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We revisit the classical dual ascent algorithm for minimization of convex functionals in the presence of linear constraints, and give convergence results which apply even for non-convex functionals. We describe limit points in terms of the…
An algorithm for the computation of global discrete conformal parametrizations with prescribed global holonomy signatures for triangle meshes was recently described in [Campen and Zorin 2017]. In this paper we provide a detailed analysis of…
Distributed optimization often requires finding the minimum of a global objective function written as a sum of local functions. A group of agents work collectively to minimize the global function. We study a continuous-time decentralized…
Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the…
We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…
This article derives lower bounds on the convergence rate of continuous-time gradient-based optimization algorithms. The algorithms are subjected to a time-normalization constraint that avoids a reparametrization of time in order to make…
Gamma Dor stars are ideal targets for studies of the innermost dynamical properties of stars, due to their rich frequency spectrum of gravito-inertial modes propagating in the radiative envelope. Recent studies found that these modes could…
Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the…
We investigate the dynamical motion of particles on a two-dimensional symmetric periodic substrate in the presence of both a dc drive along a symmetry direction of the periodic substrate and an additional circular ac drive. For large enough…
In this paper, we study minimizers of the Chon\'e--Rochet variational problem in dimension two. We first establish global $C^1$ regularity on arbitrary bounded convex domains, and then prove global $C^{1,1}$ regularity on bounded strictly…
A simple geometrical model is presented for the gravity-driven motion of a single particle on a rough inclined surface. Adopting a simple restitution law for the collisions between the particle and the surface, we arrive at a model in which…
This note studies numerical methods for solving compositional optimization problems, where the inner function is smooth, and the outer function is Lipschitz continuous, non-smooth, and non-convex but exhibits one of two special structures…
It is shown that any non-degenerate $\mathbb C$-convex domain in $\mathbb C^n$ is uniformly squeezing. It is also found the precise behavior of the squeezing function near a Dini-smooth boundary point of a plane domain.
The global Lipschitz smoothness condition underlies most convergence and complexity analyses via two key consequences: the descent lemma and the gradient Lipschitz continuity. How to study the performance of optimization algorithms in the…
Modeling folding surfaces with nonzero thickness is of practical interest for mechanical engineering. There are many existing approaches that account for material thickness in folding applications. We propose a new systematic and broadly…
This note presents a new, self-contained proof of Shahgholian's geometric theorem on quadrature surfaces using the thickness function and level set methods. By relying on a radial parametrisation and fundamental maximum principles, the…
This paper is devoted to the class of paraconvex functions and presents some of its fundamental properties, characterization, and examples that can be used for their recognition and optimization. Next, the convergence analysis of the…
In this note, we focus on smooth nonconvex optimization problems that obey: (1) all local minimizers are also global; and (2) around any saddle point or local maximizer, the objective has a negative directional curvature. Concrete…
We consider joint optimization and learning problems arising in real-time decision systems. While most existing work focuses primarily on convex, revenue-based objectives, we extend this line of research to multi-objective formulations. In…
Weight matrices in deep networks exhibit geometric continuity -- principal singular vectors of adjacent layers point in similar directions. While this property has been widely observed, its origin remains unexplained. Through experiments on…