A unified approach to evaluation algorithms for multivariate polynomials

It is known that certain Catmull-Rom splines [7] interpolate their control vertices and share many properties such as affine invariance, global smoothness, and local control with B-spline curves; they are therefore of possible interest to computer aided design. It is shown here that another property a class of Catmull-Rom splines shares with B-spline curves is that both schemes possess a simple recursive evaluation algorithm. The Catmull-Rom evaluation algorithm is constructed by combining the de Boor algorithm for evaluating B-spline curves with Neville's algorithm for evaluating Lagrange polynomials. The recursive evaluation algorithm for Catmull-Rom curves allows rapid evaluation of these curves by pipelining with specially designed hardware. Furthermore it facilitates the development of new, related curve schemes which may have useful shape parameters for altering the shape of the curve without moving the control vertices. It may also be used for constructing transformations to Bézier and B-spline form.

In this paper we develop a new approach for learning decision trees and multivariate polynomials via interpolation of multivariate polynomials. This new approach yields simple learning algorithms for multivariate polynomials and decision trees over finite fields under any constant bounded product distribution. The output hypothesis is a (single) multivariate polynomial that is an /spl epsiv/-approximation of the target under any constant bounded product distribution. The new approach demonstrates the learnability of many classes under any constant bounded product distribution and using membership queries, such as j-disjoint DNF and multivariate polynomial with bounded degree over any field. The technique shows how to interpolate multivariate polynomials with bounded term size from membership queries only. This in particular gives a learning algorithm for O(log n)-depth decision tree from membership queries only and a new learning algorithm of any multivariate polynomial over sufficiently large fields from membership queries only. We show that our results for learning from membership queries only are the best possible.

We consider the proximity testing problem for error-correcting codes which consist in evaluations of multivariate polynomials either of bounded individual degree or bounded total degree. Namely, given an oracle function $$f :L^m \rightarrow {\mathbb {F}}_q$$ , where $$L\subset {\mathbb {F}}_q$$ , a verifier distinguishes whether f is the evaluation of a low-degree polynomial or is far (in relative Hamming distance) from being one, by making only a few queries to f. This topic has been studied in the context of locally testable codes, interactive proofs, probalistically checkable proofs, and interactive oracle proofs. We present the first interactive oracle proofs of proximity (IOPP) for tensor products of Reed–Solomon codes (evaluation of polynomials with bounds on individual degrees) and for Reed–Muller codes (evaluation of polynomials with a bound on the total degree) that simultaneously achieve logarithmic query complexity, logarithmic verification time, linear oracle proof length and linear prover running time. Such low-degree polynomials play a central role in constructions of probabilistic proof systems and succinct non-interactive arguments of knowledge with zero-knowledge. For these applications, highly-efficient multivariate low-degree tests are desired, but prior probabilistic proofs of proximity required super-linear proving time. In contrast, for multivariate codes of length N, our constructions admit a prover running in time linear in N and a verifier which is logarithmic in N. Our constructions are directly inspired by the IOPP for Reed–Solomon codes of [Ben-Sasson et al., ICALP 2018] named “FRI protocol”. Compared to the FRI protocol, our IOPP for tensor products of Reed–Solomon codes achieves the same efficiency parameters. As for Reed–Muller codes, for fixed constant number of variables m, the concrete efficiency of our IOPP for Reed–Muller codes compares well, all things equal.

h-Blossoming: A new approach to algorithms and identities for h-Bernstein bases and h-Bézier curves

Chapter 12 - Multiplying with Synapses and Neurons

A straightforward discretisation of high-dimensional problems often leads to a curse of dimensions and thus the use of sparsity has become a popular tool. Efficient algorithms like the fast Fourier transform (FFT) have to be customised to these thinner discretisations and we focus on two major topics regarding the Fourier analysis of high-dimensional functions: We present stable and effective algorithms for the fast evaluation and reconstruction of multivariate trigonometric polynomials with frequencies supported on an index set \(\mathcal{I}\subset \mathbb{Z}^{d}\).KeywordsFast Fourier TransformFourier CoefficientTrigonometric PolynomialSparse GridMatrix Vector MultiplicationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Properties of generalized Ball curves and surfaces

Division algorithms for Bernstein polynomials

Robust arithmetic for multivariate Bernstein-form polynomials

Systolic algorithm for polynomial interpolation and related problems

In the recent papers, a new efficient probabilistic semi-numerical absolute (i.e., complex) factorization algorithm for multivariate polynomials with integer coefficients is given. It is based on a simple property of the monomials arising after a generic linear change of coordinates for bivariate polynomials and on a deep result of complex algebraic geometry. Here, we consider the a priori simpler problem of factorization over the field of reals. We briefly review our algorithm for complex factorization and adapt it to solving the problem on the field of reals. This allows us to spare a significant part of the computations and to improve the range of tractability. The method provides factors with approximative coefficients and eventually exact factors in a suitable real algebraic extension of \(\mathbb{Q}\). Bibliography: 15 titles.

Error-correcting decoding is generalized to multivariate sparse rational function recovery from evaluations that can be numerically inaccurate and where several evaluations can have severe errors (outliers). The generalization of the Berlekamp-Welch decoder to exact Cauchy interpolation of univariate rational functions from values with faults is by Kaltofen and Pernet in 2012. We give a different univariate solution based on structured linear algebra that yields a stable decoder with floating point arithmetic. Our multivariate polynomial and rational function interpolation algorithm combines Zippel's symbolic sparse polynomial interpolation technique [Ph.D. Thesis MIT 1979] with the numeric algorithm by Kaltofen, Yang, and Zhi [Proc. SNC 2007], and removes outliers (cleans up data) through techniques from error correcting codes. Our multivariate algorithm can build a sparse model from a number of evaluations that is linear in the sparsity of the model.

Numerical validation of compensated algorithms with stochastic arithmetic

Lin, Chung and Han introduced in 2014 the LCH-basis in order to derive FFT-based multipoint evaluation and interpolation algorithms with respect to this polynomial basis. Considering an affine space of n = 2j points, their algorithms require O(n · log2 n) operations in F2r. The LCH-basis has then been extended over finite fields of characteristic p by Lin et al. in 2016 and an n-point evaluation algorithm has been derived for n = pj with complexity O(n · logp n · p). However, the problem of interpolating polynomials represented in such a basis over FPr has not been addressed.

The Sharp GP2Y0A02YK0F is an infrared-based nonlinear distance measuring sensor unit. This sensor categorized as low-cost sensor. Since this sensor has nonlinear characteristic in output voltage made this sensor is not easy to compute the measured distance. The modelling process is one of the solutions to complete this challenge. The Newton polynomial is a robust polynomial method that used in computational purpose. However, the polynomial-based modelling methods are suffered with Runge's phenomenon especially for nonlinear model. The partial modelling method with piecewise Newton polynomials algorithm has been used to minimize the occurrence of Runge's phenomenon. The piecewise Newton polynomials method has been succeeded to generate a nonlinear model and minimize the occurrence of Runge's phenomenon. The low MSE (Mean Squared Error) level by 0.001 and error percentage by 2.38% has been achieved for the generated model. The accuracy level of the final model is 97.62%.