Related papers: Generalized companion matrix for approximate GCD
In quantum computation with continous-variable systems, quantum advantage can only be achieved if some non-Gaussian resource is available. Yet, non-Gaussian unitary evolutions and measurements suited for computation are challenging to…
We study the convergence of a variant of distributed gradient descent (DGD) on a distributed low-rank matrix approximation problem wherein some optimization variables are used for consensus (as in classical DGD) and some optimization…
In this paper, we tackle the parametric complete multiplicity problem for a univariate polynomial. Our approach to the parametric complete multiplicity problem has a significant difference from the classical method, which relies on repeated…
We present GOFMM (geometry-oblivious FMM), a novel method that creates a hierarchical low-rank approximation, "compression," of an arbitrary dense symmetric positive definite (SPD) matrix. For many applications, GOFMM enables an approximate…
Complex polynomial optimization has recently gained more and more attention in both theory and practice. In this paper, we study the optimization of a real-valued general conjugate complex form over various popular constraint sets including…
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…
The $(u,v)$-Pad\'e approximation to a function $f$ is the (unique, up to scaling) rational approximation $f(x) = P(x)/Q(x) + O(x^{u+v+1})$, where $P$ has degree $u$ and $Q$ has degree $v$. Motivated by recent work of Molin, Pazuki, and…
We introduce and study the approximation properties of $g$-polynomials, defined as linear combinations of iterated Stieltjes integrals of a constant function. Focusing on the case where the derivator $g$ has finitely many discontinuities,…
Given a graphical model (GM), computing its partition function is the most essential inference task, but it is computationally intractable in general. To address the issue, iterative approximation algorithms exploring certain local…
Using commutative algebra methods we study the generalized minimum distance function (gmd function) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. The number of solutions that a…
The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…
Given a matrix $A$, a matrix nearness problem seeks an $X$ that most closely approximates $A$ in the sense of minimizing $\lVert A - X\rVert$ under a variety of constraints on $X$. A generalized matrix nearness problem seeks the same but…
We design polynomial size, constant depth (namely, $\mathsf{AC}^0$) arithmetic formulae for the greatest common divisor (GCD) of two polynomials, as well as the related problems of the discriminant, resultant, B\'ezout coefficients,…
A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination…
The Collective Graphical Model (CGM) models a population of independent and identically distributed individuals when only collective statistics (i.e., counts of individuals) are observed. Exact inference in CGMs is intractable, and previous…
We propose a novel algebraic framework for treating probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on…
We present an encoding of a polynomial system into vanishing and non-vanishing constraints on almost-principal minors of a symmetric, principally regular matrix, such that the solvability of the system over some field is equivalent to the…
We study the shape of the F-signature function of a $d$-dimensional quotient singularity $\mathbb{k}[[ x_1,\ldots,x_d]]^G$, and we show that it is a quasi-polynomial. We prove that the second coefficient is always zero and we describe the…
We seek to approximate a composite function h(x) = g(f(x)) with a global polynomial. The standard approach chooses points x in the domain of f and computes h(x) at each point, which requires an evaluation of f and an evaluation of g. We…
Some properties of generalized convexity for sets and for functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is…