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Research Spotlights
SIAM Review ( IF 10.8 ) Pub Date : 2022-08-04 , DOI: 10.1137/22n975512 Misha E. Kilmer
SIAM Review ( IF 10.8 ) Pub Date : 2022-08-04 , DOI: 10.1137/22n975512 Misha E. Kilmer
SIAM Review, Volume 64, Issue 3, Page 623-624, August 2022.
What's in a name [brand]? In the first of three Research Spotlights articles this issue, authors Joseph D. Johnson, Adam M. Redlich, and Daniel M. Abrams address this issue by developing a mathematical model for the dynamics of competition through advertising. Specifically, in “A Mathematical Model for the Origin of Name Brands and Generics," readers are presented with a mathematical model for the dynamics of competition through advertising. Here, the terms “generic” and “name brand” refer to low and high advertising investment states, respectively. A distinguishing feature of the approach taken by the authors in this work is that they model “monopolistic competition," meaning that although there may be many suppliers of a product/service, these are distinguished only by brand and/or quality. The analysis of the existence and stability of equilibria of their ordinary differential equation system model predicts that “when advertising is relatively cheap compared to the benefit of advertising," these two advertising investment states will arise. This segmentation “contrasts starkly with (often implicit) assumptions of smooth, singly peaked functions for economic metrics." The authors note that their model predicts that segmentation should be reflected in price distributions. Indeed, although there are limitations in their model which readers are invited to consider addressing in future work, they show good qualitative agreement to this prediction on a large consumer data set. The need to solve linear least squares problems is ubiquitous in science and engineering applications, and it is at the heart of our second article, “Some Comments on Preconditioning for Normal Equations and Least Squares," authored by Andy Wathen. Iterative solvers are often preferred over direct methods for large-scale least squares as they require only the ability to perform matrix-vector products with the system matrix and its (conjugate) transpose. A preconditioner, which is an approximation to the original matrix with desirable properties (e.g., easy to apply, easy to “invert''), is usually employed to speed convergence. To be effective, the spectrum of the preconditioned normal equations operator should be appropriately clustered. Using several concrete examples as motivation, the author explores a subtle and underappreciated difficulty in designing preconditioners for least squares problems called the “matrix squaring problem." Simply put, a good approximation, $P$, to the original matrix, $B$, may not translate into having an effective preconditioner, $P^T P$, for the normal equations matrix $B^T B$. The article includes theory and discussion about when the matrix squaring problem can be expected versus when it is a nonissue in the case of invertible matrices. The author's final example shows that the matrix squaring problem can occur even in the full rank rectangular case, which should serve as a warning to practitioners that, indeed, it may not be sufficient to seek approximations to the original operator in the design of an effective preconditioner for iterative solution to the least squares problem. The final article, “Hypergraph Cuts with General Splitting Functions," by Nate Veldt, Austin R. Benson, and Jon Kleinberg, tackles a problem that is central to the study of hypergraphs. While readers may be familiar with a standard graph representation in which an edge connects exactly two vertices, in a hypergraph, an edge (a.k.a. hyperedge) refers to a grouping of (possibly) more than two vertices. This extra dimensionality associated to an edge complicates the generalization of graph splitting to hypergraphs, a fact that can be appreciated by studying the graphical illustrations provided in the article. Nevertheless, the authors provide a framework for hypergraph cuts that leads to a rich set of results and illuminates new research questions. Specifically, their framework utilizes so-called “splitting functions" which assign a penalty to each rearrangement of a hyperedge's nodes. These functions enable them to characterize the minimum $s$-$t$ cut problem of finding a minimum weight set of hyperedges to cut in order to separate (e.g., as might be required in data clustering applications) nodes $s$ and $t$ from each other. The paper contains many new contributions, among them algorithms for some variants of the hypergraph $s$-$t$ cut problem that are polynomial time and theoretical results on the NP-hardness of other variants. As this article “includes broad contributions at the intersection of graph theory, optimization, scientific computing, and other subdisciplines in applied mathematics," and offers several suggestions for follow-up research questions, it is likely to appeal to many SIREV readers.
