Singular value decomposition (SVD) has been widely used in machine learning. It lies at the root of data analysis, and it provides the mathematical basis for many data mining techniques. Recently, interest in incremental SVD has been on the rise because it is well suited to streaming data. In this paper, we propose a new algorithm of incremental SVD that is designed to improve both efficiency and accuracy during computation. More specifically, our proposed algorithm takes advantage of the special structures of arrowhead and diagonal-plus-rank-one matrices involved in updating SVD models to expedite the updating process. Moreover, because the singular values are computed independently, the proposed method can be easily parallelized. In addition, as this paper shows, increasing rank can lead to more accurate singular values in the updating process. Experimental results from synthetic and real data sets demonstrate gains in efficiency and accuracy in the updating process.
{"title":"A new method to improve the efficiency and accuracy of incremental singular value decomposition","authors":"Hansi Jiang, A. Chaudhuri","doi":"10.13001/ela.2023.7325","DOIUrl":"https://doi.org/10.13001/ela.2023.7325","url":null,"abstract":"Singular value decomposition (SVD) has been widely used in machine learning. It lies at the root of data analysis, and it provides the mathematical basis for many data mining techniques. Recently, interest in incremental SVD has been on the rise because it is well suited to streaming data. In this paper, we propose a new algorithm of incremental SVD that is designed to improve both efficiency and accuracy during computation. More specifically, our proposed algorithm takes advantage of the special structures of arrowhead and diagonal-plus-rank-one matrices involved in updating SVD models to expedite the updating process. Moreover, because the singular values are computed independently, the proposed method can be easily parallelized. In addition, as this paper shows, increasing rank can lead to more accurate singular values in the updating process. Experimental results from synthetic and real data sets demonstrate gains in efficiency and accuracy in the updating process.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46085697","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}
For a simple graph $G$, let $eta(G)$ and $c(G)$ be the nullity and the cyclomatic number of $G$, respectively. A cycle-spliced bipartite graph is a connected graph in which every block is an even cycle. It was shown by Wong et al. (2022) that for every cycle-spliced bipartite graph $G$, $0leqeta(G)leq c(G)+1$. Moreover, the extremal graphs with $eta(G) = c(G)+1$ and $eta(G) =0$, respectively, have been characterized. In this paper, we prove that there is no cycle-spliced bipartite graphs $G$ of any order with nullity $eta(G)=c(G)$. Furthermore, we also provide a structural characterization on cycle-spliced bipartite graphs $G$ with nullity $eta(G)=c(G)-1$.
{"title":"Nullities of cycle-spliced bipartite graphs","authors":"Sarula Chang, Jianxi Li, Yirong Zheng","doi":"10.13001/ela.2023.7377","DOIUrl":"https://doi.org/10.13001/ela.2023.7377","url":null,"abstract":"For a simple graph $G$, let $eta(G)$ and $c(G)$ be the nullity and the cyclomatic number of $G$, respectively. A cycle-spliced bipartite graph is a connected graph in which every block is an even cycle. It was shown by Wong et al. (2022) that for every cycle-spliced bipartite graph $G$, $0leqeta(G)leq c(G)+1$. Moreover, the extremal graphs with $eta(G) = c(G)+1$ and $eta(G) =0$, respectively, have been characterized. In this paper, we prove that there is no cycle-spliced bipartite graphs $G$ of any order with nullity $eta(G)=c(G)$. Furthermore, we also provide a structural characterization on cycle-spliced bipartite graphs $G$ with nullity $eta(G)=c(G)-1$.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42758020","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}
In this paper, the boundary generating curves and the numerical range of Kac-Sylvester matrices up to the order $9$ are characterized. Based on the obtained results and on several computational experiments performed with the Mathematica and MatLab programs, we conjecture that the found types of algebraic curves, namely ellipses and ovals, will appear for an arbitrary order.
