Related papers: Learning convex bodies is hard

Let $K$ be a convex body in $\mathbb{R} ^d$, with $d = 2,3$. We determine sharp sufficient conditions for a set $E$ composed of $1$, $2$, or $3$ points of ${\rm bd}K$, to contain at least one endpoint of a diameter of $K$ (for $d=2,3$). We…

For a two-dimensional convex body, the Kovner-Besicovitch measure of symmetry is defined as the volume ratio of the largest centrally symmetric body contained inside the body to the original body. A classical result states that the…

We extend to arbitrary finite $n$ the notion of immobilization of a convex body $O$ in $R^n$ by a finite set of points $P$ in the boundary of $O$. Because of its importance for this problem, necessary and sufficient conditions are found for…

Let $\Omega \subset \mathbb{R}^{d}$ be a convex body with everywhere positive curvature except at the origin and with the boundary $\partial \Omega$ as the graph of the function $y=|x|^{\gamma}$ in a neighborhood of the origin with $\gamma…

This work proposes a new formulation to the long-standing problem of convex decomposition through learning feature fields, enabling the first feed-forward model for open-world convex decomposition. Our method produces high-quality…

Bayesian Networks (BNs) are useful tools giving a natural and compact representation of joint probability distributions. In many applications one needs to learn a Bayesian Network (BN) from data. In this context, it is important to…

We study the fundamental problem of learning with respect to the squared loss in a convex class. The state-of-the-art sample complexity estimates in this setting rely on Rademacher complexities, which are generally difficult to control. We…

The purpose of this paper is to study the reflections of a convex body. In particular, we are interested in orthogonal reflections of its sections that can be extended to reflections of the whole body. For this reason, we need to study the…

We present an improved algorithm for {\em quasi-properly} learning convex polyhedra in the realizable PAC setting from data with a margin. Our learning algorithm constructs a consistent polyhedron as an intersection of about $t \log t$…

This paper addresses the fundamental question of when convex sets remain disjoint after random projection. We provide an analysis using ideas from high-dimensional convex geometry. For ellipsoids, we provide a bound in terms of the distance…

We consider the problem of multi-class classification, where a stream of adversarially chosen queries arrive and must be assigned a label online. Unlike traditional bounds which seek to minimize the misclassification rate, we minimize the…

We consider systems of linear partial differential equations, which contain only second and first derivatives in the $x$ variables and which are uniformly parabolic in the sense of Petrovski\v{\i} in the layer ${\mathbb R}^n\times [0,T]$.…

The complexity of a learning task is increased by transformations in the input space that preserve class identity. Visual object recognition for example is affected by changes in viewpoint, scale, illumination or planar transformations.…

{We show in this paper that two normal elliptic sections through every point of the boundary of a smooth convex body essentially characterize an ellipsoid and furthermore, that four different pairwise non-tangent elliptic sections through…

Most existing distance metric learning methods assume perfect side information that is usually given in pairwise or triplet constraints. Instead, in many real-world applications, the constraints are derived from side information, such as…

Given a convex body C in R^d containing the origin in its interior and a real number tau > 1 we seek to construct a polytope P in C with as few vertices as possible such that C in tau P. Our construction is nearly optimal for a wide range…

In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in $\mathbb{R}^n$; this theorem is a generalization of the linear programming bound for sphere packings. We…

$\newcommand{\eps}{\varepsilon}\newcommand{\tldO}{\widetilde{O}}$Consider the problem of constructing weak $\eps$-nets where the stabbing elements are lines or $k$-flats instead of points. We study this problem in the simplest setting where…

We study upper bounds on the number of lattice points for convex bodies having their centroid at the origin. For the family of simplices as well as in the planar case we obtain best possible results. For arbitrary convex bodies we provide…

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical and hyperbolic planes.…