{"title":"Second-Order Cone Representation for Convex Sets in the Plane","authors":"C. Scheiderer","doi":"10.1137/20M133717X","DOIUrl":"https://doi.org/10.1137/20M133717X","url":null,"abstract":"","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78032444","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}
Isometries and their induced symmetries are ubiquitous in the world. Taking a computational perspective, this paper considers isometries of Z (since values are discrete in digital computers), and tackles the problem of orbit computation under various isometry subgroup actions on Z. Rather than just conceptually, we aim for a practical algorithm that can partition any finite subset of Z based on the orbit relation. In this paper, instead of all subgroups of isometries, we focus on a special class of subgroups, namely atomically generated subgroups. This newly introduced notion is key to inheriting the semidirect-product structure from the whole group of isometries, and in turn, the semidirect-product structure is key to our proposed algorithm for efficient orbit computation.
{"title":"Orbit Computation for Atomically Generated Subgroups of Isometries of Zn","authors":"Haizi Yu, Igor Mineyev, L. Varshney","doi":"10.1137/20M1375127","DOIUrl":"https://doi.org/10.1137/20M1375127","url":null,"abstract":"Isometries and their induced symmetries are ubiquitous in the world. Taking a computational perspective, this paper considers isometries of Z (since values are discrete in digital computers), and tackles the problem of orbit computation under various isometry subgroup actions on Z. Rather than just conceptually, we aim for a practical algorithm that can partition any finite subset of Z based on the orbit relation. In this paper, instead of all subgroups of isometries, we focus on a special class of subgroups, namely atomically generated subgroups. This newly introduced notion is key to inheriting the semidirect-product structure from the whole group of isometries, and in turn, the semidirect-product structure is key to our proposed algorithm for efficient orbit computation.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73151314","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}
M. Bender, J. Faugère, Angelos Mantzaflaris, Elias P. Tsigaridas
Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula when we can express it as a determinant of a matrix whose elements are the coefficients of the input polynomials. We study the resultant in the context of mixed multilinear polynomial systems, that is multilinear systems with polynomials having different supports, on which determinantal formulas were not known. We construct determinantal formulas for two kind of multilinear systems related to the Multiparameter Eigenvalue Problem (MEP): first, when the polynomials agree in all but one block of variables; second, when the polynomials are bilinear with different supports, related to a bipartite graph. We use the Weyman complex to construct Koszul-type determinantal formulas that generalize Sylvester-type formulas. We can use the matrices associated to these formulas to solve square systems without computing the resultant. The combination of the resultant matrices with the eigenvalue and eigenvector criterion for polynomial systems leads to a new approach for solving MEP.
{"title":"Koszul-type determinantal formulas for families of mixed multilinear systems","authors":"M. Bender, J. Faugère, Angelos Mantzaflaris, Elias P. Tsigaridas","doi":"10.1137/20m1332190","DOIUrl":"https://doi.org/10.1137/20m1332190","url":null,"abstract":"Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula when we can express it as a determinant of a matrix whose elements are the coefficients of the input polynomials. We study the resultant in the context of mixed multilinear polynomial systems, that is multilinear systems with polynomials having different supports, on which determinantal formulas were not known. We construct determinantal formulas for two kind of multilinear systems related to the Multiparameter Eigenvalue Problem (MEP): first, when the polynomials agree in all but one block of variables; second, when the polynomials are bilinear with different supports, related to a bipartite graph. We use the Weyman complex to construct Koszul-type determinantal formulas that generalize Sylvester-type formulas. We can use the matrices associated to these formulas to solve square systems without computing the resultant. The combination of the resultant matrices with the eigenvalue and eigenvector criterion for polynomial systems leads to a new approach for solving MEP.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77818604","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}
We study the maximum likelihood (ML) degree of linear concentration models in algebraic statistics. We relate it to an intersection problem on the variety of complete quadrics. This allows us to provide an explicit, basic, albeit of high computational complexity, formula for the ML-degree. The variety of complete quadrics is an exact analog for symmetric matrices of the permutohedron variety for the diagonal matrices.
{"title":"Maximum Likelihood Degree, Complete Quadrics, and ℂ*-Action","authors":"M. Michałek, Leonid Monin, Jaroslaw A. Wisniewski","doi":"10.1137/20M1335960","DOIUrl":"https://doi.org/10.1137/20M1335960","url":null,"abstract":"We study the maximum likelihood (ML) degree of linear concentration models in algebraic statistics. We relate it to an intersection problem on the variety of complete quadrics. This allows us to provide an explicit, basic, albeit of high computational complexity, formula for the ML-degree. The variety of complete quadrics is an exact analog for symmetric matrices of the permutohedron variety for the diagonal matrices.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90174796","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}
In this paper, we address the noncommutative rank (nc-rank) computation of a linear symbolic matrix A = A1x1 + A2x2 + · · ·+ Amxm, where each Ai is an n × n matrix over a field K, and xi (i = 1, 2, . . . ,m) are noncommutative variables. For this problem, polynomial time algorithms were given by Garg, Gurvits, Oliveira, and Wigderson for K = Q, and by Ivanyos, Qiao, and Subrahmanyam for an arbitrary field K. We present a significantly different polynomial time algorithm that works on an arbitrary field K. Our algorithm is based on a combination of submodular optimization on modular lattices and convex optimization on CAT(0) spaces.