{"title":"On the numerical range of Kac-Sylvester matrices","authors":"N. Bebiano, R. Lemos, G. Soares","doi":"10.13001/ela.2023.7703","DOIUrl":"https://doi.org/10.13001/ela.2023.7703","url":null,"abstract":"In this paper, the boundary generating curves and the numerical range of Kac-Sylvester matrices up to the order $9$ are characterized. Based on the obtained results and on several computational experiments performed with the Mathematica and MatLab programs, we conjecture that the found types of algebraic curves, namely ellipses and ovals, will appear for an arbitrary order.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43380761","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}
We introduce a generalization of alternating sign matrices (ASMs) called multiASMs and develop some of their properties. Classes of multiASMs with specified row and column sum vectors $R$ and $S$ extend the classes of $(0,1)$-matrices with specified $R$ and $S$. The special case when $R=S$ is a constant vector, in particular all 2's, is treated in more detail. We also investigate the polytope spanned by a class of multiASMs. Finally, we discuss the possibility of defining a Bruhat order on a class of multiASMs.
{"title":"Multi-alternating sign matrices","authors":"R. Brualdi, G. Dahl","doi":"10.13001/ela.2023.7471","DOIUrl":"https://doi.org/10.13001/ela.2023.7471","url":null,"abstract":"We introduce a generalization of alternating sign matrices (ASMs) called multiASMs and develop some of their properties. Classes of multiASMs with specified row and column sum vectors $R$ and $S$ extend the classes of $(0,1)$-matrices with specified $R$ and $S$. The special case when $R=S$ is a constant vector, in particular all 2's, is treated in more detail. We also investigate the polytope spanned by a class of multiASMs. Finally, we discuss the possibility of defining a Bruhat order on a class of multiASMs.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43277702","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}
We apply a technique of Sinkovic and van der Holst for constructing orthogonal vector representations of a graph whose complement has given treewidth to graphs whose complement has given degeneracy.
我们将Sinkovic和van der Holst的技术应用于构造补具有树宽的图的正交向量表示到补具有退化性的图。
{"title":"Graph degeneracy and orthogonal vector representations","authors":"Lon H. Mitchell","doi":"10.13001/ela.2023.6907","DOIUrl":"https://doi.org/10.13001/ela.2023.6907","url":null,"abstract":"We apply a technique of Sinkovic and van der Holst for constructing orthogonal vector representations of a graph whose complement has given treewidth to graphs whose complement has given degeneracy.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47099055","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}
In this paper, a von Neumann-type inequality is studied on an Eaton triple $ (V,G,D) $, where $ V $ is a real inner product space, $ G $ is a compact subgroup of the orthogonal group $ O (V) $, and $ D subset V $ is a closed convex cone. By using an inner structure of an Eaton triple, a refinement of this inequality is shown. In the special case $ G = O ( V ) $, a refinement of the Cauchy-Schwarz inequality is obtained.
本文研究了Eaton三元组$ (V,G,D) $上的一个von neumann型不等式，其中$ V $是一个实内积空间，$ G $是正交组$ O (V) $的紧子群，$ D 子集V $是一个闭凸锥。利用Eaton三元组的内部结构，给出了这个不等式的一个改进。在特殊情况$ G = O (V) $下，得到了Cauchy-Schwarz不等式的一个改进。
{"title":"Refinement of von Neumann-type inequalities on product Eaton triples","authors":"M. Niezgoda","doi":"10.13001/ela.2023.7375","DOIUrl":"https://doi.org/10.13001/ela.2023.7375","url":null,"abstract":"In this paper, a von Neumann-type inequality is studied on an Eaton triple $ (V,G,D) $, where $ V $ is a real inner product space, $ G $ is a compact subgroup of the orthogonal group $ O (V) $, and $ D subset V $ is a closed convex cone. By using an inner structure of an Eaton triple, a refinement of this inequality is shown. In the special case $ G = O ( V ) $, a refinement of the Cauchy-Schwarz inequality is obtained.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43552752","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}
We present adapted Zhang neural networks (AZNN) in which the parameter settings for the exponential decay constant $eta$ and the length of the start-up phase of basic ZNN are adapted to the problem at hand. Specifically, we study experiments with AZNN for time-varying square matrix factorizations as a product of time-varying symmetric matrices and for the time-varying matrix square roots problem. Differing from generally used small $eta$ values and minimal start-up length phases in ZNN, we adapt the basic ZNN method to work with large or even gigantic $eta$ settings and arbitrary length start-ups using Euler's low accuracy finite difference formula. These adaptations improve the speed of AZNN's convergence and lower its solution error bounds for our chosen problems significantly to near machine constant or even lower levels. Parameter-varying AZNN also allows us to find full rank symmetrizers of static matrices reliably, for example, for the Kahan and Frank matrices and for matrices with highly ill-conditioned eigenvalues and complicated Jordan structures of dimensions from $n = 2$ on up. This helps in cases where full rank static matrix symmetrizers have never been successfully computed before.
