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Toric and Non-toric Bayesian Networks 环向和非环向贝叶斯网络
IF 1.2 2区 数学
SIAM Journal on Applied Algebra and Geometry Pub Date : 2022-08-12 DOI: 10.1137/22M1515690
Lisa Nicklasson
{"title":"Toric and Non-toric Bayesian Networks","authors":"Lisa Nicklasson","doi":"10.1137/22M1515690","DOIUrl":"https://doi.org/10.1137/22M1515690","url":null,"abstract":"In this paper we study Bayesian networks from a commutative algebra perspective. We characterize a class of toric Bayesian nets, and provide the first example of a Bayesian net which is proved non-toric under any linear change of variables. Concerning the class of toric Bayesian nets, we study their quadratic relations and prove a conjecture by Garcia, Stillman, and Sturmfels for this class. In addition, we give a necessary condition on the underlying directed acyclic graph for when all relations are quadratic.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74094068","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: 1
Automorphisms of Rank-One Generated Hyperbolicity Cones and Their Derivative Relaxations 秩一生成双曲锥的自同构及其导数松弛
IF 1.2 2区 数学
SIAM Journal on Applied Algebra and Geometry Pub Date : 2022-07-25 DOI: 10.1137/22m1513964
Masaru Ito, Bruno F. Lourenço
{"title":"Automorphisms of Rank-One Generated Hyperbolicity Cones and Their Derivative Relaxations","authors":"Masaru Ito, Bruno F. Lourenço","doi":"10.1137/22m1513964","DOIUrl":"https://doi.org/10.1137/22m1513964","url":null,"abstract":"A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank one, where the rank is computed with respect to the underlying hyperbolic polynomial. This is a natural class of hyperbolicity cones which are strictly more general than the ROG spectrahedral cones. In this work, we present a study of the automorphisms of ROG hyperbolicity cones and their derivative relaxations. One of our main results states that the automorphisms of the derivative relaxations are exactly the automorphisms of the original cone fixing a certain direction. As an application, we completely determine the automorphisms of the derivative relaxations of the nonnegative orthant and of the cone of positive semidefinite matrices. More generally, we also prove relations between the automorphisms of a spectral cone and the underlying permutation-invariant set, which might be of independent interest.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88322235","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: 0
Bayesian Integrals on Toric Varieties 环型上的贝叶斯积分
IF 1.2 2区 数学
SIAM Journal on Applied Algebra and Geometry Pub Date : 2022-04-13 DOI: 10.1137/22M1490569
M. Borinsky, Anna-Laura Sattelberger, B. Sturmfels, Simon Telen
{"title":"Bayesian Integrals on Toric Varieties","authors":"M. Borinsky, Anna-Laura Sattelberger, B. Sturmfels, Simon Telen","doi":"10.1137/22M1490569","DOIUrl":"https://doi.org/10.1137/22M1490569","url":null,"abstract":"We explore the positive geometry of statistical models in the setting of toric varieties. Our focus lies on models for discrete data that are parameterized in terms of Cox coordinates. We develop a geometric theory for computations in Bayesian statistics, such as evaluating marginal likelihood integrals and sampling from posterior distributions. These are based on a tropical sampling method for evaluating Feynman integrals in physics. We here extend that method from projective spaces to arbitrary toric varieties.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78099979","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: 3
Line Multiview Varieties 多视图品种
IF 1.2 2区 数学
SIAM Journal on Applied Algebra and Geometry Pub Date : 2022-03-03 DOI: 10.1137/22m1482263
Paul Breiding, Felix Rydell, Elima Shehu, Ang'elica Torres
{"title":"Line Multiview Varieties","authors":"Paul Breiding, Felix Rydell, Elima Shehu, Ang'elica Torres","doi":"10.1137/22m1482263","DOIUrl":"https://doi.org/10.1137/22m1482263","url":null,"abstract":"We present an algebraic study of line correspondences for pinhole cameras, in contrast to the thoroughly studied point correspondences. We define the line multiview variety as the Zariski closure of the image of the map projecting lines in 3-space to tuples of image lines in 2-space. We prove that in the case of generic camera matrices the line multiview variety is a determinantal variety and we provide a complete set-theoretic description for any camera arrangement. We investigate basic properties of this variety such as dimension, smoothness, and multidegree. Finally, we give experimental results for the Euclidean distance degree and robustness under noise for the triangulation of lines.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91147816","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: 10
Erratum for "Morphisms of Neural Codes" 《神经编码的形态》勘误表
IF 1.2 2区 数学
SIAM Journal on Applied Algebra and Geometry Pub Date : 2022-03-01 DOI: 10.1137/21m1450690
R. A. Jeffs
{"title":"Erratum for \"Morphisms of Neural Codes\"","authors":"R. A. Jeffs","doi":"10.1137/21m1450690","DOIUrl":"https://doi.org/10.1137/21m1450690","url":null,"abstract":"","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80616355","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: 0
Flat tori with large Laplacian eigenvalues in dimensions up to eight 具有大拉普拉斯特征值的平面环面，维数可达8
IF 1.2 2区 数学
SIAM Journal on Applied Algebra and Geometry Pub Date : 2022-02-16 DOI: 10.1137/22m1478823
C. Kao, B. Osting, J. C. Turner