{"title":"Computing the nc-Rank via Discrete Convex Optimization on CAT(0) Spaces","authors":"Masaki Hamada, H. Hirai","doi":"10.1137/20m138836x","DOIUrl":"https://doi.org/10.1137/20m138836x","url":null,"abstract":"In this paper, we address the noncommutative rank (nc-rank) computation of a linear symbolic matrix A = A1x1 + A2x2 + · · ·+ Amxm, where each Ai is an n × n matrix over a field K, and xi (i = 1, 2, . . . ,m) are noncommutative variables. For this problem, polynomial time algorithms were given by Garg, Gurvits, Oliveira, and Wigderson for K = Q, and by Ivanyos, Qiao, and Subrahmanyam for an arbitrary field K. We present a significantly different polynomial time algorithm that works on an arbitrary field K. Our algorithm is based on a combination of submodular optimization on modular lattices and convex optimization on CAT(0) spaces.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88918597","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}
Structural identifiability is a property of an ODE model with parameters that allows for the parameters to be determined from continuous noise-free data. This is natural prerequisite for practical identifiability. Conducting multiple independent experiments could make more parameters or functions of parameters identifiable, which is a desirable property to have. How many experiments are sufficient? In the present paper, we provide an algorithm to determine the exact number of experiments for multi-experiment local identifiability and obtain an upper bound that is off at most by one for the number of experiments for multi-experiment global identifiability. Interestingly, the main theoretical ingredient of the algorithm has been discovered and proved using model theory (in the sense of mathematical logic). We hope that this unexpected connection will stimulate interactions between applied algebra and model theory, and we provide a short introduction to model theory in the context of parameter identifiability. As another related application of model theory in this area, we construct a nonlinear ODE system with one output such that single-experiment and mutiple-experiment identifiability are different for the system. This contrasts with recent results about single-output linear systems. We also present a Monte Carlo randomized version of the algorithm with a polynomial arithmetic complexity. Implementation of the algorithm is provided and its performance is demonstrated on several examples. The source code is available at https://github.com/pogudingleb/ExperimentsBound.
{"title":"Multi-experiment parameter identifiability of ODEs and model theory","authors":"A. Ovchinnikov, A. Pillay, G. Pogudin, T. Scanlon","doi":"10.1137/21m1389845","DOIUrl":"https://doi.org/10.1137/21m1389845","url":null,"abstract":"Structural identifiability is a property of an ODE model with parameters that allows for the parameters to be determined from continuous noise-free data. This is natural prerequisite for practical identifiability. Conducting multiple independent experiments could make more parameters or functions of parameters identifiable, which is a desirable property to have. How many experiments are sufficient? In the present paper, we provide an algorithm to determine the exact number of experiments for multi-experiment local identifiability and obtain an upper bound that is off at most by one for the number of experiments for multi-experiment global identifiability. Interestingly, the main theoretical ingredient of the algorithm has been discovered and proved using model theory (in the sense of mathematical logic). We hope that this unexpected connection will stimulate interactions between applied algebra and model theory, and we provide a short introduction to model theory in the context of parameter identifiability. As another related application of model theory in this area, we construct a nonlinear ODE system with one output such that single-experiment and mutiple-experiment identifiability are different for the system. This contrasts with recent results about single-output linear systems. We also present a Monte Carlo randomized version of the algorithm with a polynomial arithmetic complexity. Implementation of the algorithm is provided and its performance is demonstrated on several examples. The source code is available at https://github.com/pogudingleb/ExperimentsBound.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76864317","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}
A fundamental question in computer vision is whether a set of point pairs is the image of a scene that lies in front of two cameras. Such a scene and the cameras together are known as a chiral reconstruction of the point pairs. In this paper we provide a complete classification of k point pairs for which a chiral reconstruction exists. The existence of chiral reconstructions is equivalent to the non-emptiness of certain semialgebraic sets. For up to three point pairs, we prove that a chiral reconstruction always exists while the set of five or more point pairs that do not have a chiral reconstruction is Zariski-dense. We show that for five generic point pairs, the chiral region is bounded by line segments in a Schlafli double six on a cubic surface with 27 real lines. Four point pairs have a chiral reconstruction unless they belong to two non-generic combinatorial types, in which case they may or may not.
