Related papers: Towards Mixed Gr{\"o}bner Basis Algorithms: the Mu…

This paper is concerned with linear algebra based methods for solving exactly polynomial systems through so-called Gr\"obner bases, which allow one to compute modulo the polynomial ideal generated by the input equations. This is a topical…

We present a novel, general, and unifying point of view on sparse approaches to polynomial optimization. Solving polynomial optimization problems to global optimality is a ubiquitous challenge in many areas of science and engineering.…

Many problems in scientific and engineering applications contain sparse matrices or graphs as main input objects, e.g. numerical simulations on meshes. Large inputs are abundant these days and require parallel processing for memory size and…

We consider a sparse matrix-matrix multiplication (SpGEMM) setting where one matrix is square and the other is tall and skinny. This special variant, called TS-SpGEMM, has important applications in multi-source breadth-first search,…

We review the literature on algorithms for estimating the index space in a multi-index model. The primary focus is on computationally efficient (polynomial-time) algorithms in Gaussian space, the assumptions under which consistency is…

Multiplication of polynomials is among key operations in computer algebra which plays important roles in developing techniques for other commonly used polynomial operations such as division, evaluation/interpolation, and factorization. In…

We algorithmically construct multi-output Gaussian process priors which satisfy linear differential equations. Our approach attempts to parametrize all solutions of the equations using Gr\"obner bases. If successful, a push forward Gaussian…

The computation of Gr\"obner bases is an established hard problem. By contrast with many other problems, however, there has been little investigation of whether this hardness is robust. In this paper, we frame and present results on the…

We consider the problem of mutually unbiased bases as a polynomial optimization problem over the reals. We heavily reduce it using known symmetries before exploring it using two methods, combining a number of optimization techniques. The…

This paper focuses on the computation of joint spectral radii (JSR), when the involved matrices are sparse. We provide a sparse variant of the procedure proposed by Parrilo and Jadbabaie, to compute upper bounds of the JSR by means of…

Sparse Bayesian learning (SBL) is a powerful framework for tackling the sparse coding problem. However, the most popular inference algorithms for SBL become too expensive for high-dimensional settings, due to the need to store and compute a…

Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible…

Finding the sparse representation of a signal in an overcomplete dictionary has attracted a lot of attention over the past years. This paper studies ProSparse, a new polynomial complexity algorithm that solves the sparse representation…

Algebraic cryptanalysis usually requires to recover the secret key by solving polynomial equations. Grobner bases algorithm is a well-known method to solve this problem. However, a serious drawback exists in the Grobner bases based…

We present an efficient algorithm for computing the leading monomials of a minimal Groebner basis of a generic sequence of homogeneous polynomials. Our approach bypasses costly polynomial reductions by exploiting structural properties…

We study a broad class of polynomial optimization problems whose constraints and objective functions exhibit sparsity patterns. We give two characterizations of the number of critical points to these problems, one as a mixed volume and one…

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…

Support for arithmetic in multiple precisions and number formats is becoming increasingly common in emerging high-performance architectures. From a computational scientist's perspective, our goal is to determine how and where we can safely…

In this work we address the problem of recovering sparse solutions to non linear inverse problems. We look at two variants of the basic problem, the synthesis prior problem when the solution is sparse and the analysis prior problem where…

Sparse coding is a crucial subroutine in algorithms for various signal processing, deep learning, and other machine learning applications. The central goal is to learn an overcomplete dictionary that can sparsely represent a given input…