{"title":"Adapted AZNN methods for time-varying and static matrix problems","authors":"Frank Uhlig","doi":"10.13001/ela.2023.7417","DOIUrl":"https://doi.org/10.13001/ela.2023.7417","url":null,"abstract":"We present adapted Zhang neural networks (AZNN) in which the parameter settings for the exponential decay constant $eta$ and the length of the start-up phase of basic ZNN are adapted to the problem at hand. Specifically, we study experiments with AZNN for time-varying square matrix factorizations as a product of time-varying symmetric matrices and for the time-varying matrix square roots problem. Differing from generally used small $eta$ values and minimal start-up length phases in ZNN, we adapt the basic ZNN method to work with large or even gigantic $eta$ settings and arbitrary length start-ups using Euler's low accuracy finite difference formula. These adaptations improve the speed of AZNN's convergence and lower its solution error bounds for our chosen problems significantly to near machine constant or even lower levels. Parameter-varying AZNN also allows us to find full rank symmetrizers of static matrices reliably, for example, for the Kahan and Frank matrices and for matrices with highly ill-conditioned eigenvalues and complicated Jordan structures of dimensions from $n = 2$ on up. This helps in cases where full rank static matrix symmetrizers have never been successfully computed before.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136375307","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}
Let $Phi=(G,U(mathbb{Q}),varphi)$ be a quaternion unit gain graph (or $U(mathbb{Q})$-gain graph), where $G$ is the underlying graph of $Phi$, $U(mathbb{Q})={zin mathbb{Q}: |z|=1}$ is the circle group, and $varphi:overrightarrow{E}rightarrow U(mathbb{Q})$ is the gain function such that $varphi(e_{ij})=varphi(e_{ji})^{-1}=overline{varphi(e_{ji})}$. Let $A(Phi)$ be the adjacency matrix of $Phi$ and $r(Phi)$ be the row left rank of $Phi$. In this paper, we prove that $-2c(G)leq r(Phi)-r(G)leq 2c(G)$, where $r(G)$ and $c(G)$ are the rank and the dimension of cycle space of $G$, respectively. All corresponding extremal graphs are characterized. The results will generalize the corresponding results of signed graphs (Lu et al. [20] and Wang [33]), mixed graphs (Chen et al. [7]), and complex unit gain graphs (Lu et al. [21]).