{"title":"Existence of Two View Chiral Reconstructions","authors":"Andrew Pryhuber, Rainer Sinn, Rekha R. Thomas","doi":"10.1137/20m1381848","DOIUrl":"https://doi.org/10.1137/20m1381848","url":null,"abstract":"A fundamental question in computer vision is whether a set of point pairs is the image of a scene that lies in front of two cameras. Such a scene and the cameras together are known as a chiral reconstruction of the point pairs. In this paper we provide a complete classification of k point pairs for which a chiral reconstruction exists. The existence of chiral reconstructions is equivalent to the non-emptiness of certain semialgebraic sets. For up to three point pairs, we prove that a chiral reconstruction always exists while the set of five or more point pairs that do not have a chiral reconstruction is Zariski-dense. We show that for five generic point pairs, the chiral region is bounded by line segments in a Schlafli double six on a cubic surface with 27 real lines. Four point pairs have a chiral reconstruction unless they belong to two non-generic combinatorial types, in which case they may or may not.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81967637","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}
We consider the problem of estimating the number of common complements of a family of subspaces over a finite field, in terms of the cardinality of the family and its intersection structure. We derive upper and lower bounds for this number, along with their asymptotic versions as the field size tends to infinity. We use these bounds to describe the general behavior of common complements with respect to sparsity and density, showing that the decisive property is whether or not the number of spaces to be complemented is negligible with respect to the field size. The proof techniques are based on the study of isolated vertices in certain bipartite graphs. By specializing our results to matrix spaces, we answer an open question in coding theory, proving that MRD codes in the rank metric are sparse for all parameter sets as the field grows, with only very few exceptions. We also investigate the density of MRD codes as their column length tends to infinity, obtaining a new asymptotic bound. Using properties of the Euler function from number theory, we then show that our bound improves on known results for most parameter sets. We conclude the paper by establishing two structural properties of the density function of rank-metric codes.
{"title":"Common Complements of Linear Subspaces and the Sparseness of MRD Codes","authors":"Anina Gruica, A. Ravagnani","doi":"10.1137/21m1428947","DOIUrl":"https://doi.org/10.1137/21m1428947","url":null,"abstract":"We consider the problem of estimating the number of common complements of a family of subspaces over a finite field, in terms of the cardinality of the family and its intersection structure. We derive upper and lower bounds for this number, along with their asymptotic versions as the field size tends to infinity. We use these bounds to describe the general behavior of common complements with respect to sparsity and density, showing that the decisive property is whether or not the number of spaces to be complemented is negligible with respect to the field size. The proof techniques are based on the study of isolated vertices in certain bipartite graphs. By specializing our results to matrix spaces, we answer an open question in coding theory, proving that MRD codes in the rank metric are sparse for all parameter sets as the field grows, with only very few exceptions. We also investigate the density of MRD codes as their column length tends to infinity, obtaining a new asymptotic bound. Using properties of the Euler function from number theory, we then show that our bound improves on known results for most parameter sets. We conclude the paper by establishing two structural properties of the density function of rank-metric codes.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76447637","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}
We consider two seemingly unrelated questions: the relationship between nonnegative polynomials and sums of squares on real varieties, and sparse semidefinite programming. This connection is natural when a real variety $X$ is defined by a quadratic square-free monomial ideal. In this case nonnegative polynomials and sums of squares on $X$ are also natural objects in positive semidefinite matrix completion. Nonnegative quadratic forms over $X$ naturally correspond to partially specified matrices where all of the fully specified square blocks are PSD, and sums of squares quadratic forms naturally correspond to partially specified matrices which can be completed to a PSD matrix. We show quantitative results on approximation of nonnegative polynomials by sums of squares, which leads to applications in sparse semidefinite programming.
{"title":"Sums of Squares and Sparse Semidefinite Programming","authors":"Grigoriy Blekherman, Kevin Shu","doi":"10.1137/20m1376170","DOIUrl":"https://doi.org/10.1137/20m1376170","url":null,"abstract":"We consider two seemingly unrelated questions: the relationship between nonnegative polynomials and sums of squares on real varieties, and sparse semidefinite programming. This connection is natural when a real variety $X$ is defined by a quadratic square-free monomial ideal. In this case nonnegative polynomials and sums of squares on $X$ are also natural objects in positive semidefinite matrix completion. Nonnegative quadratic forms over $X$ naturally correspond to partially specified matrices where all of the fully specified square blocks are PSD, and sums of squares quadratic forms naturally correspond to partially specified matrices which can be completed to a PSD matrix. We show quantitative results on approximation of nonnegative polynomials by sums of squares, which leads to applications in sparse semidefinite programming.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85339702","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}
We build a functorial pipeline for persistent homology. The input to this pipeline is a filtered simplicial complex indexed by any finite lattice, and the output is a persistence diagram defined as the Mobius inversion of a certain monotone integral function. We adapt the Reeb graph edit distance of Landi et. al. to each of our categories and prove that both functors in our pipeline are 1-Lipschitz making our pipeline stable.
{"title":"Edit Distance and Persistence Diagrams Over Lattices","authors":"Alex McCleary, A. Patel","doi":"10.1137/20M1373700","DOIUrl":"https://doi.org/10.1137/20M1373700","url":null,"abstract":"We build a functorial pipeline for persistent homology. The input to this pipeline is a filtered simplicial complex indexed by any finite lattice, and the output is a persistence diagram defined as the Mobius inversion of a certain monotone integral function. We adapt the Reeb graph edit distance of Landi et. al. to each of our categories and prove that both functors in our pipeline are 1-Lipschitz making our pipeline stable.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76835820","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}
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