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Showing papers on "Regular polygon published in 2020"


Proceedings ArticleDOI
14 Jun 2020
TL;DR: An algorithm for extracting and vectorizing objects in images with polygons is presented, which refines the geometry of the partition while labeling its cells by a semantic class and demonstrates its efficiency compared to existing vectorization methods.
Abstract: We present an algorithm for extracting and vectorizing objects in images with polygons. Departing from a polygonal partition that oversegments an image into convex cells, the algorithm refines the geometry of the partition while labeling its cells by a semantic class. The result is a set of polygons, each capturing an object in the image. The quality of a configuration is measured by an energy that accounts for both the fidelity to input data and the complexity of the output polygons. To efficiently explore the configuration space, we perform splitting and merging operations in tandem on the cells of the polygonal partition. The exploration mechanism is controlled by a priority queue that sorts the operations most likely to decrease the energy. We show the potential of our algorithm on different types of scenes, from organic shapes to man-made objects through floor maps, and demonstrate its efficiency compared to existing vectorization methods.

41 citations


Posted Content
TL;DR: In this paper, the authors construct global in time solutions to both the scalar and vectorial master equations in potential mean field games, when the underlying space is the whole space and so, it is not compact.
Abstract: This manuscript constructs global in time solutions to the $master\ equations$ for potential Mean Field Games. The study concerns a class of Lagrangians and initial data functions, which are $displacement\ convex$ and so, it may be in dichotomy with the class of so--called $monotone$ functions, widely considered in the literature. We construct solutions to both the scalar and vectorial master equations in potential Mean Field Games, when the underlying space is the whole space $\mathbb{R}^d$ and so, it is not compact.

39 citations


Journal ArticleDOI
Chao Li1
TL;DR: In this paper, a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature was established, answering affirmatively a dihedral rigidity conjecture by Gromov.
Abstract: The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is isometric to a flat polyhedron.

36 citations


Journal ArticleDOI
TL;DR: It is shown that Hamiltonian Cycle (and hence Hamiltonian Path ) is NP -hard on graphs of linear mim-width 1; this further hints at the expressive power of the mim- width parameter.

36 citations


Posted Content
TL;DR: In this article, a short proof of the three-dimensional Mahler conjecture for symmetric convex bodies is given, along with a self-contained proof of their two key statements.
Abstract: Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture for symmetric convex bodies. Our contributions include, in particular, simple self-contained proofs of their two key statements. The first of these is an equipartition (ham sandwich type) theorem which refines a celebrated result of Hadwiger and, as usual, can be proved using ideas from equivariant topology. The second is an inequality relating the product volume to areas of certain sections and their duals. Finally we give an alternative proof of the characterization of convex bodies that achieve the equality case and establish a new stability result.

35 citations


Posted Content
TL;DR: A convex analytic framework for ReLU neural networks is developed which elucidates the inner workings of hidden neurons and their function space characteristics and establishes a connection to $\ell_0$-$\ell_1$ equivalence for neural networks analogous to the minimal cardinality solutions in compressed sensing.
Abstract: We develop a convex analytic approach to analyze finite width two-layer ReLU networks. We first prove that an optimal solution to the regularized training problem can be characterized as extreme points of a convex set, where simple solutions are encouraged via its convex geometrical properties. We then leverage this characterization to show that an optimal set of parameters yield linear spline interpolation for regression problems involving one dimensional or rank-one data. We also characterize the classification decision regions in terms of a kernel matrix and minimum $\ell_1$-norm solutions. This is in contrast to Neural Tangent Kernel which is unable to explain predictions of finite width networks. Our convex geometric characterization also provides intuitive explanations of hidden neurons as auto-encoders. In higher dimensions, we show that the training problem can be cast as a finite dimensional convex problem with infinitely many constraints. Then, we apply certain convex relaxations and introduce a cutting-plane algorithm to globally optimize the network. We further analyze the exactness of the relaxations to provide conditions for the convergence to a global optimum. Our analysis also shows that optimal network parameters can be also characterized as interpretable closed-form formulas in some practically relevant special cases.

34 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the boundary integral expressions of Eulerian and Frechet shape derivatives for several classes of nonsmooth domains such as open sets, Lipschitz domains, polygons and curvilinear polygons, semiconvex and convex domains.