中文翻译：
研究聚光灯
SIAM 评论，第 64 卷，第 3 期，第 623-624 页，2022 年 8 月。
什么是名称 [品牌]？在本期三篇研究聚焦文章中的第一篇中，作者 Joseph D. Johnson、Adam M. Redlich 和 Daniel M. Abrams 通过开发广告竞争动态的数学模型来解决这个问题。具体而言，在“名牌和仿制药起源的数学模型”中，读者将看到一个通过广告进行竞争动态的数学模型。这里，术语“仿制药”和“名牌”指的是低广告和高广告作者在这项工作中采用的方法的一个显着特征是他们模拟了“垄断竞争”，这意味着尽管产品/服务可能有许多供应商，但这些供应商仅通过品牌和/或质量。他们的常微分方程系统模型的均衡存在性和稳定性分析预测，“当广告相对于广告的收益而言相对便宜时”，这两种广告投资状态就会出现。这种细分“与（通常是隐含的）形成鲜明对比经济指标的平滑、单峰函数的假设。” 作者指出，他们的模型预测细分应该反映在价格分布中。事实上，尽管他们的模型存在局限性，请读者考虑在未来的工作中解决这些问题，但他们对大型消费者数据集的这一预测表现出良好的定性一致性。解决线性最小二乘问题的需求在科学和工程应用中无处不在，它是我们第二篇文章的核心，“关于正态方程和最小二乘的预处理的一些评论”，作者是 Andy Wathen。对于大规模最小二乘，迭代求解器通常比直接方法更受欢迎，因为它们只需要能够与系统矩阵及其（共轭）转置。预处理器是对原始矩阵的近似，具有理想的属性（例如，易于应用，易于“反转”），通常用于加速收敛。为了有效，预处理器的频谱正规方程算子应适当聚类。作者以几个具体示例为动机，探讨了为最小二乘问题设计预条件子时一个微妙且未被充分认识的困难，称为“矩阵平方问题”。简单地说，一个很好的近似值，$P$，对于原始矩阵 $B$，对于正规方程矩阵 $B^TB$，可能无法转化为有效的预条件子 $P^TP$。这篇文章包括关于何时可以预期矩阵平方问题以及何时在可逆矩阵的情况下它不是问题的理论和讨论。作者的最后一个例子表明，即使在满秩矩形的情况下，矩阵平方问题也可能出现，这应该作为对从业者的警告，实际上，在设计有效的算子时寻求原始算子的近似值可能是不够的。用于迭代求解最小二乘问题的预处理器。Nate Veldt、Austin R. Benson 和 Jon Kleinberg 的最后一篇文章“具有一般分裂函数的超图切割”解决了超图研究的核心问题。虽然读者可能熟悉其中一条边恰好连接两个顶点的标准图形表示，但在超图中，一条边（也称为超边）指的是（可能）两个以上顶点的分组。这种与边相关的额外维度使图分裂到超图的泛化复杂化，这一事实可以通过研究文章中提供的图形插图来理解。尽管如此，作者为超图切割提供了一个框架，该框架产生了丰富的结果并阐明了新的研究问题。具体来说，他们的框架利用所谓的“分裂函数”，为超边节点的每次重新排列分配一个惩罚。这些函数使他们能够表征最小$s$-$t$ 割问题，即找到要割的最小权重集的超边，以便分离（例如，数据聚类应用程序中可能需要）节点$s$ 和$t $ 来自对方。这篇论文包含了许多新的贡献，其中包括超图$s$-$t$割问题的一些变体的算法，这些变体是多项式时间和其他变体的NP-hardness的理论结果。由于本文“包括在图论、优化、科学计算和应用数学的其他子学科交叉领域的广泛贡献”，并为后续研究问题提供了一些建议，因此它可能会吸引许多 SIREV 读者。这篇论文包含了许多新的贡献，其中包括超图$s$-$t$割问题的一些变体的算法，这些变体是多项式时间和其他变体的NP-hardness的理论结果。由于本文“包括在图论、优化、科学计算和应用数学的其他子学科交叉领域的广泛贡献”，并为后续研究问题提供了一些建议，因此它可能会吸引许多 SIREV 读者。这篇论文包含了许多新的贡献，其中包括超图$s$-$t$割问题的一些变体的算法，这些变体是多项式时间和其他变体的NP-hardness的理论结果。由于本文“包括在图论、优化、科学计算和应用数学的其他子学科交叉领域的广泛贡献”，并为后续研究问题提供了一些建议，因此它可能会吸引许多 SIREV 读者。
更新日期：2022-08-04
What's in a name [brand]? In the first of three Research Spotlights articles this issue, authors Joseph D. Johnson, Adam M. Redlich, and Daniel M. Abrams address this issue by developing a mathematical model for the dynamics of competition through advertising. Specifically, in “A Mathematical Model for the Origin of Name Brands and Generics," readers are presented with a mathematical model for the dynamics of competition through advertising. Here, the terms “generic” and “name brand” refer to low and high advertising investment states, respectively. A distinguishing feature of the approach taken by the authors in this work is that they model “monopolistic competition," meaning that although there may be many suppliers of a product/service, these are distinguished only by brand and/or quality. The analysis of the existence and stability of equilibria of their ordinary differential equation system model predicts that “when advertising is relatively cheap compared to the benefit of advertising," these two advertising investment states will arise. This segmentation “contrasts starkly with (often implicit) assumptions of smooth, singly peaked functions for economic metrics." The authors note that their model predicts that segmentation should be reflected in price distributions. Indeed, although there are limitations in their model which readers are invited to consider addressing in future work, they show good qualitative agreement to this prediction on a large consumer data