Linear Algebra and its Applications最新文献

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Multi-granularity co-clustering of multi-type data via symmetric nonnegative matrix factorization 基于对称非负矩阵分解的多类型数据的多粒度共聚类

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2026-02-15 Epub Date: 2025-11-07 DOI: 10.1016/j.laa.2025.10.028

Dongjin Choi, Haesun Park

Clustering is a fundamental task in data analysis, essential for discovering patterns and groupings in data. When dealing with multi-type data, where entities of different types are interrelated, clustering becomes more complex and requires specialized methods. Most existing clustering approaches focus on a single type of entity, potentially overlooking the rich interactions between different types. Co-clustering methods address this limitation by simultaneously clustering multiple types of entities, exploiting their interrelationships. However, current co-clustering methods may not fully capture the multi-granularity structures present in many real-world data sets, where clusters exist at varying levels of granularity.

To address this issue, we propose MG-NMF (Multi-Granularity Nonnegative Matrix Factorization), a method for multi-granularity co-clustering of multi-type data. MG-NMF integrates both intra-type and inter-type relationships through embedding entities of different types into a shared low-dimensional space. By taking as input an integrated symmetric similarity matrix that encodes the relationships among all entity types, MG-NMF simultaneously considers intra-type similarities within each type and inter-type similarities across different types. Furthermore, the framework incorporates a multi-granularity perspective, enabling the discovery of cluster structures at varying levels of granularity, from broader to more refined groupings.

The proposed method employs a symmetric nonnegative matrix factorization to obtain nonnegative embeddings in a shared space. The nonnegativity constraint ensures interpretability and captures the inherent clustering structure of the data. We present an optimization procedure based on block coordinate descent and provide convergence analysis.

We evaluate the proposed method on real-world data sets, including a hierarchical data set of scholarly entities. Experimental results indicate that MG-NMF captures hierarchical relationships between clusters at different granularity levels and achieves high-quality clustering performance. MG-NMF offers a unified framework for multi-granularity co-clustering of multi-type data, providing insights into the complex structures of real-world data sets.

聚类是数据分析中的一项基本任务，对于发现数据中的模式和分组至关重要。当处理多类型数据时，不同类型的实体是相互关联的，聚类变得更加复杂，需要专门的方法。大多数现有的聚类方法关注于单一类型的实体，可能忽略了不同类型之间的丰富交互。协同聚类方法通过同时聚类多种类型的实体，利用它们的相互关系来解决这一限制。然而，当前的共聚类方法可能无法完全捕获许多真实数据集中存在的多粒度结构，其中集群以不同的粒度级别存在。为了解决这个问题，我们提出了一种多类型数据的多粒度共聚类方法MG-NMF （Multi-Granularity non - negative Matrix Factorization）。MG-NMF通过将不同类型的实体嵌入到共享的低维空间中，整合了类型内关系和类型间关系。MG-NMF以编码所有实体类型之间关系的集成对称相似度矩阵为输入，同时考虑每种类型内的类型相似度和不同类型间的类型相似度。此外，该框架还包含了一个多粒度透视图，允许在不同粒度级别上发现集群结构，从更广泛的到更精细的分组。该方法采用对称非负矩阵分解来获得共享空间中的非负嵌入。非负性约束确保了数据的可解释性，并捕获了数据固有的聚类结构。提出了一种基于分块坐标下降的优化方法，并给出了收敛性分析。我们在真实世界的数据集上评估了所提出的方法，包括学术实体的分层数据集。实验结果表明，MG-NMF捕获了不同粒度级别聚类之间的层次关系，获得了高质量的聚类性能。MG-NMF为多类型数据的多粒度共聚类提供了一个统一的框架，提供了对现实世界数据集复杂结构的见解。

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引用次数: 0

A linear condition for non-very generic discriminantal arrangements 非非常一般的判别安排的线性条件

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2026-02-01 Epub Date: 2025-11-06 DOI: 10.1016/j.laa.2025.10.037

Simona Settepanella , So Yamagata

The discriminantal arrangement is the space of configurations of n hyperplanes in generic position in a k-dimensional space. Unlike the case

k = 1

, where it coincides with the well-known braid arrangement, the discriminantal arrangement for

k > 1

has a combinatorial structure that depends on the choice of the original n hyperplanes. It is known that this combinatorics remains constant on a Zariski-open set

Z

, but determining whether a given configuration of n generic hyperplanes belongs to

Z

has proved to be a nontrivial problem. Even providing explicit examples of configurations not contained in

Z

remains a challenging task. In this paper, building on a recent result by the present authors, we introduce the notion of weak linear independence among sets of vectors, which, when imposed, allows us to construct configurations of hyperplanes not lying in

Z

. We also present three explicit examples illustrating this construction.

