Nyström methods for solving Fredholm integral equations. Moreover, the simultaneous approximation of the Hilbert transform and its derivative (the Hadamard transform) is illustrated. For all the applications, some numerical details are given in addition to the error estimates, and the proposed approximation methods have been implemented providing numerical tests which confirm the theoretical estimates. Some open problems are also introduced

Constant & time-varying hedge ratio for FBMKLCI stock index futures

This paper examines hedging strategy in stock index futures for Kuala Lumpur Composite Index futures of Malaysia. We employed two different econometric methods such as-vector error correction model (VECM) and bivariate generalized autoregressive conditional heteroskedasticity (BGARCH) models to estimate optimal hedge ratio by using daily data of KLCI index and KLCI futures for the period from January 2012 to June 2016 amounting to a total of 1107 observations. We found that VECM model provides better results with respect to estimating hedge ratio for spot month futures and one-month futures, while BGACH shows better for distance futures. While VECM estimates time invariant hedge ratio, the BGARCH shows that hedge ratio changes over time. As such, hedger should rebalance his/her position in futures contract time to time in order to reduce risk exposure

Finding apparent horizons in numerical relativity

This paper presents a detailed discussion of the ``Newton's method'' algorithm for finding apparent horizons in 3+1 numerical relativity. We describe a method for computing the Jacobian matrix of the finite differenced H(h) function \H(\h) by symbolically differentiating the finite difference equations, giving the Jacobian elements directly in terms of the finite difference molecule coefficients used in computing \H(\h). Assuming the finite differencing scheme commutes with linearization, we show how the Jacobian elements may be computed by first linearizing the continuum H(h) equations, then finite differencing the linearized (continuum) equations. We find this symbolic differentiation method of computing the \H(\h) Jacobian to be {\em much\/} more efficient than the usual numerical-perturbation method, and also much easier to implement than is commonly thought. When solving the (discrete) \H(\h) = 0 equations, we find that Newton's method generally converges very rapidly, although there are difficulties when the initial guess contains high-spatial-frequency errors. Using 4th~order finite differencing, we find typical accuracies for the horizon position in the 10^{-5} range for \Delta \theta = \frac{\pi/2}{50}