设$Phi=(G,U(mathbb{Q}),varphi)$为四元数单位增益图(或$U(mathbb{Q})$ -增益图)，其中$G$为$Phi$的底层图，$U(mathbb{Q})={zin mathbb{Q}: |z|=1}$为圆组，$varphi:overrightarrow{E}rightarrow U(mathbb{Q})$为增益函数，使得$varphi(e_{ij})=varphi(e_{ji})^{-1}=overline{varphi(e_{ji})}$。设$A(Phi)$为$Phi$的邻接矩阵，$r(Phi)$为$Phi$的左行秩。本文证明了$-2c(G)leq r(Phi)-r(G)leq 2c(G)$，其中$r(G)$和$c(G)$分别是$G$的循环空间的秩和维数。对所有相应的极值图进行了刻画。所得结果将推广符号图(Lu et al.[20]和Wang[33])、混合图(Chen et al.[7])和复单位增益图(Lu et al.[21])的相应结果。
{"title":"Relation between the row left rank of a quaternion unit gain graph and the rank of its underlying graph","authors":"Qiannan Zhou, Yong Lu","doi":"10.13001/ela.2023.7681","DOIUrl":"https://doi.org/10.13001/ela.2023.7681","url":null,"abstract":"Let $Phi=(G,U(mathbb{Q}),varphi)$ be a quaternion unit gain graph (or $U(mathbb{Q})$-gain graph), where $G$ is the underlying graph of $Phi$, $U(mathbb{Q})={zin mathbb{Q}: |z|=1}$ is the circle group, and $varphi:overrightarrow{E}rightarrow U(mathbb{Q})$ is the gain function such that $varphi(e_{ij})=varphi(e_{ji})^{-1}=overline{varphi(e_{ji})}$. Let $A(Phi)$ be the adjacency matrix of $Phi$ and $r(Phi)$ be the row left rank of $Phi$. In this paper, we prove that $-2c(G)leq r(Phi)-r(G)leq 2c(G)$, where $r(G)$ and $c(G)$ are the rank and the dimension of cycle space of $G$, respectively. All corresponding extremal graphs are characterized. The results will generalize the corresponding results of signed graphs (Lu et al. [20] and Wang [33]), mixed graphs (Chen et al. [7]), and complex unit gain graphs (Lu et al. [21]).","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45864098","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}
We show that real, symmetric, centered (zero row sum) positive semidefinite matrices of order $n$ and rank $n-1$ with eigenvalue ratio $lambda_{max}/lambda_{min}leq n/(n-2)$ between the largest and smallest nonzero eigenvalue have nonpositive off-diagonal entries, and that this eigenvalue criterion is tight. The result is relevant in the context of matrix theory and inverse eigenvalue problems, and we discuss an application to Laplacian matrices.
{"title":"Centered PSD matrices with thin spectrum are M-matrices","authors":"K. Devriendt","doi":"10.13001/ela.2023.7051","DOIUrl":"https://doi.org/10.13001/ela.2023.7051","url":null,"abstract":"We show that real, symmetric, centered (zero row sum) positive semidefinite matrices of order $n$ and rank $n-1$ with eigenvalue ratio $lambda_{max}/lambda_{min}leq n/(n-2)$ between the largest and smallest nonzero eigenvalue have nonpositive off-diagonal entries, and that this eigenvalue criterion is tight. The result is relevant in the context of matrix theory and inverse eigenvalue problems, and we discuss an application to Laplacian matrices.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45409479","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}
It is shown that every $2n$-by-$2n$ matrix over a field $mathbb{F}$ with determinant 1 is a product of (i) four or fewer skew-involutions ($A^2 = -I$) provided $mathbb{F} neq mathbb{Z}_3$, and (ii) eight or fewer skew-involutions if $mathbb{F} = mathbb{Z}_3$ and $n > 1$. Every real symplectic matrix is a product of six real symplectic skew-involutions, and an explicit factorization of a complex symplectic matrix into two symplectic skew-involutions is given.
{"title":"Products of skew-involutions","authors":"Jesus Paolo Joven, Agnes T. Paras","doi":"10.13001/ela.2023.7709","DOIUrl":"https://doi.org/10.13001/ela.2023.7709","url":null,"abstract":"It is shown that every $2n$-by-$2n$ matrix over a field $mathbb{F}$ with determinant 1 is a product of (i) four or fewer skew-involutions ($A^2 = -I$) provided $mathbb{F} neq mathbb{Z}_3$, and (ii) eight or fewer skew-involutions if $mathbb{F} = mathbb{Z}_3$ and $n > 1$. Every real symplectic matrix is a product of six real symplectic skew-involutions, and an explicit factorization of a complex symplectic matrix into two symplectic skew-involutions is given.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48612640","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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