In this paper, we investigate the complexity of the central path of semidefinite optimization through the lens of real algebraic geometry. To that end, we propose an algorithm to compute real univariate representations describing the central path and its limit point, where the limit point is described by taking the limit of central solutions, as bounded points in the field of algebraic Puiseux series. As a result, we derive an upper bound 2 ) on the degree of the Zariski closure of the central path, when μ is sufficiently small, and for the complexity of describing the limit point, where m and n denote the number of affine constraints and size of the symmetric matrix, respectively. Furthermore, by the application of the quantifier elimination to the real univariate representations, we provide a lower bound 1/γ, with γ = 2 2), on the convergence rate of the central path.
{"title":"On the Central Path of Semidefinite Optimization: Degree and Worst-Case Convergence Rate","authors":"S. Basu, Ali Mohammad Nezhad","doi":"10.1137/21m1419933","DOIUrl":"https://doi.org/10.1137/21m1419933","url":null,"abstract":"In this paper, we investigate the complexity of the central path of semidefinite optimization through the lens of real algebraic geometry. To that end, we propose an algorithm to compute real univariate representations describing the central path and its limit point, where the limit point is described by taking the limit of central solutions, as bounded points in the field of algebraic Puiseux series. As a result, we derive an upper bound 2 ) on the degree of the Zariski closure of the central path, when μ is sufficiently small, and for the complexity of describing the limit point, where m and n denote the number of affine constraints and size of the symmetric matrix, respectively. Furthermore, by the application of the quantifier elimination to the real univariate representations, we provide a lower bound 1/γ, with γ = 2 2), on the convergence rate of the central path.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"79 1","pages":"299-318"},"PeriodicalIF":1.2,"publicationDate":"2021-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80112138","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}
Timothy Duff, Viktor Korotynskiy, T. Pajdla, Margaret H. Regan
We consider Galois/monodromy groups arising in computer vision applications, with a view towards building more efficient polynomial solvers. The Galois/monodromy group allows us to decide when a given problem decomposes into algebraic subproblems, and whether or not it has any symmetries. Tools from numerical algebraic geometry and computational group theory allow us to apply this framework to classical and novel reconstruction problems. We consider three classical cases--3-point absolute pose, 5-point relative pose, and 4-point homography estimation for calibrated cameras--where the decomposition and symmetries may be naturally understood in terms of the Galois/monodromy group. We then show how our framework can be applied to novel problems from absolute and relative pose estimation. For instance, we discover new symmetries for absolute pose problems involving mixtures of point and line features. We also describe a problem of estimating a pair of calibrated homographies between three images. For this problem of degree 64, we can reduce the degree to 16; the latter better reflecting the intrinsic difficulty of algebraically solving the problem. As a byproduct, we obtain new constraints on compatible homographies, which may be of independent interest.
{"title":"Galois/monodromy groups for decomposing minimal problems in 3D reconstruction","authors":"Timothy Duff, Viktor Korotynskiy, T. Pajdla, Margaret H. Regan","doi":"10.1137/21m1422872","DOIUrl":"https://doi.org/10.1137/21m1422872","url":null,"abstract":"We consider Galois/monodromy groups arising in computer vision applications, with a view towards building more efficient polynomial solvers. The Galois/monodromy group allows us to decide when a given problem decomposes into algebraic subproblems, and whether or not it has any symmetries. Tools from numerical algebraic geometry and computational group theory allow us to apply this framework to classical and novel reconstruction problems. We consider three classical cases--3-point absolute pose, 5-point relative pose, and 4-point homography estimation for calibrated cameras--where the decomposition and symmetries may be naturally understood in terms of the Galois/monodromy group. We then show how our framework can be applied to novel problems from absolute and relative pose estimation. For instance, we discover new symmetries for absolute pose problems involving mixtures of point and line features. We also describe a problem of estimating a pair of calibrated homographies between three images. For this problem of degree 64, we can reduce the degree to 16; the latter better reflecting the intrinsic difficulty of algebraically solving the problem. As a byproduct, we obtain new constraints on compatible homographies, which may be of independent interest.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"9 1","pages":"740-772"},"PeriodicalIF":1.2,"publicationDate":"2021-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81488150","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 present results on the number of linear regions of the functions that can be represented by artificial feedforward neural networks with maxout units. A rank-k maxout unit is a function computing the maximum of $k$ linear functions. For networks with a single layer of maxout units, the linear regions correspond to the upper vertices of a Minkowski sum of polytopes. We obtain face counting formulas in terms of the intersection posets of tropical hypersurfaces or the number of upper faces of partial Minkowski sums, along with explicit sharp upper bounds for the number of regions for any input dimension, any number of units, and any ranks, in the cases with and without biases. Based on these results we also obtain asymptotically sharp upper bounds for networks with multiple layers.