判别排列是k维空间中n个超平面在一般位置上的构型空间。与k=1的情况不同，k=1与众所周知的辫状排列一致，k>；1的判别排列具有组合结构，取决于原始n个超平面的选择。已知该组合在zariski开集Z上保持不变，但确定n个泛型超平面的给定构型是否属于Z已被证明是一个非平凡问题。即使提供Z中不包含的配置的显式示例仍然是一项具有挑战性的任务。在本文中，基于作者最近的一个结果，我们引入了向量集之间的弱线性无关的概念，当施加它时，我们可以构造不在z中的超平面的构型。我们还给出了三个明确的例子来说明这种构造。

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引用次数: 0

Spectral extremal problem for the odd prism 奇棱镜的光谱极值问题

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-24 DOI: 10.1016/j.laa.2025.10.025

Xinhui Duan, Lu Lu

The spectral Turán number

spex (n, F)

denotes the maximum spectral radius

λ (G)

of an F-free graph G of order n. This paper determines

spex (n, C_{2 k + 1}^{□})

for sufficiently large n, establishing the unique extremal graph. Here,

C_{2 k + 1}^{□}

is the odd prism, which is the Cartesian product

C_{2 k + 1} □ K_{2}

, where the Cartesian product

G □ F

has vertex set

V (G) \times V (F)

, and edges between

(u_{1}, v_{1})

and

(u_{2}, v_{2})

if either

u_{1} = u_{2}

and

v_{1} v_{2} \in E (F)

, or

v_{1} = v_{2}

and

u_{1} u_{2} \in E (G)

谱Turán数spex（n，F）表示n阶无F图G的最大谱半径λ(G)。当n足够大时，确定了spex（n,C2k+1□），建立了唯一极值图。这里，C2k+1□是奇棱镜，它是笛卡尔积C2k+1□K2，其中笛卡尔积G□F具有顶点集V(G)×V(F)，并且在（u1,v1）和（u2,v2）之间有边，如果u1=u2和v1v2∈E(F)，或者v1=v2和u1u2∈E(G)。

引用次数: 0

Universal bound on the eigenvalues of 2-positive trace-preserving maps 2-正保持迹映射特征值的全称界

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-24 DOI: 10.1016/j.laa.2025.10.022

Frederik vom Ende , Dariusz Chruściński , Gen Kimura , Paolo Muratore-Ginanneschi

We prove an upper bound on the trace of any 2-positive, trace-preserving map in terms of its smallest eigenvalue. We show that this spectral bound is tight, and that 2-positivity is necessary for this inequality to hold in general. Moreover, we use this to infer a similar bound for generators of one-parameter semigroups of 2-positive trace-preserving maps. With this approach we generalize known results for completely positive trace-preserving dynamics while providing a significantly simpler proof that is entirely algebraic.

我们用最小特征值证明了任意2正、保迹映射的迹的上界。我们证明了这个谱界是紧的，并且2正性对于这个不等式一般成立是必要的。此外，我们还利用这一理论推导出了2正迹保持映射的单参数半群的生成器的类似界。用这种方法，我们推广了已知的完全正迹保持动力学的结果，同时提供了一个明显更简单的证明，这是完全代数的。

引用次数: 0

Corrigendum to “Nilpotent linear spaces and Albert's Problem” [Linear Algebra Appl. 518 (2017) 57–78] “幂零线性空间与阿尔伯特问题”的勘误表[线性代数应用，518 (2017):57-78]

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-28 DOI: 10.1016/j.laa.2025.10.010

Juan C. Gutierrez Fernandez , E.O. Quintero Vanegas

In our article Nilpotent Linear Spaces and Albert's Problem [Linear Algebra Appl. 518 (2017) 57–78], the proof of Theorem 6 was incomplete, as a case was omitted. Here we supply the missing argument. The statement of Theorem 6, and all subsequent results depending on it, remain valid.

在我们的文章《幂零线性空间与阿尔伯特问题》[线性代数应用，518(2017)57-78]中，定理6的证明是不完整的，因为省略了一个情况。在这里，我们提供缺失的论据。定理6的陈述，以及所有依赖于它的后续结果，仍然有效。

引用次数: 0

Uniform hypertrees with maximum nullity 具有最大零的一致超树

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-17 DOI: 10.1016/j.laa.2025.10.016

Ya-Nan Zheng

Let

A

be the adjacency tensor of a k-uniform hypergraph H. The nullity of H is the multiplicity of the eigenvalue zero in the spectrum of H, i.e., the algebraic multiplicity of the eigenvalue zero of

A

. A connected and acyclic hypergraph is called a hypertree. In this paper, by exploring the relationship between the nullity of k-uniform hypertrees and the nullity of their subhypergraphs, we study the extremal nullity of k-uniform hypertrees. We prove that the k-uniform hyperstar

S_{m}^{k}

attains uniquely the maximum nullity among all k-uniform hypertrees with m edges.

设A为k-一致超图H的邻接张量，H的零性是H谱中特征值零的多重性，即A的特征值零的代数多重性。连通无环超图称为超树。本文通过探讨k-一致超树的零性与其子超图的零性之间的关系，研究了k-一致超树的极值零性。证明了k-一致超星Smk在所有有m条边的k-一致超树中唯一地达到极大零。

引用次数: 0

On the core-nilpotent decomposition of unicyclic graphs 单环图的核幂零分解

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-30 DOI: 10.1016/j.laa.2025.10.034

Daniel A. Jaume , Cristian Panelo , Maikon M. Toledo , Micaela E. Vega

In this work we show, through the null decomposition of unicyclic graphs given by Allem et al. (2020), that the core-nilpotent decomposition of the adjacency matrix of a unicyclic graph, can be obtained directly from the graph itself.