29 citations


Journal ArticleDOI
TL;DR: A pure cutting plane algorithm is constructed which is shown to converge if the initial relaxation is a polyhedron, and a theory of outer-product-free sets, where S is the set of real, symmetric matrices of the form $$xx^T$$ x x T .
Abstract: This paper introduces cutting planes that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set $$S\cap P$$ , where S is a closed set, and P is a polyhedron. Given an oracle that provides the distance from a point to S, we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron. These cuts are generated from convex forbidden zones, or S-free sets, derived from the oracle. We also consider the special case of polynomial optimization. Accordingly we develop a theory of outer-product-free sets, where S is the set of real, symmetric matrices of the form $$xx^T$$ . All maximal outer-product-free sets of full dimension are shown to be convex cones and we identify several families of such sets. These families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently. Computational experiments demonstrate the promise of our approach.

28 citations


Posted Content
TL;DR: This paper focuses on how the results of decentralized distributed convex optimization can be explained based on optimal algorithms for the non-distributed setup, and provides recent results that have not been published yet.
Abstract: In the last few years, the theory of decentralized distributed convex optimization has made significant progress. The lower bounds on communications rounds and oracle calls have appeared, as well as methods that reach both of these bounds. In this paper, we focus on how these results can be explained based on optimal algorithms for the non-distributed setup. In particular, we provide our recent results that have not been published yet, and that could be found in details only in arXiv preprints.

25 citations


Posted Content
TL;DR: The results improve upon the iteration complexity of the first-order Mirror Prox method of Nemirovski and the second-order method of Monteiro and Svaiter and give improved convergence rates for constrained convex-concave min-max problems and monotone variational inequalities with higher-order smoothness.
Abstract: We provide improved convergence rates for constrained convex-concave min-max problems and monotone variational inequalities with higher-order smoothness. In min-max settings where the $p^{th}$-order derivatives are Lipschitz continuous, we give an algorithm HigherOrderMirrorProx that achieves an iteration complexity of $O(1/T^{\frac{p+1}{2}})$ when given access to an oracle for finding a fixed point of a $p^{th}$-order equation. We give analogous rates for the weak monotone variational inequality problem. For $p>2$, our results improve upon the iteration complexity of the first-order Mirror Prox method of Nemirovski [2004] and the second-order method of Monteiro and Svaiter [2012]. We further instantiate our entire algorithm in the unconstrained $p=2$ case.

23 citations


Posted Content
Qin Deng1
TL;DR: In this article, it was shown that a metric measure space (X,d,m) satisfying the finite Riemannian curvature-dimension condition is non-branching and tangent cones from the same sequence of rescalings are Holder continuous along the interior of every geodesic.
Abstract: In this paper we prove that a metric measure space $(X,d,m)$ satisfying the finite Riemannian curvature-dimension condition ${\sf RCD}(K,N)$ is non-branching and that tangent cones from the same sequence of rescalings are Holder continuous along the interior of every geodesic in $X$. More precisely, we show that the geometry of balls of small radius centred in the interior of any geodesic changes in at most a Holder continuous way along the geodesic in pointed Gromov-Hausdorff distance. This improves a result in the Ricci limit setting by Colding-Naber where the existence of at least one geodesic with such properties between any two points is shown. As in the Ricci limit case, this implies that the regular set of an ${\sf RCD}(K,N)$ space has $m$-a.e. constant dimension, a result already established by Brue-Semola, and is $m$-a.e convex. It also implies that the top dimension regular set is weakly convex and, therefore, connected. In proving the main theorems, we develop in the ${\sf RCD}(K,N)$ setting the expected second order interpolation formula for the distance function along the Regular Lagrangian flow of some vector field using its covariant derivative.

Journal ArticleDOI
TL;DR: In this article, it was shown that a planar graph is realized as the 1-skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if the graph admits a Hamiltonian cycle.
Abstract: We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph $$\Gamma $$ is realized as the 1-skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if $$\Gamma $$ is realized as the 1-skeleton of a polyhedron inscribed in the sphere and $$\Gamma $$ admits a Hamiltonian cycle. This answers a question asked by Steiner in 1832. Rivin characterized convex polyhedra inscribed in the sphere by studying the geometry of ideal polyhedra in hyperbolic space. We study the case of the hyperboloid and the cylinder by parameterizing the space of convex ideal polyhedra in anti-de Sitter geometry and in half-pipe geometry. Just as the cylinder can be seen as a degeneration of the sphere and the hyperboloid, half-pipe geometry is naturally a limit of both hyperbolic and anti-de Sitter geometry. We promote a unified point of view to the study of the three cases throughout.

