Isodiametry, Variance, and Regular Simplices from Particle Interactions

CHAPTER X - PROBLEMS IN HIGHER SPACES

The optimal transport problem aims at finding an optimal way to transport a given probability measure into another. In contrast to the well-known Monge–Kantorovich problem, the ramified optimal transportation problem aims at modeling a tree-typed branching transport network by an optimal transport path between two given probability measures. An essential feature of such a transport path is to favor group transportation in a large amount. In previous works, the author has studied the existence of optimal transport paths between probability measures as well as their interior regularity, that is away from the supports of the measures. Such an optimal transport path may be understood as a 1-dimensional rectifiable current that has boundary the difference of the 2 measures and that minimizes a suitable cost functional. In this article, we study the regularity of such an optimal transport path nearby its boundary. Motivated from observing the vein structure of a tree leaf, we show that each superlevel set of an optimal transport path is locally supported on a bi-Lipschitz graph, which is a finite union of bi-Lipschitz curves.

Agraph G is a diameter graph in ℝd if its vertex set is a finite subset in ℝd of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ℝ4 contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ℝ4, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1.

Given two probability measures $\mu$ and $\nu$ in “convex order” on $\mathbb{R}^{d}$, we study the profile of one-step martingale plans $\pi$ on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ that optimize the expected value of the modulus of their increment among all martingales having $\mu$ and $\nu$ as marginals. While there is a great deal of results for the real line (i.e., when $d=1$), much less is known in the richer and more delicate higher-dimensional case that we tackle in this paper. We show that many structural results can be obtained, provided the initial measure $\mu$ is absolutely continuous with respect to the Lebesgue measure. One such a property is that $\mu$-almost every $x$ in $\mathbb{R}^{d}$ is transported by the optimal martingale plan into a probability measure $\pi_{x}$ concentrated on the extreme points of the closed convex hull of its support. This will be established for the distance cost $c(x,y)=\vert x-y\vert $ in the two-dimensional case, and also for any $d\geq3$ as long as the marginals are in “subharmonic order.” In some cases, $\pi_{x}$ is supported on the vertices of a $k(x)$-dimensional polytope, such as when the target measure is discrete. Duality plays a crucial role in our approach, even though, in contrast to standard optimal transports, the dual extremal problem may not be attained in general. We show however that “martingale supporting” Borel subsets of $\mathbb{R}^{d}\times\mathbb{R}^{d}$ can be decomposed into a collection of mutually disjoint components by means of a “convex paving” of the source space, in such a way that when the martingale is optimal for a general cost function, each of the components then supports a restricted optimal martingale transport whose dual problem is attained. This decomposition is used to obtain structural results in cases where global duality is not attained. On the other hand, it shows that certain “optimal martingale supporting” Borel sets can be viewed as higher-dimensional versions of Nikodym-type sets. The paper focuses on the distance cost, but much of the results hold for general Lipschitz cost functions.

Given a determinate (multivariate) probability measure μ, we characterize Gaussian mixtures ν ϕ which minimize the Wasserstein distance W 2 (μ,ν ϕ ) to μ when the mixing probability measure ϕ on the parameters (m,Σ) of the Gaussians is supported on a compact set S. (i) We first show that such mixtures are optimal solutions of a particular optimal transport (OT) problem where the marginal ν ϕ of the OT problem is also unknown via the mixing measure variable ϕ. Next (ii) by using a well-known specific property of Gaussian measures, this optimal transport is then viewed as a Generalized Moment Problem (GMP) and if the set S of mixture parameters (m,Σ) is a basic compact semi-algebraic set, we provide a “mesh-free” numerical scheme to approximate as closely as desired the optimal distance by solving a hierarchy of semidefinite relaxations of increasing size. In particular, we neither assume that the mixing measure is finitely supported nor that the variance is the same for all components. If the original measure μ is not a Gaussian mixture with parameters (m,Σ)∈S, then a strictly positive distance is detected at a finite step of the hierarchy. If the original measure μ is a Gaussian mixture with parameters (m,Σ)∈S, then all semidefinite relaxations of the hierarchy have same zero optimal value. Moreover if the mixing measure is atomic with finite support, its components can sometimes be extracted from an optimal solution at some semidefinite relaxation of the hierarchy when Curto & Fialkow’s flatness condition holds for some moment matrix.

