Approximate computation of zero-dimensional polynomial ideals

Journal of Algebra and Its Applications, 2019

In this paper, we consider the problem of computing all possible order ideals and also sets connected to 1, and the corresponding border bases, for the vanishing ideal of a given finite set of points. In this context, two different approaches are discussed: based on the Buchberger–Möller Algorithm [H. M. Möller and B. Buchberger, The construction of multivariate polynomials with preassigned zeros, EUROCAM ’82 Conf., Computer Algebra, Marseille/France 1982, Lect. Notes Comput. Sci. 144, (1982), pp. 24–31], we first propose a new algorithm to compute all possible order ideals and the corresponding border bases for an ideal of points. The second approach involves adapting the Farr–Gao Algorithm [J. B. Farr and S. Gao, Computing Gröbner bases for vanishing ideals of finite sets of points, in 16th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC-16, Las Vegas, NV, USA (Springer, Berlin, 2006), pp. 118–127] for finding all sets connected to 1, as well as ...

Journal of Symbolic Computation, 2011

We construct an explicit minimal strong Gröbner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m ≥ 2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Gröbner basis is independent of the monomial order and that the set of leading terms of the constructed Gröbner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Gröbner basis in Z/m[x1, x2, . . . , xn] along the prime factorization of m. The obtained results are not only of mathematical interest but have immediate applications in formal verification of data paths for microelectronic systems-on-chip.

arXiv (Cornell University), 2022

The vanishing ideal of a set of points = {x 1 ,. .. , x } ⊆ ℝ is the set of polynomials that evaluate to 0 over all points x ∈ and admits an efficient representation by a finite subset of generators. In practice, to accommodate noise in the data, algorithms that construct generators of the approximate vanishing ideal are widely studied but their computational complexities remain expensive. In this paper, we scale up the oracle approximate vanishing ideal algorithm (OAVI), the only generator-constructing algorithm with known learning guarantees. We prove that the computational complexity of OAVI is not superlinear, as previously claimed, but linear in the number of samples. In addition, we propose two modifications that accelerate OAVI's training time: Our analysis reveals that replacing the pairwise conditional gradients algorithm, one of the solvers used in OAVI, with the faster blended pairwise conditional gradients algorithm leads to an exponential speed-up in the number of features. Finally, using a new inverse Hessian boosting approach, intermediate convex optimization problems can be solved almost instantly, improving OAVI's training time by multiple orders of magnitude in a variety of numerical experiments. 1. A set ⊆ ℝ is algebraic if it is the set of common roots of a finite set of polynomials.

Journal of Symbolic Computation, 2010

From the numerical point of view, given a set X ⊂ R n of s points whose coordinates are known with only limited precision, each set e X of s points whose elements differ from those of X of a quantity less than the data uncertainty can be considered equivalent to X. We present an algorithm that, given X and a tolerance ε on the data error, computes a set G of polynomials such that each element of G "almost vanishing" at X and at all its equivalent sets e X. Even if G is not, in the general case, a basis of the vanishing ideal I(X), we show that, differently from the basis of I(X) that can be greatly influenced by the data uncertainty, G can determine a geometrical configuration simultaneously characterizing the set X and all its equivalent sets e X.

ACM Communications in Computer Algebra, 2006

A contemporary and exciting application of Gröbner bases is their use in computational biology, particularly in the reverse engineering of gene regulatory networks from experimental data. In this setting, the data are typically limited to tens of points, while the number of genes or variables is potentially in the thousands. As such data sets vastly underdetermine the biological network, many models may fit the same data and reverse engineering programs often require the use of methods for choosing parsimonious models. Gröbner bases have recently been employed as a selection tool for polynomial dynamical systems that are characterized by maps in a vector space over a finite field. While there are numerous existing algorithms to compute Gröbner bases, to date none has been specifically designed to cope with large numbers of variables and few distinct data points. In this paper, we present an algorithm for computing Gröbner bases of zero-dimensional ideals that is optimized for the ca...

2007

We present an algorithm for computing Gröbner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies a set of essential variables, which reduces the time complexity with respect to the number of indeterminates, and then uses PLU decompositions to reduce the time complexity with respect to the number of points. This gives a theoretical upper bound for its time complexity that is an order of magnitude lower than the known one for the standard Buchberger-Möller algorithm if the number of indeterminates is much larger than the number of points. Comparison of implementations of our algorithm and the standard Buchberger-Möller algorithm in Macaulay 2 confirm the theoretically predicted speedup. This work is motivated by recent applications of Gröbner bases to the problem of network reconstruction in molecular biology.

2007

We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner basis is independent of the monomial order and that the set of leading terms of the constructed Groebner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Groebner basis in Z/m[x_1,x_2,...,x_n] along the prime factorization of m. The obtained results are not only of mathematical interest but have immediate applications in formal verification of data paths for microelectronic systems-on-chip.

Proceedings of the 1993 international symposium on Symbolic and algebraic computation - ISSAC '93, 1993

ArXiv, 2016

Two models were recently proposed to explore the robust hardness of Gr\"obner basis computation. Given a polynomial system, both models allow an algorithm to selectively ignore some of the polynomials: the algorithm is only responsible for returning a Gr\"obner basis for the ideal generated by the remaining polynomials. For the $q$-Fractional Gr\"obner Basis Problem the algorithm is allowed to ignore a constant $(1-q)$-fraction of the polynomials (subject to one natural structural constraint). Here we prove a new strongest-parameter result: even if the algorithm is allowed to choose a $(3/10-\epsilon)$-fraction of the polynomials to ignore, and need only compute a Gr\"obner basis with respect to some lexicographic order for the remaining polynomials, this cannot be accomplished in polynomial time (unless $P=NP$). This statement holds even if every polynomial has maximum degree 3. Next, we prove the first robust hardness result for polynomial systems of maximum de...

2008

Given a numerical approximation to a point p on the set V of common zeroes of a set of multivariate polynomials with complex coefficients, this article presents an efficient method to compute the maximum dimension of the irreducible components of V which pass through p, i.e., a local dimension test. Such a test, used to filter out the so-called "junk points," is a crucial element in the numerical irreducible decomposition algorithms of Sommese, Verschelde, and Wampler. Computational evidence presented in this article illustrates that, with this new local-dimension test, "junk-point filtering" is no longer a bottleneck in the computation of a numerical irreducible decomposition. For moderate size examples this results in well over an order of magnitude improvement in the computation of a numerical irreducible decomposition. Also, to compute the irreducible components of a fixed dimension, it is no longer necessary to compute the numerical irreducible decomposition of all higher dimensions.