Posted Content
TL;DR: In this paper, it was shown that if a neighborhood of the boundary of the billiard domain has a convex caustics of rotation numbers in the interval (0, 1/4) then the boundary curve is an ellipse.
Abstract: In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric $C^2$-smooth convex planar billiards. We assume that the domain $\mathcal A$ between the invariant curve of $4$-periodic orbits and the boundary of the phase cylinder is foliated by $C^0$-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. Other versions of Birkhoff-Poritsky conjecture follow from this result. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a $C^1$-smooth foliation by convex caustics of rotation numbers in the interval (0; 1/4] then the boundary curve is an ellipse. In the language of first integrals one can assert that {if the billiard inside a centrally-symmetric $C^2$-smooth convex curve $\gamma$ admits a $C^1$-smooth first integral which is not singular on $\mathcal A$, then the curve $\gamma$ is an ellipse. } The main ingredients of the proof are : (1) the non-standard generating function for convex billiards discovered in [8], [10]; (2) the remarkable structure of the invariant curve consisting of $4$-periodic orbits; and (3) the integral-geometry approach initiated in [6], [7] for rigidity results of circular billiards. Surprisingly, we establish a Hopf-type rigidity for billiard in ellipse.

Posted Content
TL;DR: In this article, it was shown that for monotone toric domains in arbitrary dimension, the Gromov width agrees with the first equivariant capacity, and that all normalized symplectic capacities agree on convex domains.
Abstract: A strong version of a conjecture of Viterbo asserts that all normalized symplectic capacities agree on convex domains. We review known results showing that certain specific normalized symplectic capacities agree on convex domains. We also review why all normalized symplectic capacities agree on $S^1$-invariant convex domains. We introduce a new class of examples called "monotone toric domains", which are not necessarily convex, and which include all dynamically convex toric domains in four dimensions. We prove that for monotone toric domains in four dimensions, all normalized symplectic capacities agree. For monotone toric domains in arbitrary dimension, we prove that the Gromov width agrees with the first equivariant capacity. We also study a family of examples of non-monotone toric domains and determine when the conclusion of the strong Viterbo conjecture holds for these examples. Along the way we compute the cylindrical capacity of a large class of "weakly convex toric domains" in four dimensions.

Posted Content
TL;DR: The desingularization of the Thomsom polygon for the point vortex system, that is, point vortices located at the vertex of a regular polygon with N sides, is proposed and the study of the contour dynamics equation combined with the application of the infinite dimensional Implicit Function theorem and the well--chosen of the function spaces.
Abstract: This paper deals with the existence of $N$ vortex patches located at the vertex of a regular polygon with $N$ sides that rotate around the center of the polygon at a constant angular velocity. That is done for Euler and (SQG)$_\beta$ equations, with $\beta\in(0,1)$, but may be also extended to more general models. The idea is the desingularization of the Thomsom polygon for the $N$ point vortex system, that is, $N$ point vortices located at the vertex of a regular polygon with $N$ sides. The proof is based on the study of the contour dynamics equation combined with the application of the infinite dimensional Implicit Function theorem and the well--chosen of the function spaces.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions.
Abstract: We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions. As an improvement upon our previous work (Krejciřik and Lotoreichik J. Convex Anal. 25, 319–337, 2018), we show that under either a constraint of fixed perimeter or area, the maximiser within the class of exteriors of simply connected planar sets is always the exterior of a disk, without the need of convexity assumption. Moreover, we generalise the result to disconnected compact planar sets. Namely, we prove that under a constraint of fixed average value of the perimeter over all the connected components, the maximiser within the class of disconnected compact planar sets, consisting of finitely many simply connected components, is again a disk. In higher dimensions, we prove a completely new result that the lowest point in the spectrum is maximised by the exterior of a ball among all sets exterior to bounded convex sets satisfying a constraint on the integral of a dimensional power of the mean curvature of their boundaries. Furthermore, it follows that the critical coupling at which the lowest point in the spectrum becomes a discrete eigenvalue emerging from the essential spectrum is minimised under the same constraint by the critical coupling for the exterior of a ball.

