Showing papers on "Regular polygon published in 2019"

On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes

Abstract: Stochastic gradient descent is the method of choice for large scale optimization of machine learning objective functions. Yet, its performance is greatly variable and heavily depends on the choice of the stepsizes. This has motivated a large body of research on adaptive stepsizes. However, there is currently a gap in our theoretical understanding of these methods, especially in the non-convex setting. In this paper, we start closing this gap: we theoretically analyze in the convex and non-convex settings a generalized version of the AdaGrad stepsizes. We show sufficient conditions for these stepsizes to achieve almost sure asymptotic convergence of the gradients to zero, proving the first guarantee for generalized AdaGrad stepsizes in the non-convex setting. Moreover, we show that these stepsizes allow to automatically adapt to the level of noise of the stochastic gradients in both the convex and non-convex settings, interpolating between O(1/T) and O(1/sqrt(T)), up to logarithmic terms.

The Fuglede conjecture for convex domains is true in all dimensions

Abstract: A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $\Omega$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $\Omega \subset \mathbb{R}^d$ the "tiling implies spectral" part of the conjecture is in fact true. To the contrary, the "spectral implies tiling" direction of the conjecture for convex bodies was proved only in $\mathbb{R}^2$, and also in $\mathbb{R}^3$ under the a priori assumption that $\Omega$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $\Omega$ is a polytope) and could not be treated using the previously developed techniques. In this paper we fully settle Fuglede's conjecture for convex bodies affirmatively in all dimensions, i.e. we prove that if a convex body $\Omega \subset \mathbb{R}^d$ is a spectral set then $\Omega$ is a convex polytope which can tile the space by translations. To prove this we introduce a new technique, involving a construction from crystallographic diffraction theory, which allows us to establish a geometric "weak tiling" condition necessary for a set $\Omega \subset \mathbb{R}^d$ to be spectral.

Finite-Sum Smooth Optimization with SARAH

Abstract: The total complexity (measured as the total number of gradient computations) of a stochastic first-order optimization algorithm that finds a first-order stationary point of a finite-sum smooth nonconvex objective function $F(w)=\frac{1}{n} \sum_{i=1}^n f_i(w)$ has been proven to be at least $\Omega(\sqrt{n}/\epsilon)$ for $n \leq \mathcal{O}(\epsilon^{-2})$ where $\epsilon$ denotes the attained accuracy $\mathbb{E}[ \| abla F(\tilde{w})\|^2] \leq \epsilon$ for the outputted approximation $\tilde{w}$ (Fang et al., 2018). In this paper, we provide a convergence analysis for a slightly modified version of the SARAH algorithm (Nguyen et al., 2017a;b) and achieve total complexity that matches the lower-bound worst case complexity in (Fang et al., 2018) up to a constant factor when $n \leq \mathcal{O}(\epsilon^{-2})$ for nonconvex problems. For convex optimization, we propose SARAH++ with sublinear convergence for general convex and linear convergence for strongly convex problems; and we provide a practical version for which numerical experiments on various datasets show an improved performance.

Constructing Convex Inner Approximations of Steady-State Security Regions

Abstract: We propose a scalable optimization framework for estimating convex inner approximations of the steady-state security sets. The framework is based on Brouwer fixed point theorem applied to a fixed-point form of the power flow equations. It establishes a certificate for the self-mapping of a polytope region constructed around a given feasible operating point. This certificate is based on the explicit bounds on the nonlinear terms that hold within the self-mapped polytope. The shape of the polytope is adapted to find the largest approximation of the steady-state security region. While the corresponding optimization problem is nonlinear and non-convex, every feasible solution found by local search defines a valid inner approximation. The number of variables scales linearly with the system size, and the general framework can naturally be applied to other nonlinear equations with affine dependence on inputs. Test cases, with the system sizes up to 1354 buses, are used to illustrate the scalability of the approach. The results show that the approximated regions are not unreasonably conservative and that they cover substantial fractions of the true steady-state security regions for most medium-sized test cases.

Ancient asymptotically cylindrical flows and applications.

