Related papers: Finding Minimum Volume Circumscribing Ellipsoids U…
This paper introduces Least Volume (LV)--a simple yet effective regularization method inspired by geometric intuition--that reduces the number of latent dimensions required by an autoencoder without prior knowledge of the dataset's…
In this note, we first try to prove a uniform lower bound of nodal volume in elliptic homogenization setting. This lower bound is far from optimal. But, we can prove a constant lower bound in dimension two. Motivated by the proof, we extend…
We find an optimal upper bound on the volume of the John ellipsoid of a $k$-dimensional section of the $n$-dimensional cube, and an optimal lower bound on the volume of the L\"owner ellipsoid of a projection of the $n$-dimensional…
Computing the copositive minimum of a strictly copositive quadratic form is a natural generalization of computing the arithmetical minimum of a positive definite one. In this paper we show that this generalized problem is NP-complete.…
Star-shaped bodies are an important nonconvex generalization of convex bodies (e.g., linear programming with violations). Here we present an efficient algorithm for sampling a given star-shaped body. The complexity of the algorithm grows…
We introduce a method to improve the tractability of the well-known Sample Average Approximation (SAA) without compromising important theoretical properties, such as convergence in probability and the consistency of an independent and…
This paper addresses a multi-scale finite element method for second order linear elliptic equations with arbitrarily rough coefficient. We propose a local oversampling method to construct basis functions that have optimal local…
We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…
In this paper, we propose an inexact proximal Newton-type method for nonconvex composite problems. We establish the global convergence rate of the order $\mathcal{O}(k^{-1/2})$ in terms of the minimal norm of the KKT residual mapping and…
The Minimum Volume Covering Ellipsoid (MVCE) problem, characterised by $n$ observations in $d$ dimensions where $n \gg d$, can be computationally very expensive in the big data regime. We apply methods from randomised numerical linear…
Reachable sets for a dynamical system describe collections of system states that can be reached in finite time, subject to system dynamics. They can be used to guarantee goal satisfaction in controller design or to verify that unsafe…
Given a hypothesis space, the large volume principle by Vladimir Vapnik prioritizes equivalence classes according to their volume in the hypothesis space. The volume approximation has hitherto been successfully applied to binary learning…
In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper and the lower level problems are polynomials. We present methods for finding its global minimizers and…
This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the…
This paper tackles the challenging problem of finding global optimal solutions for two-stage stochastic programs with continuous decision variables and nonconvex recourse functions. We introduce a two-phase approach. The first phase…
This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach is formalised as attempting to solve a low-rank…
The complexity of Philip Wolfe's method for the minimum Euclidean-norm point problem over a convex polytope has remained unknown since he proposed the method in 1974. The method is important because it is used as a subroutine for one of the…
We consider the problem of computing the Lebesgue volume of compact basic semi-algebraic sets. In full generality, it can be approximated as closely as desired by a converging hierarchy of upper bounds obtained by applying the Moment-SOS…
John's fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ in $\mathbb{R}^d$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids by…
The subspace approximation problem Subspace($k$,$p$) asks for a $k$-dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the $\ell_{p}$ norm…