相关论文: Multiplication of sparse Laurent polynomials and P…

Solving systems of polynomial equations is a central problem in nonlinear and computational algebra. Since Buchberger's algorithm for computing Gr\"obner bases in the 60s, there has been a lot of progress in this domain. Moreover, these…

The Strassen algorithm and Winograd's variant accelerate matrix multiplication by using fewer arithmetic operations than standard matrix multiplication. Although many papers have been published to accelerate single- as well as…

We study the problem of computing matrix chain multiplications in a distributed computing cluster. In such systems, performance is often limited by the straggler problem, where the slowest worker dominates the overall computation latency.…

Multivariate global polynomial approximations - such as polynomial chaos or stochastic collocation methods - are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses…

This paper proposes an improved quasi-Newton penalty decomposition algorithm for the minimization of continuously differentiable functions, possibly nonconvex, over sparse symmetric sets. The method solves a sequence of penalty subproblems…

High-dimensional sparse data present computational and statistical challenges for supervised learning. We propose compact linear sketches for reducing the dimensionality of the input, followed by a single layer neural network. We show that…

We present a recursive way to partition hypergraphs which creates and exploits hypergraph geometry and is suitable for many-core parallel architectures. Such partitionings are then used to bring sparse matrices in a recursive Bordered Block…

We propose a non-commutative algorithm for multiplying 2x2 matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast algorithms, and a time complexity…

Toric (or sparse) elimination theory is a framework developped during the last decades to exploit monomial structures in systems of Laurent polynomials. Roughly speaking, this amounts to computing in a \emph{semigroup algebra}, \emph{i.e.}…

In this paper we propose an algorithm for recovering sparse orthogonal polynomials using stochastic collocation. Our approach is motivated by the desire to use generalized polynomial chaos expansions (PCE) to quantify uncertainty in models…

This work presents a new algorithm for matrix power series which is near-sparse, that is, there are a large number of near-zero elements in it. The proposed algorithm uses a filtering technique to improve the sparsity of the matrices…

In this paper, we propose several dictionary learning algorithms for sparse representations that also impose specific structures on the learned dictionaries such that they are numerically efficient to use: reduced number of…

We propose a stochastic variance reduced optimization algorithm for solving sparse learning problems with cardinality constraints. Sufficient conditions are provided, under which the proposed algorithm enjoys strong linear convergence…

The classical work of (Arora et al., 1999) provides a scheme that gives, for any $\epsilon>0$, a polynomial time $1-\epsilon$ approximation algorithm for dense instances of a family of $\mathcal{NP}$-hard problems, such as Max-CUT and…

Generalized sparse matrix-matrix multiplication (or SpGEMM) is a key primitive for many high performance graph algorithms as well as for some linear solvers, such as algebraic multigrid. Here we show that SpGEMM also yields efficient…

We consider the problem of interpolating a sparse multivariate polynomial over a finite field, represented with a black box. Building on the algorithm of Ben-Or and Tiwari for interpolating polynomials over rings with characteristic zero,…

Sparse optimization is a central problem in machine learning and computer vision. However, this problem is inherently NP-hard and thus difficult to solve in general. Combinatorial search methods find the global optimal solution but are…

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.…

We describe an algorithm to factor sparse multivariate polynomials using O(d) bivariate factorizations where d is the number of variables. This algorithm is implemented in the Giac/Xcas computer algebra system.

In this paper, we propose two new interpolation algorithms for sparse multivariate polynomials represented by a straight-line program(SLP). Both of our algorithms work over any finite fields $F_q$ with large characteristic. The first one is…