{"title":"Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums","authors":"Guido Montúfar, Yue Ren, Leon Zhang","doi":"10.1137/21m1413699","DOIUrl":"https://doi.org/10.1137/21m1413699","url":null,"abstract":"We present results on the number of linear regions of the functions that can be represented by artificial feedforward neural networks with maxout units. A rank-k maxout unit is a function computing the maximum of $k$ linear functions. For networks with a single layer of maxout units, the linear regions correspond to the upper vertices of a Minkowski sum of polytopes. We obtain face counting formulas in terms of the intersection posets of tropical hypersurfaces or the number of upper faces of partial Minkowski sums, along with explicit sharp upper bounds for the number of regions for any input dimension, any number of units, and any ranks, in the cases with and without biases. Based on these results we also obtain asymptotically sharp upper bounds for networks with multiple layers.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"7 1","pages":"618-649"},"PeriodicalIF":1.2,"publicationDate":"2021-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75835135","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}
For given integers $n$ and $d$, both at least 2, we consider a homogeneous multivariate polynomial $f_d$ of degree $d$ in variables indexed by the edges of the complete graph on $n$ vertices and coefficients depending on cardinalities of certain unions of edges. Cardinaels, Borst and Van Leeuwaarden (arXiv:2111.05777, 2021) asked whether $f_d$, which arises in a model of job-occupancy in redundancy scheduling, attains its minimum over the standard simplex at the uniform probability vector. Brosch, Laurent and Steenkamp [SIAM J. Optim. 31 (2021), 2227--2254] proved that $f_d$ is convex over the standard simplex if $d=2$ and $d=3$, implying the desired result for these $d$. We give a symmetry reduction to show that for fixed $d$, the polynomial is convex over the standard simplex (for all $ngeq 2$) if a constant number of constant matrices (with size and coefficients independent of $n$) are positive semidefinite. This result is then used in combination with a computer-assisted verification to show that the polynomial $f_d$ is convex for $dleq 9$.
{"title":"Symmetry Reduction to Optimize a Graph-based Polynomial From Queueing Theory","authors":"Sven Polak","doi":"10.1137/21M1413298","DOIUrl":"https://doi.org/10.1137/21M1413298","url":null,"abstract":"For given integers $n$ and $d$, both at least 2, we consider a homogeneous multivariate polynomial $f_d$ of degree $d$ in variables indexed by the edges of the complete graph on $n$ vertices and coefficients depending on cardinalities of certain unions of edges. Cardinaels, Borst and Van Leeuwaarden (arXiv:2111.05777, 2021) asked whether $f_d$, which arises in a model of job-occupancy in redundancy scheduling, attains its minimum over the standard simplex at the uniform probability vector. Brosch, Laurent and Steenkamp [SIAM J. Optim. 31 (2021), 2227--2254] proved that $f_d$ is convex over the standard simplex if $d=2$ and $d=3$, implying the desired result for these $d$. We give a symmetry reduction to show that for fixed $d$, the polynomial is convex over the standard simplex (for all $ngeq 2$) if a constant number of constant matrices (with size and coefficients independent of $n$) are positive semidefinite. This result is then used in combination with a computer-assisted verification to show that the polynomial $f_d$ is convex for $dleq 9$.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"31 1","pages":"243-266"},"PeriodicalIF":1.2,"publicationDate":"2021-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82649077","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 study the locus of singular tuples of a complex valued multisymmetric tensor. The main problem that we focus on is: given the set of singular tuples of some general tensor, which are all the tensors that admit those same singular tuples. Assume that the triangular inequality holds, that is exactly the condition such that the dual variety to the Segre-Veronese variety is an hypersurface, or equivalently, the hyperdeterminant exists. We show in such case that, when at least one component has degree odd, this tensor is projectively unique. On the other hand, if all the degrees are even, the fiber is an $1$-dimensional space.