Journal ArticleDOI
TL;DR: In this paper, the authors considered carbon and silicon polyprismanes, the special type of single-walled nanotubes with an extremely small cross-section in the form of a regular polygon.
Abstract: Polyprismanes are the special type of single-walled nanotubes with an extremely small cross-section in the form of a regular polygon. In the presented study, we considered carbon and silicon polypr...

Journal ArticleDOI
TL;DR: In this article, the minimal area problem for homology classes of curves was formulated as a local convex program and an equivalent dual program involving maximization of a concave functional was derived.
Abstract: The minimal-area problem that defines string diagrams in closed string field theory asks for the metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least $$2\pi $$ . This is an extremal length problem in conformal geometry as well as a problem in systolic geometry. We consider the analogous minimal-area problem for homology classes of curves and, with the aid of calibrations and the max flow-min cut theorem, formulate it as a local convex program. We derive an equivalent dual program involving maximization of a concave functional. These two programs give new insights into the form of the minimal-area metric and are amenable to numerical solution. We explain how the homology problem can be modified to provide the solution to the original homotopy problem.

Posted Content
TL;DR: The quality of arbitrary sampling points are characterized via the L_\gamma(\Omega)-norm of the distance function $\rm{dist}(\cdot,P)$, which improves upon previous characterizations based on the covering radius of $P.
Abstract: We show that independent and uniformly distributed sampling points are as good as optimal sampling points for the approximation of functions from the Sobolev space $W_p^s(\Omega)$ on bounded convex domains $\Omega\subset \mathbb{R}^d$ in the $L_q$-norm if $q

Journal ArticleDOI
01 Apr 2020
TL;DR: In this article, the fixed points of convex contraction and generalized convex contractions were studied and the assumption of continuity condition in [11] was replaced by a relatively weaker condition of k-continuity under various settings.
Abstract: Istr$$\check{a}$$tescu (Lib Math 1:151–163, 1981) introduced the notion of convex contraction. He proved that each convex contraction has a unique fixed point on a complete metric space. In this paper we study fixed points of convex contraction and generalized convex contractions. We show that the assumption of continuity condition in [11] can be replaced by a relatively weaker condition of k-continuity under various settings. On this way a new and distinct solution to the open problem of Rhoades (Contemp Math 72:233–245, 1988) is found. Several examples are given to illustrate our results.

Posted ContentDOI
TL;DR: The volesti as discussed by the authors package is a C++ package with an R interface that provides efficient, scalable algorithms for volume estimation, uniform and Gaussian sampling from convex polytopes.
Abstract: Sampling from high dimensional distributions and volume approximation of convex bodies are fundamental operations that appear in optimization, finance, engineering and machine learning. In this paper we present volesti, a C++ package with an R interface that provides efficient, scalable algorithms for volume estimation, uniform and Gaussian sampling from convex polytopes. volesti scales to hundreds of dimensions, handles efficiently three different types of polyhedra and provides non existing sampling routines to R. We demonstrate the power of volesti by solving several challenging problems using the R language.

Posted Content
TL;DR: In this paper, the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones has been studied, and it has been shown that there is a unique positive discrete harmonic function for these processes up to a multiplicative constant.
Abstract: We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up to a multiplicative constant); in other words, the Martin boundary reduces to a singleton.

Posted Content
TL;DR: In this article, the authors prove the log-Brunn-Minkowski conjecture for convex bodies with symmetries to independent hyperplanes, and discuss the equality case and the uniqueness of the solution of the related case of the logarithmic Minkowski problem.
Abstract: We prove the log-Brunn-Minkowski conjecture for convex bodies with symmetries to $n$ independent hyperplanes, and discuss the equality case and the uniqueness of the solution of the related case of the logarithmic Minkowski problem. We also clarify a small gap in the known argument classifying the equality case of the log-Brunn-Minkowski conjecture for unconditional convex bodies.

Posted Content
TL;DR: In this article, it was shown that for any constant constant constant √ n, there exist continuous weak Mikado flows with disjoint supports in space and time that satisfy local energy inequality and strictly dissipate the total kinetic energy.
Abstract: We show that for any $\al<\frac 17$ there exist $\al$-Holder continuous weak solutions of the three-dimensional incompressible Euler equation, which satisfy the local energy inequality and strictly dissipate the total kinetic energy. The proof relies on the convex integration scheme and the main building blocks of the solution are various Mikado flows with disjoint supports in space and time.

