Related papers: Realizing Metric Spaces with Convex Obstacles
Minimizing a convex function of a measure with a sparsity-inducing penalty is a typical problem arising, e.g., in sparse spikes deconvolution or two-layer neural networks training. We show that this problem can be solved by discretizing the…
The triangulations of a surface $\Sigma$ with a prescribed set of vertices can be endowed with a graph structure $\mathcal{F}(\Sigma)$. Its edges connect two triangulations that differ by a single arc. It is known that, when $\Sigma$ is a…
We study the query complexity of geodesically convex (g-convex) optimization on a manifold. To isolate the effect of that manifold's curvature, we primarily focus on hyperbolic spaces. In a variety of settings (smooth or not; strongly…
An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two…
In many robotics applications, it is necessary to compute not only the distance between the robot and the environment, but also its derivative - for example, when using control barrier functions. However, since the traditional Euclidean…
Parametrized families of PDEs arise in various contexts such as inverse problems, control and optimization, risk assessment, and uncertainty quantification. In most of these applications, the number of parameters is large or perhaps even…
Many methods in learning from demonstration assume that the demonstrator has knowledge of the full environment. However, in many scenarios, a demonstrator only sees part of the environment and they continuously replan as they gather…
Quantifying robustness in a single measure for the purposes of model selection, development of adversarial training methods, and anticipating trends has so far been elusive. The simplest metric to consider is the number of trainable…
We conjecture that a convex polytope is uniquely determined up to isometry by its edge-graph, edge lengths and the collection of distances of its vertices to some arbitrary interior point, across all dimensions and all combinatorial types.…
Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance d_w(p, q) between two points p,q belonging to S is defined as w(p) + d(p, q) + w(q) if p \ne q and it is…
Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We overview the theory of classical MDS, along with its…
Metric spaces $(X, d)$ are ubiquitous objects in mathematics and computer science that allow for capturing (pairwise) distance relationships $d(x, y)$ between points $x, y \in X$. Because of this, it is natural to ask what useful…
Metric data plays an important role in various settings such as metric-based indexing, clustering, classification, and approximation algorithms in general. Due to measurement error, noise, or an inability to completely gather all the data,…
The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces.…
The Procrustes distance is used to quantify the similarity or dissimilarity of (3-dimensional) shapes, and extensively used in biological morphometrics. Typically each (normalized) shape is represented by N landmark points, chosen to be…
We study the problem of supervised learning a metric space under discriminative constraints. Given a universe $X$ and sets ${\cal S}, {\cal D}\subset {X \choose 2}$ of similar and dissimilar pairs, we seek to find a mapping $f:X\to Y$, into…
Given a convex potential in a space with convex obstacles, an artificial potential is used to navigate to the minimum of the natural potential while avoiding collisions. The artificial potential combines the natural potential with…
We present a novel approach to understand geometric-incompatibility-induced rigidity in under-constrained materials, including sub-isostatic 2D spring networks and 2D and 3D vertex models for dense biological tissues. We show that in all…
The tight span, or injective envelope, is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded. The concept has stimulated a large body of theory and has…
The reachability problem is a central decision problem for formal verification based on vector addition systems with states (VASS), which are equivalent to Petri nets and form one of the most studied and applied models of concurrency.…