Abstract: In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in $\mathbb{R}^{n+1}$ for all $n\geq 3$: we show that if a mean curvature flow $\{M_t\}$ in $\mathbb{R}^{n+1}$ has an $S^{n-1}\times \mathbb{R}$ singularity at $(x_0,t_0)$, then there exists an $\varepsilon=\varepsilon(x_0,t_0)>0$ such that $M_t\cap B(x_0,\varepsilon)$ is mean-convex for all $t\in(t_0-\varepsilon^2,t_0+\varepsilon^2)$. As in the case $n=2$, which was resolved by the first three authors in arXiv:1810.08467, the existence of such a mean-convex neighborhood follows from classifying a certain class of ancient Brakke flows that arise as potential blowup limits near a neck singularity. Specifically, we prove that any ancient unit-regular integral Brakke flow with a cylindrical blowdown must be either a round shrinking cylinder, a translating bowl soliton, or an ancient oval. In particular, combined with a prior result of the last two authors, we obtain uniqueness of mean curvature flow through neck singularities. The main difficulty in addressing the higher dimensional case is in promoting the spectral analysis on the cylinder to global geometric properties of the solution. Most crucially, due to the potential wide variety of self-shrinking flows with entropy lower than the cylinder when $n\geq 3$, smoothness does not follow from the spectral analysis by soft arguments. This precludes the use of the classical moving plane method to derive symmetry. To overcome this, we introduce a novel variant of the moving plane method, which we call "moving plane method without assuming smoothness" - where smoothness and symmetry are established in tandem.

Fundamental Barriers to High-Dimensional Regression with Convex Penalties

Abstract: In high-dimensional regression, we attempt to estimate a parameter vector ${\boldsymbol \beta}_0\in{\mathbb R}^p$ from $n\lesssim p$ observations $\{(y_i,{\boldsymbol x}_i)\}_{i\le n}$ where ${\boldsymbol x}_i\in{\mathbb R}^p$ is a vector of predictors and $y_i$ is a response variable. A well-estabilished approach uses convex regularizers to promote specific structures (e.g. sparsity) of the estimate $\widehat{\boldsymbol \beta}$, while allowing for practical algorithms. Theoretical analysis implies that convex penalization schemes have nearly optimal estimation properties in certain settings. However, in general the gaps between statistically optimal estimation (with unbounded computational resources) and convex methods are poorly understood. We show that, in general, a large gap exists between the best performance achieved by \emph{any convex regularizer} and the optimal statistical error. Remarkably, we demonstrate that this gap is generic as soon as we try to incorporate very simple structural information about the empirical distribution of the entries of ${\boldsymbol \beta}_0$. Our results follow from a detailed study of standard Gaussian designs, a setting that is normally considered particularly friendly to convex regularization schemes such as the Lasso. We prove a lower bound on the estimation error achieved by any convex regularizer which is invariant under permutations of the coordinates of its argument. This bound is expected to be generally tight, and indeed we prove tightness under certain conditions. Further, it implies a gap with respect to Bayes-optimal estimation that can be precisely quantified and persists if the prior distribution of the signal ${\boldsymbol \beta}_0$ is known to the statistician. Our results provide rigorous evidence towards a broad conjecture regarding computational-statistical gaps in high-dimensional estimation.

Halfspace depth and floating body

Abstract: Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Maximum halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the maximum depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies of measures used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.

Neural Codes, Decidability, and a New Local Obstruction to Convexity

Abstract: Given an intersection pattern of arbitrary sets in Euclidean space, is there an arrangement of convex open sets in Euclidean space that exhibits the same intersections? This question is combinatori...

Uniqueness of convex ancient solutions to mean curvature flow in $${\mathbb {R}}^3$$ R 3

Abstract: A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension 3 which have positive sectional curvature and are $$\kappa $$ -noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in $${\mathbb {R}}^3$$ , and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in $${\mathbb {R}}^3$$ which is strictly convex and noncollapsed.

