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Kronecker coefficients encode the tensor products of complex irreducible representations of symmetric groups. Their stability properties have been considered recently by several authors (Vallejo, Pak and Panova, Stembridge). We describe a…
In the kernel density estimation (KDE) problem one is given a kernel $K(x, y)$ and a dataset $P$ of points in a Euclidean space, and must prepare a data structure that can quickly answer density queries: given a point $q$, output a…
This paper presents an algorithm, Voted Kernel Regularization , that provides the flexibility of using potentially very complex kernel functions such as predictors based on much higher-degree polynomial kernels, while benefitting from…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
We propose a strategy for the generation of fast and accurate versions of non-commutative recursive matrix multiplication algorithms. To generate these algorithms, we consider matrix and tensor norm bounds governing the stability and…
For various $2\leq n,m \leq 6$, we propose some new algorithms for multiplying an $n\times m$ matrix with an $m \times 6$ matrix over a possibly noncommutative coefficient ring.
Large-scale optimization problems that seek sparse solutions have become ubiquitous. They are routinely solved with various specialized first-order methods. Although such methods are often fast, they usually struggle with not-so-well…
Tensor decompositions are promising tools for big data analytics as they bring multiple modes and aspects of data to a unified framework, which allows us to discover complex internal structures and correlations of data. Unfortunately most…
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…
Consider a sparse multivariate polynomial f with integer coefficients. Assume that f is represented as a "modular black box polynomial", e.g. via an algorithm to evaluate f at arbitrary integer points, modulo arbitrary positive integers.…
Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…
In this paper a novel hybrid approach for compensating the distortion of any interpolation has been proposed. In this hybrid method, a modular approach was incorporated in an iterative fashion. By using this approach we can get drastic…
We study the problem of space and time efficient evaluation of a nonparametric estimator that approximates an unknown density. In the regime where consistent estimation is possible, we use a piecewise multivariate polynomial interpolation…
Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral…
New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high…
We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via…
We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete…
Additive Fourier Transform is sdudied. A fast multiplication algorithm for polynomials over the binary field is given. The bit complexity of the algorithm is $O(n(log n)(\log\log n)^2)$.
A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra…
Hyperbolic polynomials is a class of real-roots polynomials that has wide range of applications in theoretical computer science. Each hyperbolic polynomial also induces a hyperbolic cone that is of particular interest in optimization due to…