Empirical optimal transport (OT) plans and distances provide effective tools to compare and statistically match probability measures defined on a given ground space. Fundamental to this are distributional limit laws, and we derive a central limit theorem for the empirical OT distance of circular data. Our limit results require only mild assumptions in general and include prominent examples such as the von Mises or wrapped Cauchy family. Most notably, no assumptions are required when data are sampled from the probability measure to be compared with, which is in strict contrast to the real line. A bootstrap principle follows immediately as our proof relies on Hadamard differentiability of the OT functional. This paves the way for a variety of statistical inference tasks and is exemplified for asymptotic OT-based goodness of fit testing for circular distributions. We discuss numerical implementation, consistency and investigate its statistical power. For testing uniformity, it turns out that this approach performs particularly well for unimodal alternatives and is almost as powerful as Rayleigh's test, the most powerful invariant test for von Mises alternatives. For regimes with many modes, the circular OT test is less powerful which is explained by the shape of the corresponding transport plan.

D. L. Carleton is a former staff member of Santa Fe Junior College in Gainesville, Florida, where he had taught mathematics from 1969 to 1971. He is continuing his studies now at the University of Texas at Austin, where he is doing research in the relationships of mathematics and art. In the plane, we know that the isosceles triangle of maximum area which can be inscribed in a circle of given radius is actually an equilateral triangle. Similarly, the rectangle of maximum area which can be inscribed in a circle of given radius is a square. In these two cases, the maximum areas are defined by regular polygons. When we consider this observation together with the fact that these two polygons generate a right circular cone and a right circular cylinder, an interesting question arises. Can the right circular cone of maximum volume and the right circular cylinder of maximum volume which can be inscribed in a sphere of given radius be defined in terms of regular polygons? The purpose of this article is to answer this question in the affirmative. In particular, this article will: (1) use the regular tetrahedron to define the right circular cone of maximum volume which can be inscribed in a sphere of given radius; and (2) use the regular octahedron to define the right circular cylinder of maximum volume which can be inscribed in a sphere of given radius. We will verify statement (1) by showing that the vertex angle of the cone of maximum volume is the dihedral angle of the regular tetrahedron. One way to find the value of this angle is to express the volume of the cone as a function of its vertex angle and the radius of the sphere, where the radius is treated as a constant. Figure 1 is a cross-section of a right circular cone inscribed in a sphere, where the vertex angle of the cone is 0, the center of the sphere is 0, and the length of the radius of the sphere is R. The radius of the base of the cone and the height of the cone are AC and BC, respectively. Since the triangle AOB is an isosceles triangle, the angle BA O is equal to 0 /2 which means that the angle A OC is equal to

Traditional geodetic network optimization deals with static and discrete control points. The modern space geodetic network is, on the other hand, composed of moving control points in space (satellites) and on the Earth (ground stations). The network configuration composed of these facilities is essentially dynamic and continuous. Moreover, besides the position parameter which needs to be estimated, other geophysical information or signals can also be extracted from the continuous observations. The dynamic (continuous) configuration of the space network determines whether a particular frequency of signals can be identified by this system. In this paper, we employ the functional analysis and graph theory to study the dynamic configuration of space geodetic networks, and mainly focus on the optimal estimation of the position and clock-offset parameters. The principle of the D-optimization is introduced in the Hilbert space after the concept of the traditional discrete configuration is generalized from the finite space to the infinite space. It shows that the D-optimization developed in the discrete optimization is still valid in the dynamic configuration optimization, and this is attributed to the natural generalization of least squares from the Euclidean space to the Hilbert space. Then, we introduce the principle of D-optimality invariance under the combination operation and rotation operation, and propose some D-optimal simplex dynamic configurations: (1) (Semi) circular configuration in 2-dimensional space; (2) the D-optimal cone configuration and D-optimal helical configuration which is close to the GPS constellation in 3-dimensional space. The initial design of GPS constellation can be approximately treated as a combination of 24 D-optimal helixes by properly adjusting the ascending node of different satellites to realize a so-called Walker constellation. In the case of estimating the receiver clock-offset parameter, we show that the circular configuration, the symmetrical cone configuration and helical curve configuration are still D-optimal. It shows that the given total observation time determines the optimal frequency (repeatability) of moving known points and vice versa, and one way to improve the repeatability is to increase the rotational speed. Under the Newton’s law of motion, the frequency of satellite motion determines the orbital altitude. Furthermore, we study three kinds of complex dynamic configurations, one of which is the combination of D-optimal cone configurations and a so-called Walker constellation composed of D-optimal helical configuration, the other is the nested cone configuration composed of n cones, and the last is the nested helical configuration composed of n orbital planes. It shows that an effective way to achieve high coverage is to employ the configuration composed of a certain number of moving known points instead of the simplex configuration (such as D-optimal helical configuration), and one can use the D-optimal simplex solutions or D-optimal complex configurations in any combination to achieve powerful configurations with flexile coverage and flexile repeatability. Alternately, how to optimally generate and assess the discrete configurations sampled from the continuous one is discussed. The proposed configuration optimization framework has taken the well-known regular polygons (such as equilateral triangle and quadrangular) in two-dimensional space and regular polyhedrons (regular tetrahedron, cube, regular octahedron, regular icosahedron, or regular dodecahedron) into account. It shows that the conclusions made by the proposed technique are more general and no longer limited by different sampling schemes. By the conditional equation of D-optimal nested helical configuration, the relevance issues of GNSS constellation optimization are solved and some examples are performed by GPS constellation to verify the validation of the newly proposed optimization technique. The proposed technique is potentially helpful in maintenance and quadratic optimization of single GNSS of which the orbital inclination and the orbital altitude change under the precession, as well as in optimally nesting GNSSs to perform global homogeneous coverage of the Earth.