{"title":"On Tensors That Are Determined by Their Singular Tuples","authors":"Ettore Teixeira Turatti","doi":"10.1137/21M1412980","DOIUrl":"https://doi.org/10.1137/21M1412980","url":null,"abstract":"In this paper we study the locus of singular tuples of a complex valued multisymmetric tensor. The main problem that we focus on is: given the set of singular tuples of some general tensor, which are all the tensors that admit those same singular tuples. Assume that the triangular inequality holds, that is exactly the condition such that the dual variety to the Segre-Veronese variety is an hypersurface, or equivalently, the hyperdeterminant exists. We show in such case that, when at least one component has degree odd, this tensor is projectively unique. On the other hand, if all the degrees are even, the fiber is an $1$-dimensional space.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"47 1","pages":"319-338"},"PeriodicalIF":1.2,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78812267","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}
Suppose that $Gamma=(V,E)$ is a graph with vertices $V$, edges $E$, a free group action on the vertices $mathbb{Z}^d curvearrowright V$ with finitely many orbits, and a linear operator $D$ on the Hilbert space $l^2(V)$ such that $D$ commutes with the group action. Fix $lambda in mathbb{R}$ in the pure-point spectrum of $D$ and consider the vector space of all eigenfunctions of finite support $K$. Then $K$ is a non-trivial finitely generated module over the ring of Laurent polynomials, and the density of $lambda$ is given by an Euler-characteristic type formula by taking a finite free resolution of $K$. Furthermore, these claims generalize under suitable assumptions to the non-commutative setting of a finite generated amenable group acting on the vertices freely with finitely many orbits, and commuting with the operator $D$.
假设$Gamma=(V,E)$是一个图，它有顶点$V$，边$E$，顶点$mathbb{Z}^d curvearrowright V$上有一个有有限多个轨道的自由群作用，希尔伯特空间$l^2(V)$上有一个线性算子$D$，使得$D$与群作用交换。将$lambda in mathbb{R}$固定在$D$的纯点谱中，考虑有限支持$K$的所有特征函数的向量空间。然后$K$是Laurent多项式环上的非平凡有限生成模，并且$lambda$的密度通过取$K$的有限自由分辨率由欧拉特征型公式给出。此外，在适当的假设下，这些主张推广到一个有限生成的可服从群的非交换设置，该群自由作用于具有有限多个轨道的顶点上，并与算子$D$交换。
{"title":"On the Density of Eigenvalues on Periodic Graphs","authors":"C. Kravaris","doi":"10.1137/22m1505268","DOIUrl":"https://doi.org/10.1137/22m1505268","url":null,"abstract":"Suppose that $Gamma=(V,E)$ is a graph with vertices $V$, edges $E$, a free group action on the vertices $mathbb{Z}^d curvearrowright V$ with finitely many orbits, and a linear operator $D$ on the Hilbert space $l^2(V)$ such that $D$ commutes with the group action. Fix $lambda in mathbb{R}$ in the pure-point spectrum of $D$ and consider the vector space of all eigenfunctions of finite support $K$. Then $K$ is a non-trivial finitely generated module over the ring of Laurent polynomials, and the density of $lambda$ is given by an Euler-characteristic type formula by taking a finite free resolution of $K$. Furthermore, these claims generalize under suitable assumptions to the non-commutative setting of a finite generated amenable group acting on the vertices freely with finitely many orbits, and commuting with the operator $D$.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"2 1","pages":"585-609"},"PeriodicalIF":1.2,"publicationDate":"2021-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90862853","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 introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the"life span"or"persistence"of a homology class. As a case study, we analyze the power filtration on unweighted graphs, and provide explicit bounds for the life spans of homology classes in persistence diagrams in all dimensions.
