Related papers: An optimization problem on the sphere

Inspired by the boolean discrepancy problem, we study the following optimization problem which we term \textsc{Spherical Discrepancy}: given $m$ unit vectors $v_1, \dots, v_m$, find another unit vector $x$ that minimizes $\max_i \langle x,…

We study efficient distributed algorithms for the fundamental problem of principal component analysis and leading eigenvector computation on the sphere, when the data are randomly distributed among a set of computational nodes. We propose a…

We consider the problem of federated learning in a one-shot setting in which there are $m$ machines, each observing $n$ sample functions from an unknown distribution on non-convex loss functions. Let $F:[-1,1]^d\to\mathbb{R}$ be the…

This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point…

In this paper we investigate the Erd\"os/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been…

We consider the problem of approximating a semialgebraic set with a sublevel-set of a polynomial function. In this setting, it is standard to seek a minimum volume outer approximation and/or maximum volume inner approximation. As there is…

This paper studies the problem of distributed Riemannian optimization over a network of agents whose cost functions are geodesically smooth but possibly geodesically non-convex. Extending a well-known distributed optimization strategy…

As observed by Auderset et al. (2005) and Wiesel (2012), viewing covariance matrices as elements of a Riemannian manifold and using the concept of geodesic convexity provide useful tools for studying M-estimators of multivariate scatter. In…

This research studies a non-convex geometric optimization problem arising from the field of optical wireless power transfer. In the considered optimization problem, the cost function is a sum of negatively and fractionally powered distances…

This paper establishes a strict mathematical relationship between an arbitrary continuous function on a compact set and its global minima, like the well-known first order optimality condition for convex and differentiable functions. By…

Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics. In…

Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent…

This paper establishes several new facts on generalized polyhedral convex sets and shows how they can be used in vector optimization. Among other things, a scalarization formula for the efficient solution sets of generalized vector…

A framework is introduced for sequentially solving convex stochastic minimization problems, where the objective functions change slowly, in the sense that the distance between successive minimizers is bounded. The minimization problems are…

Recent numerical experiments have demonstrated that the choice of optimization geometry used during training can impact generalization performance when learning expressive nonlinear model classes such as deep neural networks. These…

We show that in the setting of proper metric spaces one obtains a solution of the classical two-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area (in the sense of convex geometry) has…

We consider a stochastic version of the proximal point algorithm for optimization problems posed on a Hilbert space. A typical application of this is supervised learning. While the method is not new, it has not been extensively analyzed in…

Consider a set P of N random points on the unit sphere of dimension $d-1$, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that…

We investigate robust optimization problems defined for maximizing convex functions. For finite uncertainty set, we develop a geometric branch-and-bound algorithmic approach to solve this problem. The geometric branch-and-bound algorithm…

Weight optimization of frame structures with continuous cross-section parametrization is a challenging non-convex problem that has traditionally been solved by local optimization techniques. Here, we exploit its inherent semi-algebraic…