Optimum harvesting area of convex and concave polygon field for path planning of robot combine harvester

Abstract: This paper presents an optimum harvesting area of a convex and concave polygon for the path planning of a robot combine harvester. A convenient optimum harvesting area for a convex and concave polygon is proposed. The notion is that path planning specifically for a robot combine harvester is required to choose the crop field optimum harvesting area; otherwise, crop losses may occur during harvesting of the field. For a safe turning margin of the robot combine harvester, the surrounding crop near the boundary zone is cut twice or thrice by manual operation. However, this surrounding cutting crop is not exactly straight, and sometimes it is curved or meanders. In addition, path planning with a conventional AB point method in order to take a corner position from the global positioning system by visual observation is a time-consuming operation. A curved or meandering crop is not cut and left in the field during harvesting, and the harvesting area is not optimum. Therefore, a suitable N-polygon algorithm and split of convex hull and cross-point method for determining the optimum harvesting area for path planning are proposed, which reduce the crop losses in the field. The results show that this developed algorithm estimates the optimum harvesting area for a convex or concave polygon field and its corner vertices, takes all crop portions, and reduces crop losses. It is also illustrated that the working path calculated based on the corner vertices minimizes the total operational processing time.

On Minkowski difference-based contact detection in discrete/discontinuous modelling of convex polygons/polyhedra

Abstract: Contact detection for convex polygons/polyhedra has been a critical issue in discrete/discontinuous modelling, such as the discrete element method (DEM) and the discontinuous deformation analysis (DDA). The recently developed 3D contact theory for polyhedra in DDA depends on the so-called entrance block of two polyhedra and reduces the contact to evaluate the distance between the reference point to the corresponding entrance block, but effective implementation is still lacking.,In this paper, the equivalence of the entrance block and the Minkowski difference of two polyhedra is emphasised and two well-known Minkowski difference-based contact detection and overlap computation algorithms, GJK and expanding polytope algorithm (EPA), are chosen as the possible numerical approaches to the 3D contact theory for DDA, and also as alternatives for computing polyhedral contact features in DEM. The key algorithmic issues are outlined and their important features are highlighted.,Numerical examples indicate that the average number of updates required in GJK for polyhedral contact is around 6, and only 1 or 2 iterations are needed in EPA to find the overlap and all the relevant contact features when the overlap between polyhedra is small.,The equivalence of the entrance block in DDA and the Minkowski difference of two polyhedra is emphasised; GJK- and EPA-based contact algorithms are applied to convex polyhedra in DEM; energy conservation is guaranteed for the contact theory used; and numerical results demonstrate the effectiveness of the proposed methodologies.

Convex Decomposition for a Coverage Path Planning for Autonomous Vehicles: Interior Extension of Edges

Abstract: To cover an area of interest by an autonomous vehicle, such as an Unmanned Aerial Vehicle (UAV), planning a coverage path which guides the unit to cover the area is an essential process. However, coverage path planning is often problematic, especially when the boundary of the area is complicated and the area contains several obstacles. A common solution for this situation is to decompose the area into disjoint convex sub-polygons and to obtain coverage paths for each sub-polygon using a simple back-and-forth pattern. Aligned with the solution approach, we propose a new convex decomposition method which is simple and applicable to any shape of target area. The proposed method is designed based on the idea that, given an area of interest represented as a polygon, a convex decomposition of the polygon mainly occurs at the points where an interior angle between two edges of the polygon is greater than 180 degrees. The performance of the proposed method is demonstrated by comparison with existing convex decomposition methods using illustrative examples.

Universal Functionals in Density Functional Theory

Abstract: In this chapter we first review the Levy-Lieb functional, which gives the lowest kinetic and interaction energy that can be reached with all possible quantum states having a given density. We discuss two possible convex generalizations of this functional, corresponding to using mixed canonical and grand-canonical states, respectively. We present some recent works about the local density approximation, in which the functionals get replaced by purely local functionals constructed using the uniform electron gas energy per unit volume. We then review the known upper and lower bounds on the Levy-Lieb functionals. We start with the kinetic energy alone, then turn to the classical interaction alone, before we are able to put everything together. An appendix is devoted to the Hohenberg-Kohn theorem and the role of many-body unique continuation in its proof.

