Related papers: Minimum volume semialgebraic sets for robust estim…
This work provides closed-form solutions and minimum achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of bounded linear operators the previous results…
The $k$-local Hamiltonian problem is a central model for quantum many-body systems and Hamiltonian complexity. Semidefinite programming and noncommutative sum-of-squares hierarchies provide systematic certificates for ground-state energies,…
Let $K$ be a convex body in ${\mathbb R}^n$, and let $\Pi_1({\mathbb R}^n)$ be the space of polynomials in $n$ variables of degree at most $1$. Given an $(n+1)$-element set $Y\subset K$ in general position, we let $P_Y$ denote the Lagrange…
We study the problem of approximating the mixed volume $V(P_1^{(\alpha_1)}, \dots, P_k^{(\alpha_k)})$ of an $k$-tuple of convex polytopes $(P_1, \dots, P_k)$, each of which is defined as the convex hull of at most $m_0$ points in…
This work studies the problem of maximizing a higher degree real homogeneous multivariate polynomial over the unit sphere. This problem is equivalent to finding the leading eigenvalue of the associated symmetric tensor of higher order,…
Hydro unit commitment is the problem of maximizing water use efficiency while minimizing start-up costs in the daily operation of multiple hydro plants, subject to constraints on short-term reservoir operation, and long-term goals. A…
Consider the closed convex hull $K$ of a monomial curve given parametrically as $(t^{m_1},\ldots,t^{m_n})$, with the parameter $t$ varying in an interval $I$. We show, using constructive arguments, that $K$ admits a lifted semidefinite…
Robust mean estimation is one of the most important problems in statistics: given a set of samples in $\mathbb{R}^d$ where an $\alpha$ fraction are drawn from some distribution $D$ and the rest are adversarially corrupted, we aim to…
We study some numerical methods for solving second order elliptic problem with interface. We introduce an immersed interface finite element method based on the `broken' $P_1$-nonconforming piecewise linear polynomials on interface…
We introduce the problem of computing a parsing where each phrase is of length at most $m$ and which minimizes the zeroth order entropy of parsing. Based on the recent theoretical results we devise a heuristic for this problem. The solution…
Determining the number of embeddings of Laman graph frameworks is an open problem which corresponds to understanding the solutions of the resulting systems of equations. In this paper we investigate the bounds which can be obtained from the…
We consider the problem of globally minimizing the sum of many rational functions over a given compact semialgebraic set. The number of terms can be large (10 to 100), the degree of each term should be small (up to 10), and the number of…
The volume function V(t) of a compact set S\in R^d is just the Lebesgue measure of the set of points within a distance to S not larger than t. According to some classical results in geometric measure theory, the volume function turns out to…
We consider optimization problems with polynomial inequality constraints in non-commuting variables. These non-commuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial…
We derive concentration inequalities for the supremum norm of the difference between a kernel density estimator (KDE) and its point-wise expectation that hold uniformly over the selection of the bandwidth and under weaker conditions on the…
From a numerical analysis perspective, assessing the robustness of l1-minimization is a fundamental issue in compressed sensing and sparse regularization. Yet, the recovery guarantees available in the literature usually depend on a priori…
Let $K \subset \mathbb R^n$ be a convex body with barycenter at the origin. We show there is a simplex $S \subset K$ having also barycenter at the origin such that $\left(\frac{vol(S)}{vol(K)}\right)^{1/n} \geq \frac{c}{\sqrt{n}},$ where…
We construct a quasi-polynomial time deterministic approximation algorithm for computing the volume of an independent set polytope with restrictions. Randomized polynomial time approximation algorithms for computing the volume of a convex…
We present a unified theoretical framework for parametric low-rank approximation, a research area devoted to the development of efficient algorithms that act as adaptive alternatives of traditional methods such as Singular Value…
We introduce a particular optimization problem that minimizes the sum of a non-convex quadratic function and logarithmic barrier-functions in a $\ell_\infty$-trust-region (i.e. cube). Our paper covers three topics. We explain the relevance…