set. The need to solve linear least squares problems is ubiquitous in science and engineering applications, and it is at the heart of our second article, “Some Comments on Preconditioning for Normal Equations and Least Squares," authored by Andy Wathen. Iterative solvers are often preferred over direct methods for large-scale least squares as they require only the ability to perform matrix-vector products with the system matrix and its (conjugate) transpose. A preconditioner, which is an approximation to the original matrix with desirable properties (e.g., easy to apply, easy to “invert''), is usually employed to speed convergence. To be effective, the spectrum of the preconditioned normal equations operator should be appropriately clustered. Using several concrete examples as motivation, the author explores a subtle and underappreciated difficulty in designing preconditioners for least squares problems called the “matrix squaring problem." Simply put, a good approximation, $P$, to the original matrix, $B$, may not translate into having an effective preconditioner, $P^T P$, for the normal equations matrix $B^T B$. The article includes theory and discussion about when the matrix squaring problem can be expected versus when it is a nonissue in the case of invertible matrices. The author's final example shows that the matrix squaring problem can occur even in the full rank rectangular case, which should serve as a warning to practitioners that, indeed, it may not be sufficient to seek approximations to the original operator in the design of an effective preconditioner for iterative solution to the least squares problem. The final article, “Hypergraph Cuts with General Splitting Functions," by Nate Veldt, Austin R. Benson, and Jon Kleinberg, tackles a problem that is central to the study of hypergraphs. While readers may be familiar with a standard graph representation in which an edge connects exactly two vertices, in a hypergraph, an edge (a.k.a. hyperedge) refers to a grouping of (possibly) more than two vertices. This extra dimensionality associated to an edge complicates the generalization of graph splitting to hypergraphs, a fact that can be appreciated by studying the graphical illustrations provided in the article. Nevertheless, the authors provide a framework for hypergraph cuts that leads to a rich set of results and illuminates new research questions. Specifically, their framework utilizes so-called “splitting functions" which assign a penalty to each rearrangement of a hyperedge's nodes. These functions enable them to characterize the minimum $s$-$t$ cut problem of finding a minimum weight set of hyperedges to cut in order to separate (e.g., as might be required in data clustering applications) nodes $s$ and $t$ from each other. The paper contains many new contributions, among them algorithms for some variants of the hypergraph $s$-$t$ cut problem that are polynomial time and theoretical results on the NP-hardness of other variants. As this article “includes broad contributions at the intersection of graph theory, optimization, scientific computing, and other subdisciplines in applied mathematics," and offers several suggestions for follow-up research questions, it is likely to appeal to many SIREV readers.