Sharp fundamental gap estimate on convex domains of sphere

Abstract: In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space [3] and conjectured similar results hold for spaces with constant sectional curvature. We prove the conjecture for the sphere. Namely when $D$, the diameter of a convex domain in the unit $\mathbb{S}^n$ sphere, is $\leq \frac{\pi}{2}$, the gap is greater than the gap of the corresponding 1-dim sphere model. We also prove the gap is $\geq 3 \frac{\pi^2}{D^2}$ when $n \geq 3$, giving a sharp bound. As in [3], the key is to prove a super log-concavity of the first eigenfunction.

On the local systolic optimality of Zoll contact forms

Abstract: We prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (i) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (ii) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (iii) a generalization of Gromov's non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones.

Shape effects on graphene magnetoplasmons

Abstract: Graphene plasmons possess a number of unique optical properties and have been demonstrated to enable a variety of applications in the THz frequency range and the infrared frequency range, where active tunability through electrostatic gating plays an important role. In addition, graphene plasmons can be tuned by a static magnetic field, resulting in so called graphene magnetoplasmons. Here, we investigate the excitation of magnetoplasmons in graphene nanostructures with different shapes. We show that in the presence of a static magnetic field, plasmonic dipolar modes will split and the splitting is symmetrical in regular polygons. The splitting depends not on the size but only on the number of sides of the regular polygons. Larger splitting will occur in regular polygons with more sides, where the maximum splitting is achieved in circular disks. We further introduce a simple Lorentz model that could provide an excellent description of optical excitations in regular polygons. Finally, we examine the magnetoplasmons in the shapes without rotational symmetry, such as rectangles, where the symmetry of the splitting breaks as well.

A Hybrid Analytical-Numerical Technique for Elliptic PDEs

Abstract: Recent work has given rise to a novel and simple numerical technique for solving elliptic boundary value problems formulated in convex polygons in two dimensions. The method, based on the unified t...

Optimized ellipse packings in regular polygons

Abstract: We present model development and numerical solution approaches to the problem of packing a general set of ellipses without overlaps into an optimized polygon. Specifically, for a given set of ellipses, and a chosen integer m ≥ 3, we minimize the apothem of the regular m-polygon container. Our modeling and solution strategy is based on the concept of embedded Lagrange multipliers. To solve models with up to n ≤ 10 ellipses, we use the LGO solver suite for global–local nonlinear optimization. In order to reduce increasing runtimes, for model instances with 10 ≤ n ≤ 20 ellipses, we apply local search launching the Ipopt solver from selected random starting points. The numerical results demonstrate the applicability of our modeling and optimization approach to a broad class of highly non-convex ellipse packing problems, by consistently returning good quality feasible solutions in all (231) illustrative model instances considered here.

Packing ellipses in an optimized convex polygon

Abstract: Packing ellipses with arbitrary orientation into a convex polygonal container which has a given shape is considered. The objective is to find a minimum scaling (homothetic) coefficient for the polygon still containing a given collection of ellipses. New phi-functions and quasi phi-functions to describe non-overlapping and containment constraints are introduced. The packing problem is then stated as a continuous nonlinear programming problem. A solution approach is proposed combining a new starting point algorithm and a new modification of the LOFRT procedure (J Glob Optim 65(2):283–307, 2016) to search for locally optimal solutions. Computational results are provided to demonstrate the efficiency of our approach. The computational results are presented for new problem instances, as well as for instances presented in the recent paper ( http://www.optimization-online.org/DB_FILE/2016/03/5348.pdf , 2016).

Euclidean Combinatorial Configurations: Typology and Applications

Abstract: In this paper, a concept of Euclidean combinatorial configuration as a mapping of a set of certain objects into a point of Euclidean space is introduced. A typology of Euclidean combinatorial configurations sets is offered based on their main structural and geometric characteristics. Optimization problems over vertex-located sets of Euclidean combinatorial configurations are studied, in particular, their equivalent formulation with convex both objective function and functional constraints is constructed.