{"title":"Geometric Approaches on Persistent Homology","authors":"Henry Adams, Baris Coskunuzer","doi":"10.1137/21M1422914","DOIUrl":"https://doi.org/10.1137/21M1422914","url":null,"abstract":"We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the\"life span\"or\"persistence\"of a homology class. As a case study, we analyze the power filtration on unweighted graphs, and provide explicit bounds for the life spans of homology classes in persistence diagrams in all dimensions.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"63 1","pages":"685-710"},"PeriodicalIF":1.2,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86125144","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 consider the Waring rank of monomials over the real and the rational numbers. We give a new upper bound for it by establishing a way in which one can take a structured apolar set for any given monomial $X_0^{a_0}X_1^{a_1}cdots X_n^{a_n}$ ($a_i>0$). This bound coincides with the real Waring rank in the case $n=1$ and in the case $min(a_i)=1$, which are all the known cases for the real rank of monomials. Our bound is also lower than any other known general bounds for the real Waring rank. Since all of the constructions are still valid over the rational numbers, this provides a new result for the rational Waring rank of any monomial as well. Some examples and computational implementation for potential use are presented in the end.
{"title":"A New Bound for the Waring Rank of Monomials","authors":"Kangjin Han, Hyunsuk Moon","doi":"10.1137/21M1390736","DOIUrl":"https://doi.org/10.1137/21M1390736","url":null,"abstract":"In this paper we consider the Waring rank of monomials over the real and the rational numbers. We give a new upper bound for it by establishing a way in which one can take a structured apolar set for any given monomial $X_0^{a_0}X_1^{a_1}cdots X_n^{a_n}$ ($a_i>0$). This bound coincides with the real Waring rank in the case $n=1$ and in the case $min(a_i)=1$, which are all the known cases for the real rank of monomials. Our bound is also lower than any other known general bounds for the real Waring rank. Since all of the constructions are still valid over the rational numbers, this provides a new result for the rational Waring rank of any monomial as well. Some examples and computational implementation for potential use are presented in the end.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"156 1-2","pages":"407-431"},"PeriodicalIF":1.2,"publicationDate":"2021-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72447733","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 provide a complete taxonomic characterization of robust hierarchical clustering methods for directed networks following an axiomatic approach. We begin by introducing three practical properties associated with the notion of robustness in hierarchical clustering: linear scale preservation, stability, and excisiveness. Linear scale preservation enforces imperviousness to change in units of measure whereas stability ensures that a bounded perturbation in the input network entails a bounded perturbation in the clustering output. Excisiveness refers to the local consistency of the clustering outcome. Algorithmically, excisiveness implies that we can reduce computational complexity by only clustering a subset of our data while theoretically guaranteeing that the same hierarchical outcome would be observed when clustering the whole dataset. In parallel to these three properties, we introduce the concept of representability, a generative model for describing clustering methods through the specification of their action on a collection of networks. Our main result is to leverage this generative model to give a precise characterization of all robust -- i.e., excisive, linear scale preserving, and stable -- hierarchical clustering methods for directed networks. We also address the implementation of our methods and describe an application to real data.
{"title":"Robust Hierarchical Clustering for Directed Networks: An Axiomatic Approach","authors":"G. Carlsson, Facundo M'emoli, Santiago Segarra","doi":"10.1137/20m1359201","DOIUrl":"https://doi.org/10.1137/20m1359201","url":null,"abstract":"We provide a complete taxonomic characterization of robust hierarchical clustering methods for directed networks following an axiomatic approach. We begin by introducing three practical properties associated with the notion of robustness in hierarchical clustering: linear scale preservation, stability, and excisiveness. Linear scale preservation enforces imperviousness to change in units of measure whereas stability ensures that a bounded perturbation in the input network entails a bounded perturbation in the clustering output. Excisiveness refers to the local consistency of the clustering outcome. Algorithmically, excisiveness implies that we can reduce computational complexity by only clustering a subset of our data while theoretically guaranteeing that the same hierarchical outcome would be observed when clustering the whole dataset. In parallel to these three properties, we introduce the concept of representability, a generative model for describing clustering methods through the specification of their action on a collection of networks. Our main result is to leverage this generative model to give a precise characterization of all robust -- i.e., excisive, linear scale preserving, and stable -- hierarchical clustering methods for directed networks. We also address the implementation of our methods and describe an application to real data.","PeriodicalId":48489,"journal":{"name":"SIAM Journal on Applied Algebra and Geometry","volume":"27 1","pages":"675-700"},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78884685","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}
{"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":"127 1","pages":"114-139"},"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}
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