中文翻译：
研究聚光灯
SIAM 评论，第 64 卷，第 3 期，第 623-624 页，2022 年 8 月。
什么是名称 [品牌]？在本期三篇研究聚焦文章中的第一篇中，作者 Joseph D. Johnson、Adam M. Redlich 和 Daniel M. Abrams 通过开发广告竞争动态的数学模型来解决这个问题。具体而言，在“名牌和仿制药起源的数学模型”中，读者将看到一个通过广告进行竞争动态的数学模型。这里，术语“仿制药”和“名牌”指的是低广告和高广告作者在这项工作中采用的方法的一个显着特征是他们模拟了“垄断竞争”，这意味着尽管产品/服务可能有许多供应商，但这些供应商仅通过品牌和/或质量。他们的常微分方程系统模型的均衡存在性和稳定性分析预测，“当广告相对于广告的收益而言相对便宜时”，这两种广告投资状态就会出现。这种细分“与（通常是隐含的）形成鲜明对比经济指标的平滑、单峰函数的假设。” 作者指出，他们的模型预测细分应该反映在价格分布中。事实上，尽管他们的模型存在局限性，请读者考虑在未来的工作中解决这些问题，但他们对大型消费者数据集的这一预测表现出良好的定性一致性。解决线性最小二乘问题的需求在科学和工程应用中无处不在，它是我们第二篇文章的核心，“关于正态方程和最小二乘的预处理的一些评论”，作者是 Andy Wathen。对于大规模最小二乘，迭代求解器通常比直接方法更受欢迎，因为它们只需要能够与系统矩阵及其（共轭）转置。预处理器是对原始矩阵的近似，具有理想的属性（例如，易于应用，易于“反转”），通常用于加速收敛。为了有效，预处理器的频谱正规方程算子应适当聚类。作者以几个具体示例为动机，探讨了为最小二乘问题设计预条件子时一个微妙且未被充分认识的困难，称为“矩阵平方问题”。简单地说，一个很好的近似值，$P$，对于原始矩阵 $B$，对于正规方程矩阵 $B^TB$，可能无法转化为有效的预条件子 $P^TP$。这篇文章包括关于何时可以预期矩阵平方问题以及何时在可逆矩阵的情况下它不是问题的理论和讨论。作者的最后一个例子表明，即使在满秩矩形的情况下，矩阵平方问题也可能出现，这应该作为对从业者的警告，实际上，在设计有效的算子时寻求原始算子的近似值可能是不够的。用于迭代求解最小二乘问题的预处理器。Nate Veldt、Austin R. Benson 和 Jon Kleinberg 的最后一篇文章“具有一般分裂函数的超图切割”解决了超图研究的核心问题。虽然读者可能熟悉其中一条边恰好连接两个顶点的标准图形表示，但在超图中，一条边（也称为超边）指的是（可能）两个以上顶点的分组。这种与边相关的额外维度使图分裂到超图的泛化复杂化，这一事实可以通过研究文章中提供的图形插图来理解。尽管如此，作者为超图切割提供了一个框架，该框架产生了丰富的结果并阐明了新的研究问题。具体来说，他们的框架利用所谓的“分裂函数”，为超边节点的每次重新排列分配一个惩罚。这些函数使他们能够表征最小$s$-$t$ 割问题，即找到要割的最小权重集的超边，以便分离（例如，数据聚类应用程序中可能需要）节点$s$ 和$t $ 来自对方。这篇论文包含了许多新的贡献，其中包括超图$s$-$t$割问题的一些变体的算法，这些变体是多项式时间和其他变体的NP-hardness的理论结果。由于本文“包括在图论、优化、科学计算和应用数学的其他子学科交叉领域的广泛贡献”，并为后续研究问题提供了一些建议，因此它可能会吸引许多 SIREV 读者。这篇论文包含了许多新的贡献，其中包括超图$s$-$t$割问题的一些变体的算法，这些变体是多项式时间和其他变体的NP-hardness的理论结果。由于本文“包括在图论、优化、科学计算和应用数学的其他子学科交叉领域的广泛贡献”，并为后续研究问题提供了一些建议，因此它可能会吸引许多 SIREV 读者。这篇论文包含了许多新的贡献，其中包括超图$s$-$t$割问题的一些变体的算法，这些变体是多项式时间和其他变体的NP-hardness的理论结果。由于本文“包括在图论、优化、科学计算和应用数学的其他子学科交叉领域的广泛贡献”，并为后续研究问题提供了一些建议，因此它可能会吸引许多 SIREV 读者。
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