{"title":"OUP accepted manuscript","authors":"","doi":"10.1093/imaiai/iaac002","DOIUrl":"https://doi.org/10.1093/imaiai/iaac002","url":null,"abstract":"","PeriodicalId":45437,"journal":{"name":"Information and Inference-A Journal of the Ima","volume":"35 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72863158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"OUP accepted manuscript","authors":"","doi":"10.1093/imaiai/iaac011","DOIUrl":"https://doi.org/10.1093/imaiai/iaac011","url":null,"abstract":"","PeriodicalId":45437,"journal":{"name":"Information and Inference-A Journal of the Ima","volume":"37 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80590832","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"OUP accepted manuscript","authors":"","doi":"10.1093/imaiai/iaac015","DOIUrl":"https://doi.org/10.1093/imaiai/iaac015","url":null,"abstract":"","PeriodicalId":45437,"journal":{"name":"Information and Inference-A Journal of the Ima","volume":"11 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88692244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"OUP accepted manuscript","authors":"","doi":"10.1093/imaiai/iaac009","DOIUrl":"https://doi.org/10.1093/imaiai/iaac009","url":null,"abstract":"","PeriodicalId":45437,"journal":{"name":"Information and Inference-A Journal of the Ima","volume":"16 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89527684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"OUP accepted manuscript","authors":"","doi":"10.1093/imaiai/iaac014","DOIUrl":"https://doi.org/10.1093/imaiai/iaac014","url":null,"abstract":"","PeriodicalId":45437,"journal":{"name":"Information and Inference-A Journal of the Ima","volume":"84 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78926489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"OUP accepted manuscript","authors":"","doi":"10.1093/imaiai/iaac006","DOIUrl":"https://doi.org/10.1093/imaiai/iaac006","url":null,"abstract":"","PeriodicalId":45437,"journal":{"name":"Information and Inference-A Journal of the Ima","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90201390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"OUP accepted manuscript","authors":"","doi":"10.1093/imaiai/iaac017","DOIUrl":"https://doi.org/10.1093/imaiai/iaac017","url":null,"abstract":"","PeriodicalId":45437,"journal":{"name":"Information and Inference-A Journal of the Ima","volume":"170 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81124091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"OUP accepted manuscript","authors":"","doi":"10.1093/imaiai/iaab028","DOIUrl":"https://doi.org/10.1093/imaiai/iaab028","url":null,"abstract":"","PeriodicalId":45437,"journal":{"name":"Information and Inference-A Journal of the Ima","volume":"57 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80501605","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Carlos Am'endola, M. Drton, Alexandros Grosdos, R. Homs, Elina Robeva
In this paper, we study linear non-Gaussian graphical models from the perspective of algebraic statistics. These are acyclic causal models in which each variable is a linear combination of its direct causes and independent noise. The underlying directed causal graph can be identified uniquely via the set of second and third-order moments of all random vectors that lie in the corresponding model. Our focus is on finding the algebraic relations among these moments for a given graph. We show that when the graph is a polytree, these relations form a toric ideal. We construct explicit trek-matrices associated to 2-treks and 3-treks in the graph. Their entries are covariances and third-order moments and their $2$-minors define our model set-theoretically. Furthermore, we prove that their 2-minors also generate the vanishing ideal of the model. Finally, we describe the polytopes of third-order moments and the ideals for models with hidden variables.
