SIAM Journal on Applied Algebra and Geometry最新文献_第6页

Computing the Conformal Barycenter 计算共形质心

IF 1.2 2区数学

SIAM Journal on Applied Algebra and Geometry Pub Date : 2020-04-08 DOI: 10.1137/21M1449282

J. Cantarella, Henrik Schumacher

{"title":"Computing the Conformal Barycenter","authors":"J. Cantarella, Henrik Schumacher","doi":"10.1137/21M1449282","DOIUrl":"https://doi.org/10.1137/21M1449282","url":null,"abstract":"The conformal barycenter of a point cloud on the sphere at infinity of the Poincare ball model of hyperbolic space is a hyperbolic analogue of the geometric median of a point cloud in Euclidean space. It was defined by Douady and Earle as part of a construction of a conformally natural way to extend homeomorphisms of the circle to homeomorphisms of the disk, and it plays a central role in Millson and Kapovich's model of the configuration space of cyclic linkages with fixed edgelengths. \u0000In this paper we consider the problem of computing the conformal barycenter. Abikoff and Ye have given an iterative algorithm for measures on $mathbb{S}^1$ which is guaranteed to converge. We analyze Riemannian versions of Newton's method computed in the intrinsic geometry of the Poincare ball model. We give Newton-Kantorovich (NK) conditions under which we show that Newton's method with fixed step size is guaranteed to converge quadratically to the conformal barycenter for measures on any $mathbb{S}^d$ (including infinite-dimensional spheres). For measures given by $n$ atoms on a finite dimensional sphere which obey the NK conditions, we give an explicit linear bound on the computation time required to approximate the conformal barycenter to fixed error. We prove that our NK conditions hold for all but exponentially few $n$ atom measures. For all measures with a unique conformal barycenter we show that a regularized Newton's method with line search will always converge (eventually superlinearly) to the conformal barycenter. Though we do not have hard time bounds for this algorithm, experiments show that it is extremely efficient in practice and in particular much faster than the Abikoff-Ye iteration.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"70 1","pages":"503-530"},"PeriodicalIF":1.2,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87427417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2

Invariant Theory and Scaling Algorithms for Maximum Likelihood Estimation 极大似然估计的不变量理论和缩放算法

IF 1.2 2区数学

SIAM Journal on Applied Algebra and Geometry Pub Date : 2020-03-30 DOI: 10.1137/20M1328932

Carlos Am'endola, Kathlén Kohn, Philipp Reichenbach, A. Seigal

引用次数: 29

Abstract Vergleichsstellensätze for Preordered Semifields and Semirings I 优秀的闪纯粹和西蒙

IF 1.2 2区数学

SIAM Journal on Applied Algebra and Geometry Pub Date : 2020-03-30 DOI: 10.1137/22M1498413

T. Fritz

引用次数: 6

Multi-Trek Separation in Linear Structural Equation Models 线性结构方程模型中的多重分离

IF 1.2 2区数学

SIAM Journal on Applied Algebra and Geometry Pub Date : 2020-01-28 DOI: 10.1137/20M1316470

Elina Robeva, Jean-Baptiste Seby

引用次数: 5

The Distributions of Functions Related to Parametric Integer Optimization 参数整数优化相关函数的分布

IF 1.2 2区数学

SIAM Journal on Applied Algebra and Geometry Pub Date : 2019-12-31 DOI: 10.1137/19M1275954

Timm Oertel, Joseph Paat, R. Weismantel

引用次数: 16

Representations of Partial Leaf Sets in Phylogenetic Tree Space 系统发育树空间中部分叶集的表示

IF 1.2 2区数学

SIAM Journal on Applied Algebra and Geometry Pub Date : 2019-12-10 DOI: 10.1137/18m1235855

Gillian Grindstaff, Megan Owen

引用次数: 3

Whitney Numbers of Combinatorial Geometries and Higher-Weight Dowling Lattices 组合几何的Whitney数与高权重Dowling格

IF 1.2 2区数学

SIAM Journal on Applied Algebra and Geometry Pub Date : 2019-09-23 DOI: 10.1137/20m1382635

A. Ravagnani

{"title":"Whitney Numbers of Combinatorial Geometries and Higher-Weight Dowling Lattices","authors":"A. Ravagnani","doi":"10.1137/20m1382635","DOIUrl":"https://doi.org/10.1137/20m1382635","url":null,"abstract":"We study the Whitney numbers of the first kind of combinatorial geometries. The first part of the paper is devoted to general results relating the Mobius functions of nested atomistic lattices, extending some classical theorems in combinatorics. We then specialize our results to restriction geometries, i.e., to sublattices $mathcal{L}(A)$ of the lattice of subspaces of an $mathbb{F}_q$-linear space, say $X$, generated by a set of projective points $A subseteq X$. In this context, we introduce the notion of subspace distribution, and show that partial knowledge of the latter is equivalent to partial knowledge of the Whitney numbers of $mathcal{L}(A)$. This refines a classical result by Dowling. \u0000The most interesting applications of our results are to be seen in the theory of higher-weight Dowling lattices (HWDLs), to which we dovote the second and most substantive part of the paper. These combinatorial geometries were introduced by Dowling in 1971 in connection with fundamental problems in coding theory, and further studied, among others, by Zaslavsky, Bonin, Kung, Brini, and Games. To date, still very little is known about these lattices. In particular, the techniques to compute their Whitney numbers have not been discovered yet. In this paper, we bring forward the theory of HWDLs, computing their Whitney numbers for new infinite families of parameters. Moreover, we show that the second Whitney numbers of HWDLs are polynomials in the underlying field size $q$, whose coefficients are expressions involving the Bernoulli numbers. This reveals a new link between combinatorics, coding theory, and number theory. We also study the asymptotics of the Whitney numbers of HWDLs as the field size grows, giving upper bounds and exact estimates in some cases. In passing, we obtain new results on the density functions of error-correcting codes.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"66 1","pages":"156-189"},"PeriodicalIF":1.2,"publicationDate":"2019-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72788425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 6

p-Adic Integral Geometry p进积分几何

IF 1.2 2区数学

SIAM Journal on Applied Algebra and Geometry Pub Date : 2019-08-13 DOI: 10.1137/19m1284737

Avinash Kulkarni, A. Lerário

引用次数: 16

Convex Algebraic Geometry of Curvature Operators 曲率算子的凸代数几何

IF 1.2 2区数学

SIAM Journal on Applied Algebra and Geometry Pub Date : 2019-08-10 DOI: 10.1137/20M1350777

R. G. Bettiol, Mario Kummer, R. Mendes

引用次数: 9

The Algebraic Boundary of the Sonc-Cone 声波锥的代数边界