Posted Content
TL;DR: In this article, the dual Orlicz-Minkowski problem was studied in convex geometry and a new existence result of smooth even solutions to this problem for smooth even measures was obtained.
Abstract: In this paper we study the dual Orlicz-Minkowski problem, which is a generalization of the dual Minkowski problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial vectors, we obtain a new existence result of smooth even solutions to this problem for smooth even measures.

Book
28 Aug 2020
TL;DR: In this paper, the authors present a set of sets of convex sets and functions for area measures to valuations, based on the Brunn-Minkowski theory of geometrical formulae.
Abstract: Preface -- Preliminaries and Notation -- 1. Convex Sets -- 2. Convex Functions -- 3. Brunn-Minkowski Theory -- 4. From Area Measures to Valuations -- 5. Integral Geometric Formulas.-6. Solutions of Selected Exercises -- References -- Index.

Journal ArticleDOI
TL;DR: In this paper, first-order methods for minimization of a convex function on a simple convex set were proposed, assuming that the objective function is a composite function given as a sum of a set of simples.
Abstract: In this paper, we propose new first-order methods for minimization of a convex function on a simple convex set. We assume that the objective function is a composite function given as a sum of a sim...

Posted Content
TL;DR: The connection between the existence of lifts of a convex set and certain structured factorizations of its associated slack operator is explained, and a uniform approach to the construction of spectrahedral lifts of convex sets is described.
Abstract: This paper presents a selected tour through the theory and applications of lifts of convex sets. A lift of a convex set is a higher-dimensional convex set that projects onto the original set. Many convex sets have lifts that are dramatically simpler to describe than the original set. Finding such simple lifts has significant algorithmic implications, particularly for optimization problems. We consider both the classical case of polyhedral lifts, described by linear inequalities, as well as spectrahedral lifts, defined by linear matrix inequalities, with a focus on recent developments related to spectrahedral lifts. Given a convex set, ideally we would either like to find a (low-complexity) polyhedral or spectrahedral lift, or find an obstruction proving that no such lift is possible. To this end, we explain the connection between the existence of lifts of a convex set and certain structured factorizations of its associated slack operator. Based on this characterization, we describe a uniform approach, via sums of squares, to the construction of spectrahedral lifts of convex sets and illustrate the method on several families of examples. Finally, we discuss two flavors of obstruction to the existence of lifts: one related to facial structure, and the other related to algebraic properties of the set in question. Rather than being exhaustive, our aim is to illustrate the richness of the area. We touch on a range of different topics related to the existence of lifts, and present many examples of lifts from different areas of mathematics and its applications.

Posted Content
TL;DR: Under some mild assumptions, it is shown that NL-IAPIAL generates an approximate stationary solution of the aforementioned problem in at most ${\cal O}(\log(1/\rho)/ \rho^{3})$ inner iterations, where $\rho>0$ is a given tolerance.
Abstract: This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving smooth nonconvex composite optimization problems with nonlinear $\cal K$-convex constraints, i.e., the constraints are convex with respect to the order given by a closed convex cone $\cal K$. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a Lagrange multiplier update. Under some mild assumptions, it is shown that NL-IAPIAL generates an approximate stationary solution of the constrained problem in ${\cal O}(\log(1/\rho)/\rho^{3})$ inner iterations, where $\rho>0$ is a given tolerance. Numerical experiments are also given to illustrate the computational efficiency of the proposed method.

Posted Content
TL;DR: In this article, a variational construction of special solutions to the generalized surface quasi-geostrophic equations is provided, which take the form of N vortex patches with N-fold symmetry, which are steady in a uniformly rotating frame.
Abstract: We provide a variational construction of special solutions to the generalized surface quasi-geostrophic equations. These solutions take the form of N vortex patches with N-fold symmetry , which are steady in a uniformly rotating frame. Moreover, we investigate their asymptotic properties when the size of the corresponding patches vanishes. In this limit, we prove these solutions to be a desingularization of N Dirac masses with the same intensity, located on the N vertices of a regular polygon rotating at a constant angular velocity.