Related papers: Mixed Determinants and the Kadison-Singer problem
Polynomial algebra offers a standard approach to handle several problems in geometric modeling. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the…
We propose two novel conditional gradient-based methods for solving structured stochastic convex optimization problems with a large number of linear constraints. Instances of this template naturally arise from SDP-relaxations of…
This paper describes a numerical method for finding good packings in Grassmannian manifolds equipped with various metrics. This investigation also encompasses packing in projective spaces. In each case, producing a good packing is…
A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Some classical polynomial-time optimization problems, such as spanning tree and forest, shortest path,…
We develop an analytical framework for Boolean Promise Constraint Satisfaction Problems (PCSPs) that studies polymorphisms through the notion of influence from Fourier analysis of Boolean functions. Extending the work of Brakensiek,…
We consider a learning-augmented framework for NP-hard permutation problems. The algorithm has access to predictions telling, given a pair $u,v$ of elements, whether $u$ is before $v$ or not in an optimal solution. Building on the work of…
The property of balance (in the sense of Feder and Mihail) is investigated in the context of paving matroids. The following examples are exhibited: (a) a class of ``sparse'' paving matroids that are balanced, but at the same time rich…
The discriminant of a multivariate polynomial with indeterminate coefficients is not necessarily a hypersurface, and characterizing its codimension was an open problem for quite a while. We resolve this problem for the discriminants of…
The Kaczmarz algorithm is popular for iteratively solving an overdetermined system of linear equations. The traditional Kaczmarz algorithm can approximate the solution in few sweeps through the equations but a randomized version of the…
One fundamental goal of high-dimensional statistics is to detect or recover planted structure (such as a low-rank matrix) hidden in noisy data. A growing body of work studies low-degree polynomials as a restricted model of computation for…
Propensity score matching (PSM) is the de-facto standard for estimating causal effects in observational studies. We show that PSM and its implementations are susceptible to several major drawbacks and illustrate these findings using a case…
Hidden-variable resultant methods are a class of algorithms for solving multidimensional polynomial rootfinding problems. In two dimensions, when significant care is taken, they are competitive practical rootfinders. However, in higher…
We develop certain combinatorial tools for the study of discriminants of general systems of polynomial equations. Applying these tools in a sequel paper, we completely classify components of such discriminants, generalizing the classical…
Most data sets comprise of measurements on continuous and categorical variables. In regression and classification Statistics literature, modeling high-dimensional mixed predictors has received limited attention. In this paper we study the…
This paper describes the expected characteristic polynomial of the commutator of randomly rotated matrices, in the context of the finite free probability theory initiated by Marcus, Spielman, and Srivastava. The key technical features are…
Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement…
We study metric projections onto cones in the Wasserstein space of probability measures, defined by stochastic orders. Dualities for backward and forward projections are established under general conditions. Dual optimal solutions and their…
Sequences of parametrized Lyapunov equations can be encountered in many application settings. Moreover, solutions of such equations are often intermediate steps of an overall procedure whose main goal is the computation of…
We consider the problem of finding for a given $N$-tuple of polynomials (real or complex) the closest $N$-tuple that has a common divisor of degree at least $d$. Extended weighted Euclidean seminorm of the coefficients is used as a measure…
The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [B. Despr\'es, Numer.…