在这项工作中，我们通过Allem等人（2020）给出的单环图的零分解表明，单环图邻接矩阵的核幂零分解可以直接从图本身获得。

引用次数: 0

Non-linear maps preserving ascent or descent of product of operators 保持运算符乘积上升或下降的非线性映射

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-27 DOI: 10.1016/j.laa.2025.10.026

Rabi Marzouki, Khalid Souilah

In this article, we provide a complete description of all maps on the algebra of all bounded linear operators acting on an infinite-dimensional complex Banach space, that leave invariant the ascent, or descent, under the product of two operators.

在这篇文章中，我们提供了作用于无限维复Banach空间的所有有界线性算子在代数上的所有映射的完整描述，这些映射在两个算子的乘积下使上升或下降保持不变。

引用次数: 0

The main reasons for matrices multiplying to zero 矩阵乘以为零的主要原因

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2026-02-01 Epub Date: 2025-11-06 DOI: 10.1016/j.laa.2025.11.001

Jakub Koncki , Richárd Rimányi

We provide an explicit description of the maximal-dimensional components of the variety parametrizing sequences of matrices of prescribed sizes whose product is zero.

我们给出了乘积为零的规定大小矩阵的各种参数化序列的最大维分量的显式描述。

引用次数: 0

The signless Laplacian spectral Turán problems for color-critical graphs 色临界图的无符号拉普拉斯谱Turán问题

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications

Pub Date : 2026-02-01 Epub Date: 2025-11-05 DOI: 10.1016/j.laa.2025.10.036

Jian Zheng , Yongtao Li , Honghai Li

<div><div>The well-known Turán theorem states that if G is an n-vertex <math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math>-free graph, then <math><mi>e</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>e</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo></math>, with equality if and only if G is the r-partite Turán graph <math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub></math>. A graph F is called color-critical if it contains an edge whose deletion reduces its chromatic number. Extending the Turán theorem, Simonovits (1968) proved that for any color-critical graph F with <math><mi>χ</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><mi>r</mi><mo>+</mo><mn>1</mn></math> and sufficiently large n, the Turán graph <math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub></math> is the unique graph that attains the maximum number of edges over all n-vertex F-free graphs. Subsequently, Nikiforov (2009) [40] proved a spectral version of Simonovits' theorem in terms of the adjacency spectral radius. In this paper, we show an extension of Simonovits' theorem for the signless Laplacian spectral radius. We prove that for any color-critical graph F with <math><mi>χ</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>≥</mo><mn>4</mn></math>, if n is sufficiently large and G is an F-free graph on n vertices, then <math><mi>q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>q</mi><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo></math>, with equality if and only if <math><mi>G</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub></math>. Our approach is to establish a signless Laplacian spectral version of the criterion of Keevash, Lenz and Mubayi (2014) [26]. Consequently, we determine the signless Laplacian spectral extremal graphs for generalized books and even wheels. As an application, our result gives an upper bound on the degree power of an F-free graph. We show that if n is sufficiently large and G is an F-free graph on n vertices with m edges, then <math><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>v</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>r</mi></mrow></mfrac><mo>)</mo><mi>m</mi><mi>n</mi></math>, with equality if and only if G is a regular T

著名的Turán定理指出，如果G是一个n顶点无Kr+1的图，那么e(G)≤e(Tn,r)，当且仅当G是r部Turán图Tn，r时相等。如果图F中有一条边的删除会减少其色数，则称为颜色临界。推广Turán定理，Simonovits（1968）证明了对于任意χ(F)=r+1且n足够大的色临界图F， Turán图Tn，r是在所有n顶点的无F图上达到最大边数的唯一图。随后，Nikiforov(2009)[40]从邻接谱半径的角度证明了Simonovits定理的谱版本。本文给出了Simonovits定理在无符号拉普拉斯谱半径上的推广。证明了对于任意χ(F)=r+1≥4的色临界图F，如果n足够大且G是n个顶点上的无F图，则q(G)≤q(Tn,r)，且当且仅当G=Tn，r相等。我们的方法是建立Keevash， Lenz和Mubayi(2014)[26]标准的无符号拉普拉斯谱版本。因此，我们确定了广义书本和偶数车轮的无符号拉普拉斯谱极值图。作为一个应用，我们的结果给出了无f图的次幂的上界。我们证明了如果n足够大，并且G是一个有n个顶点和m条边的无f图，那么∑v∈v (G)d2(v)≤2(1−1r)mn，当且仅当G是正则Turán图Tn，r时相等。这延伸了Nikiforov和Rousseau（2004）的结果。最后，我们提出了两个有趣的猜想，以刺激这一方向的进一步研究。

引用次数: 0

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