Use of Decoys to Optimize an All-Atom Force Field Including Hydration

We describe an active contour based local energy minimization point distribution, behavior of orientation, relaxation iterative algorithm that estimates feature points of static target in image sequence. In our approach, we build a contour model of a target to get some of low-energy points kernels. The use of snake based line model results in more reliable convergence of the point local energy minimization. The algorithm uses auto-relation relation to give the behavior of orientation in a local kernel's window. It uses local window-based probability method to refine the current corresponding relations of scene kernels. Results are illustrated on real outdoor image sequence.

We introduce a new paradigm, `measure synchronization', for synchronizing graphs with measure-valued edges. We formulate this problem as maximization of the cycle-consistency in the space of probability measures over relative rotations. In particular, we aim at estimating marginal distributions of absolute orientations by synchronizing the `conditional' ones, which are defined on the Riemannian manifold of quaternions. Such graph optimization on distributions-on-manifolds enables a natural treatment of multimodal hypotheses, ambiguities and uncertainties arising in many computer vision applications such as SLAM, SfM, and object pose estimation. We first formally define the problem as a generalization of the classical rotation graph synchronization, where in our case the vertices denote probability measures over rotations. We then measure the quality of the synchronization by using Sinkhorn divergences, which reduces to other popular metrics such as Wasserstein distance or the maximum mean discrepancy as limit cases. We propose a nonparametric Riemannian particle optimization approach to solve the problem. Even though the problem is non-convex, by drawing a connection to the recently proposed sparse optimization methods, we show that the proposed algorithm converges to the global optimum in a special case of the problem under certain conditions. Our qualitative and quantitative experiments show the validity of our approach and we bring in new perspectives to the study of synchronization.

Nous analysons le probleme du transport optimal entre deux mesures aleatoires et equivariantes et demontrons des conditions qui garantissent l’existence d’une solution de type Monge. En outre, nous demontrons que l’equivariance apparait naturellement dans ce contexte en prouvant que les couplages optimaux classiques dans des ensembles bornes convergent vers le couplage optimal dans tout l’espace. Finalement nous demontrons des conditions suffisantes pour que le cout au sens $L^{p}$ soit fini en introduisant une metrique appropriee.

Natural product and synthetic macrocycles are chemically and topologically diverse. An efficient, accurate, and general method for generating macrocycle conformations would enable structure-based design of macrocycle drugs or host-guest complexes. Computational sampling also provides insight into transiently populated states, complementing crystallographic and NMR data. Here, we report a new algorithm, BRIKARD, that addresses this challenge through computational algebraic geometry and inverse kinematics together with local energy minimization. BRIKARD is demonstrated on 67 diverse macrocycles with structural data, encompassing various ring topologies. We find this approach enumerates diverse structures with macrocyclic RMSD < 1.0 Å to the experimental conformation for 85% of our data set in contrast to success rates of 67-75% with other approaches, while for the subset of 21 more challenging compounds in the data set, these rates are 57% and 10-29%, respectively. Because the algorithm can be efficiently run in parallel on many processors, exhaustive conformational sampling of complex cycles can be obtained in minutes rather than hours: with a 40 processor implementation protocol, BRIKARD samples the conformational diversity of a potential energy landscape in a median of 1.3 minutes of wallclock time, much faster than 3.1-10.3 hours necessary with current programs. By rigorously testing BRIKARD on a broad range of scaffolds with highly complex ring systems, we push the frontiers of macrocycle sampling to encompass multiring compounds, including those with more than 50 ring atoms and up to seven interlaced flexible rings.

Structural studies of vinblastine alkaloids by exciton coupled circular dichroism

In the past few years, 3D face modeling has gained significant attention. Reliable modeling of the face is necessary for good performance of a 3D assisted face recognition system. In this paper, we model 3D human face from binocular stereo images. The stereo matching problem has been formulated as an energy minimization problem that progressively propagates depth from reliable regions to unreliable regions by an annealing scheme. The concept of smooth 2D grid is used to enable regularizing the final depth solution. In the energy equation, area based matching is used for the data term. The smoothness term relies on annealing. The 2D grid facilitates the estimation of the smoothness parameter, from a correlation profile obtained from neighborhood of the grid node. It is evident from 3D modeling results that the proposed algorithm performs well on images of human faces.