Related papers: A complete algorithm to find exact minimal polynom…
Over the past a few years, research and development has made significant progresses on big data analytics. A fundamental issue for big data analytics is the efficiency. If the optimal solution is unable to attain or not required or has a…
In this paper, an exact algorithm in polynomial time is developed to solve unrestricted binary quadratic programs. The computational complexity is $O\left( n^{\frac{15}{2}}\right) $, although very conservative, it is sufficient to prove…
Binary quadratic programming problems have attracted much attention in the last few decades due to their potential applications. This type of problems are NP-hard in general, and still considered a challenge in the design of efficient…
We consider problems with multiple linear objectives and linear constraints and use Adjustable Robust Optimization and Polynomial Optimization as tools to approximate the Pareto set with polynomials of arbitrarily large degree. The main…
Recent years have witnessed the introduction and development of extremely fast rational function algorithms. Many ideas in this realm arose from polynomial-based linear-algebraic algorithms. However, polynomial approximation is occasionally…
Given a multivariate real (or complex) polynomial $p$ and a domain $\cal D$, we would like to decide whether an algorithm exists to evaluate $p(x)$ accurately for all $x \in {\cal D}$ using rounded real (or complex) arithmetic. Here…
We study the minimization of fixed-degree polynomials over the simplex. This problem is well-known to be NP-hard, as it contains the maximum stable set problem in graph theory as a special case. In this paper, we consider a rational…
While leverage score sampling provides powerful tools for approximating solutions to large least squares problems, the cost of computing exact scores and sampling often prohibits practical application. This paper addresses this challenge by…
We give an algorithm for computing approximate PSD factorizations of nonnegative matrices. The running time of the algorithm is polynomial in the dimensions of the input matrix, but exponential in the PSD rank and the approximation error.…
We establish efficient approximate counting algorithms for several natural problems in local lemma regimes. In particular, we consider the probability of intersection of events and the dimension of intersection of subspaces. Our approach is…
The goal of this paper is to understand how exponential-time approximation algorithms can be obtained from existing polynomial-time approximation algorithms, existing parameterized exact algorithms, and existing parameterized approximation…
In this paper we address the problem of finding well approximating lattices for a given finite set $A$ of points in ${\mathbb R}^n$. More precisely, we search for $\v{o},\v{d_1}, \dots,\v{d_n}\in \mathbb{R}^n$ such that $\v{a}-\v{o}$ is…
We show the existence of a deep neural network capable of approximating a wide class of high-dimensional approximations. The construction of the proposed neural network is based on a quasi-optimal polynomial approximation. We show that this…
We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank…
We provide experimental evaluation of a number of known and new algorithms for approximate computation of Monroe's and Chamberlin-Courant's rules. Our experiments, conducted both on real-life preference-aggregation data and on synthetic…
Quaternion optimization has attracted significant interest due to its broad applications, including color face recognition, video compression, and signal processing. Despite the growing literature on quadratic and matrix quaternion…
We study the theoretical and practical aspects of computing braids described by approximate descriptions of paths in the plane. Exact algorithms rely on the lexicographic ordering of the points in the plane, which is unstable under…
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…
We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real or the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of…
Matrices with low numerical rank are omnipresent in many signal processing and data analysis applications. The pivoted QLP (p-QLP) algorithm constructs a highly accurate approximation to an input low-rank matrix. However, it is…