Localized Gaussian width of $M$-convex hulls with applications to Lasso and convex aggregation

Abstract: Upper and lower bounds are derived for the Gaussian mean width of a convex hull of $M$ points intersected with a Euclidean ball of a given radius. The upper bound holds for any collection of extreme points bounded in Euclidean norm. The upper bound and the lower bound match up to a multiplicative constant whenever the extreme points satisfy a one sided Restricted Isometry Property. An appealing aspect of the upper bound is that no assumption on the covariance structure of the extreme points is needed. This aspect is especially useful to study regression problems with anisotropic design distributions. We provide applications of this bound to the Lasso estimator in fixed-design regression, the Empirical Risk Minimizer in the anisotropic persistence problem, and the convex aggregation problem in density estimation.

Zeroth-order Stochastic Compositional Algorithms for Risk-Aware Learning

Abstract: We present Free-MESSAGEp, the first zeroth-order algorithm for convex mean-semideviation-based risk-aware learning, which is also the first three-level zeroth-order compositional stochastic optimization algorithm, whatsoever. Using a non-trivial extension of Nesterov's classical results on Gaussian smoothing, we develop the Free-MESSAGEp algorithm from first principles, and show that it essentially solves a smoothed surrogate to the original problem, the former being a uniform approximation of the latter, in a useful, convenient sense. We then present a complete analysis of the Free-MESSAGEp algorithm, which establishes convergence in a user-tunable neighborhood of the optimal solutions of the original problem, as well as explicit convergence rates for both convex and strongly convex costs. Orderwise, and for fixed problem parameters, our results demonstrate no sacrifice in convergence speed compared to existing first-order methods, while striking a certain balance among the condition of the problem, its dimensionality, as well as the accuracy of the obtained results, naturally extending previous results in zeroth-order risk-neutral learning.

CvxPnPL: A Unified Convex Solution to the Absolute Pose Estimation Problem from Point and Line Correspondences.

Abstract: We present a new convex method to estimate 3D pose from mixed combinations of 2D-3D point and line correspondences, the Perspective-n-Points-and-Lines problem (PnPL). We merge the contributions of each point and line into a unified Quadratic Constrained Quadratic Problem (QCQP) and then relax it into a Semi Definite Program (SDP) through Shor's relaxation. This makes it possible to gracefully handle mixed configurations of points and lines. Furthermore, the proposed relaxation allows us to recover a finite number of solutions under ambiguous configurations. In such cases, the 3D pose candidates are found by further enforcing geometric constraints on the solution space and then retrieving such poses from the intersections of multiple quadrics. Experiments provide results in line with the best performing state of the art methods while providing the flexibility of solving for an arbitrary number of points and lines.

Convex trigonometry with applications to sub-Finsler geometry

Abstract: A new convenient method of describing flat convex compact sets is proposed. It generalizes classical trigonometric functions $\sin$ and $\cos$. Apparently, this method may be very useful for explicit description of solutions of optimal control problems with two-dimensional control. Using this method a series of sub-Finsler problems with two-dimensional control lying in an arbitrary convex set $\Omega$ is investigated. Namely, problems on the Heisenberg, Engel, and Cartan groups and also Grushin's and Martinet's cases are considered. A particular attention is paid to the case when $\Omega$ is a polygon.

Marked Length Spectral determination of analytic chaotic billiards with axial symmetries

Abstract: We consider billiards obtained by removing from the plane finitely many strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift, which provides a natural labeling of periodic orbits. We show that under suitable symmetry and genericity assumptions, the Marked Length Spectrum determines the geometry of the billiard table.

Quantitative Stability for Anisotropic Nearly Umbilical Hypersurfaces

Abstract: We prove qualitative and quantitative stability of the following rigidity theorem: the only anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $$n \ge 2$$ , $$p\in (1, \, +\infty )$$ and $$\Sigma $$ an n-dimensional, closed hypersurface in $$\mathbb {R}^{n+1}$$ , which is the boundary of a convex, open set. We show that if the $$L^p$$ -norm of the trace-free part of the anisotropic second fundamental form is small, then $$\Sigma $$ must be $$W^{2, \, p}$$ -close to the Wulff shape, with a quantitative estimate.