{"title":"Flat tori with large Laplacian eigenvalues in dimensions up to eight","authors":"C. Kao, B. Osting, J. C. Turner","doi":"10.1137/22m1478823","DOIUrl":"https://doi.org/10.1137/22m1478823","url":null,"abstract":"We consider the optimization problem of maximizing the $k$-th Laplacian eigenvalue, $lambda_{k}$, over flat $d$-dimensional tori of fixed volume. For $k=1$, this problem is equivalent to the densest lattice sphere packing problem. For larger $k$, this is equivalent to the NP-hard problem of finding the $d$-dimensional (dual) lattice with longest $k$-th shortest lattice vector. As a result of extensive computations, for $d leq 8$, we obtain a sequence of flat tori, $T_{k,d}$, each of volume one, such that the $k$-th Laplacian eigenvalue of $T_{k,d}$ is very large; for each (finite) $k$ the $k$-th eigenvalue exceeds the value in (the $kto infty$ asymptotic) Weyl's law by a factor between 1.54 and 2.01, depending on the dimension. Stationarity conditions are derived and numerically verified for $T_{k,d}$ and we describe the degeneration of the tori as $k to infty$.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77110368","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: 0
The Polyhedral Geometry of Pivot Rules and Monotone Paths 轴心规则和单调路径的多面体几何
IF 1.2 2区 数学
SIAM Journal on Applied Algebra and Geometry Pub Date : 2022-01-13 DOI: 10.1137/22m1475910
Alexander E. Black, J. D. De Loera, Niklas Lütjeharms, Raman Sanyal
{"title":"The Polyhedral Geometry of Pivot Rules and Monotone Paths","authors":"Alexander E. Black, J. D. De Loera, Niklas Lütjeharms, Raman Sanyal","doi":"10.1137/22m1475910","DOIUrl":"https://doi.org/10.1137/22m1475910","url":null,"abstract":"Motivated by the analysis of the performance of the simplex method we study the behavior of families of pivot rules of linear programs. We introduce normalized-weight pivot rules which are fundamental for the following reasons: First, they are memory-less, in the sense that the pivots are governed by local information encoded by an arborescence. Second, many of the most used pivot rules belong to that class, and we show this subclass is critical for understanding the complexity of all pivot rules. Finally, normalized-weight pivot rules can be parametrized in a natural continuous manner. We show the existence of two polytopes, the pivot rule polytopes and the neighbotopes, that capture the behavior of normalized-weight pivot rules on polytopes and linear programs. We explain their face structure in terms of multi-arborescences. We compute upper bounds on the number of coherent arborescences, that is, vertices of our polytopes. Beyond optimization, our constructions provide new perspectives on classical geometric combinatorics. We introduce a normalized-weight pivot rule, we call the max-slope pivot rule which generalizes the shadow-vertex pivot rule. The corresponding pivot rule polytopes and neighbotopes refine monotone path polytopes of Billera--Sturmfels. Moreover special cases of our polytopes yield permutahedra, associahedra, and multiplihedra. For the greatest improvement pivot rules we draw connections to sweep polytopes and polymatroids.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64315867","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: 3
Learning Polytopes with Fixed Facet Directions 学习多边形与固定面方向
IF 1.2 2区 数学
SIAM Journal on Applied Algebra and Geometry Pub Date : 2022-01-10 DOI: 10.1137/22m1481695
M. Dostert, Katharina Jochemko
{"title":"Learning Polytopes with Fixed Facet Directions","authors":"M. Dostert, Katharina Jochemko","doi":"10.1137/22m1481695","DOIUrl":"https://doi.org/10.1137/22m1481695","url":null,"abstract":"We consider the task of reconstructing polytopes with fixed facet directions from finitely many support function evaluations. We show that for a fixed simplicial normal fan the least-squares estimate is given by a convex quadratic program. We study the geometry of the solution set and give a combinatorial characterization for the uniqueness of the reconstruction in this case. We provide an algorithm that, under mild assumptions, converges to the unknown input shape as the number of noisy support function evaluations increases. We also discuss limitations of our results if the restriction on the normal fan is removed.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83378009","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: 0
The Subrank of a Complex Symmetric Tensor Can Exceed its Symmetric Subrank 复对称张量的子分支可以超越它的对称子分支
IF 1.2 2区 数学
SIAM Journal on Applied Algebra and Geometry Pub Date : 2022-01-01 DOI: 10.1137/21m1465494
Y. Shitov
{"title":"The Subrank of a Complex Symmetric Tensor Can Exceed its Symmetric Subrank","authors":"Y. Shitov","doi":"10.1137/21m1465494","DOIUrl":"https://doi.org/10.1137/21m1465494","url":null,"abstract":"","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89226789","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: 1
Symmetrically Colored Gaussian Graphical Models with Toric Vanishing Ideals 具有环面消失理想的对称彩色高斯图形模型
IF 1.2 2区 数学
SIAM Journal on Applied Algebra and Geometry Pub Date : 2021-11-29 DOI: 10.1137/21M1466943
Jane Ivy Coons, Aida Maraj, Pratik Misra, Miruna-Stefana Sorea
{"title":"Symmetrically Colored Gaussian Graphical Models with Toric Vanishing Ideals","authors":"Jane Ivy Coons, Aida Maraj, Pratik Misra, Miruna-Stefana Sorea","doi":"10.1137/21M1466943","DOIUrl":"https://doi.org/10.1137/21M1466943","url":null,"abstract":"A colored Gaussian graphical model is a linear concentration model in which equalities among the concentrations are specified by a coloring of an underlying graph. The model is called RCOP if this coloring is given by the edge and vertex orbits of a subgroup of the automorphism group of the graph. We show that RCOP Gaussian graphical models on block graphs are toric in the space of covariance matrices and we describe Markov bases for them. To this end, we learn more about the combinatorial structure of these models and their connection with Jordan algebras.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78924586","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: 1
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