Related papers: Metric Optimization in Penner Coordinates
Every convex optimization problem has a dual problem. The $p$-Dirichlet problem in metric measure spaces is an optimization problem whose solutions are $p$-harmonic functions. What is its dual problem? In this paper, we give an answer to…
The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. When the distance information is incomplete, the…
Random embeddings project high-dimensional spaces to low-dimensional ones; they are careful constructions which allow the approximate preservation of key properties, such as the pair-wise distances between points. Often in the field of…
The Continuous p-Dispersion Problem (CpDP) with boundary constraints asks for the placement of a fixed number of points in a compact subset of Euclidean space such that the minimum distance between any two points, as well as the points and…
Regularization and interior point approaches offer valuable perspectives to address constrained nonlinear optimization problems in view of control applications. This paper discusses the interactions between these techniques and proposes an…
Many computational algorithms applied to geometry operate on discrete representations of shape. It is sometimes necessary to first simplify, or coarsen, representations found in modern datasets for practicable or expedited processing. The…
We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies for wide kind of geometries: domains…
The notions of discrete conformality on triangle meshes have rich mathematical theories and wide applications. The related notions of discrete uniformizations on triangle meshes, suggest efficient methods for computing the uniformizations…
Sequential Convex Programming (SCP) has recently gained popularity as a tool for trajectory optimization due to its sound theoretical properties and practical performance. Yet, most SCP-based methods for trajectory optimization are…
Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take…
By generalizing the notion of linearization, a concept originally arising from microlocal analysis and symbolic calculus, to diffeological spaces, we make a first proposal setting for optimization problems in this category. We show how…
Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include…
We investigate the approximation of Monge--Kantorovich problems on general compact metric spaces, showing that optimal values, plans and maps can be effectively approximated via a fully discrete method. First we approximate optimal values…
Ordinal Embedding places n objects into R^d based on comparisons such as "a is closer to b than c." Current optimization-based approaches suffer from scalability problems and an abundance of low quality local optima. We instead consider a…
Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…
In practice, optimization tasks have some structure that allows developing new algorithms for every problem with faster convergence rates. Using the structure of optimization tasks, we can propose algorithms with more optimistic convergence…
Optimizing over the set of orthogonal matrices is a central component in problems like sparse-PCA or tensor decomposition. Unfortunately, such optimization is hard since simple operations on orthogonal matrices easily break orthogonality,…
Standard convolutions are prevalent in image processing and deep learning, but their fixed kernels limits adaptability. Several deformation strategies of the reference kernel grid have been proposed. Yet, they lack a unified theoretical…
We study the problem of finding the one-dimensional structure in a given data set. In other words we consider ways to approximate a given measure (data) by curves. We consider an objective functional whose minimizers are a regularization of…