{"title":"Third-order moment varieties of linear non-Gaussian graphical models","authors":"Carlos Am'endola, M. Drton, Alexandros Grosdos, R. Homs, Elina Robeva","doi":"10.1093/imaiai/iaad007","DOIUrl":"https://doi.org/10.1093/imaiai/iaad007","url":null,"abstract":"\u0000 In this paper, we study linear non-Gaussian graphical models from the perspective of algebraic statistics. These are acyclic causal models in which each variable is a linear combination of its direct causes and independent noise. The underlying directed causal graph can be identified uniquely via the set of second and third-order moments of all random vectors that lie in the corresponding model. Our focus is on finding the algebraic relations among these moments for a given graph. We show that when the graph is a polytree, these relations form a toric ideal. We construct explicit trek-matrices associated to 2-treks and 3-treks in the graph. Their entries are covariances and third-order moments and their $2$-minors define our model set-theoretically. Furthermore, we prove that their 2-minors also generate the vanishing ideal of the model. Finally, we describe the polytopes of third-order moments and the ideals for models with hidden variables.","PeriodicalId":45437,"journal":{"name":"Information and Inference-A Journal of the Ima","volume":"47 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90964388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The standard simplex in $mathbb{R}^{n}$, also known as the probability simplex, is the set of nonnegative vectors whose entries sum up to 1. It frequently appears as a constraint in optimization problems that arise in machine learning, statistics, data science, operations research and beyond. We convert the standard simplex to the unit sphere and thus transform the corresponding constrained optimization problem into an optimization problem on a simple, smooth manifold. We show that Karush-Kuhn-Tucker points and strict-saddle points of the minimization problem on the standard simplex all correspond to those of the transformed problem, and vice versa. So, solving one problem is equivalent to solving the other problem. Then, we propose several simple, efficient and projection-free algorithms using the manifold structure. The equivalence and the proposed algorithm can be extended to optimization problems with unit simplex, weighted probability simplex or $ell _{1}$-norm sphere constraints. Numerical experiments between the new algorithms and existing ones show the advantages of the new approach. Open source code is available at https://github.com/DanielMckenzie/HadRGD.
$mathbb{R}^{n}$中的标准单纯形，也称为概率单纯形，是其项之和为1的非负向量的集合。它经常作为约束出现在机器学习、统计学、数据科学、运筹学等领域的优化问题中。我们将标准单纯形转化为单位球，从而将相应的约束优化问题转化为简单光滑流形上的优化问题。证明了标准单纯形上最小化问题的Karush-Kuhn-Tucker点和严格鞍点都对应于变换问题的Karush-Kuhn-Tucker点和严格鞍点，反之亦然。所以，解决一个问题等于解决另一个问题。然后，我们利用流形结构提出了几种简单、高效、无投影的算法。该等价性和所提出的算法可以推广到具有单位单纯形、加权概率单纯形或$ well _{1}$-范数球面约束的优化问题。通过与现有算法的对比实验，证明了新算法的优越性。开源代码可从https://github.com/DanielMckenzie/HadRGD获得。
{"title":"From the simplex to the sphere: faster constrained optimization using the Hadamard parametrization","authors":"Qiuwei Li, Daniel Mckenzie, W. Yin","doi":"10.1093/imaiai/iaad017","DOIUrl":"https://doi.org/10.1093/imaiai/iaad017","url":null,"abstract":"\u0000 The standard simplex in $mathbb{R}^{n}$, also known as the probability simplex, is the set of nonnegative vectors whose entries sum up to 1. It frequently appears as a constraint in optimization problems that arise in machine learning, statistics, data science, operations research and beyond. We convert the standard simplex to the unit sphere and thus transform the corresponding constrained optimization problem into an optimization problem on a simple, smooth manifold. We show that Karush-Kuhn-Tucker points and strict-saddle points of the minimization problem on the standard simplex all correspond to those of the transformed problem, and vice versa. So, solving one problem is equivalent to solving the other problem. Then, we propose several simple, efficient and projection-free algorithms using the manifold structure. The equivalence and the proposed algorithm can be extended to optimization problems with unit simplex, weighted probability simplex or $ell _{1}$-norm sphere constraints. Numerical experiments between the new algorithms and existing ones show the advantages of the new approach. Open source code is available at https://github.com/DanielMckenzie/HadRGD.","PeriodicalId":45437,"journal":{"name":"Information and Inference-A Journal of the Ima","volume":"66 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78101118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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