Related papers: Multivariate sparse interpolation using randomized…
In this paper, we study how to quickly compute the <-minimal monomial interpolating basis for a multivariate polynomial interpolation problem. We address the notion of "reverse" reduced basis of linearly independent polynomials and design…
A nonstandard application of bivariate polynomial interpolation is discussed: the implicitization of a rational algebraic curve given by its parametric equations. Three different approaches using the same interpolation space are considered,…
The usual univariate interpolation problem of finding a monic polynomial f of degree n that interpolates n given values is well understood. This paper studies a variant where f is required to be composite, say, a composition of two…
Simply put, a sparse polynomial is one whose zero coefficients are not explicitly stored. Such objects are ubiquitous in exact computing, and so naturally we would like to have efficient algorithms to handle them. However, with this compact…
This paper considers the extension of classical Lagrange interpolation in one real or complex variable to "polynomials of one quaternionic variable". To do this we develop some aspects of the theory of such polynomials. We then give a…
We propose a fast greedy algorithm to compute sparse representations of signals from continuous dictionaries that are factorizable, i.e., with atoms that can be separated as a product of sub-atoms. Existing algorithms strongly reduce the…
In this paper, we build up a framework for sparse interpolation. We first investigate the theoretical limit of the number of unisolvent points for sparse interpolation under a general setting and try to answer some basic questions of this…
In this paper, we present a probabilistic algorithm to multiply two sparse polynomials almost as efficiently as two dense univariate polynomials with a result of approximately the same size. The algorithm depends on unproven heuristics that…
In this paper, we give new sparse interpolation algorithms for black box univariate and multivariate rational functions h=f/g whose coefficients are integers with an upper bound. The main idea is as follows: choose a proper integer beta and…
We give a new probabilistic algorithm for interpolating a "sparse" polynomial f given by a straight-line program. Our algorithm constructs an approximation f* of f, such that their difference probably has at most half the number of terms of…
In this paper, we propose new deterministic and Monte Carlo interpolation algorithms for sparse multivariate polynomials represented by straight-line programs. Let $f$ be an $n$-variate polynomial given by a straight-line program, which has…
We present a new rational approximation algorithm based on the empirical interpolation method for interpolating a family of parametrized functions to rational polynomials with invariant poles, leading to efficient numerical algorithms for…
Schubert polynomials were discovered by A. Lascoux and M. Sch\"utzenberger in the study of cohomology rings of flag manifolds in 1980's. These polynomials generalize Schur polynomials, and form a linear basis of multivariate polynomials. In…
Kernel based regularized interpolation is a well known technique to approximate a continuous multivariate function using a set of scattered data points and the corresponding function evaluations, or data values. This method has some…
We present sparse interpolation algorithms for recovering a polynomial with $\le B$ terms from $N$ evaluations at distinct values for the variable when $\le E$ of the evaluations can be erroneous. Our algorithms perform exact arithmetic in…
In this paper, we demonstrate that many of the computational tools for univariate orthogonal polynomials have analogues for a family of bivariate orthogonal polynomials on the triangle, including Clenshaw's algorithm and sparse…
This paper introduces a new multivariate convolutional sparse coding based on tensor algebra with a general model enforcing both element-wise sparsity and low-rankness of the activations tensors. By using the CP decomposition, this model…
In some fields such as Mathematics Mechanization, automated reasoning and Trustworthy Computing etc., exact results are needed. Symbolic computations are used to obtain the exact results. Symbolic computations are of high complexity. In…
We consider space-saving versions of several important operations on univariate polynomials, namely power series inversion and division, division with remainder, multi-point evaluation, and interpolation. Now-classical results show that…
We present a highly scalable algorithm for multiplying sparse multivariate polynomials represented in a distributed format. This algo- rithm targets not only the shared memory multicore computers, but also computers clusters or specialized…