Tight Coefficients of Averaged Operators via Scaled Relative Graph

Huang, Xinmeng; Ryu, Ernest K.; Yin, Wotao

doi:10.48550/arxiv.1912.01593

Cited by 3 publications

(3 citation statements)

References 22 publications

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“…Prior work [12,9] focused on the SRG of nonlinear multi-valued operators. For linear operators, Ryu, Hannah, and Yin [12] established G(A) includes Λ(A), as stated in Fact 1, but did not characterize when and how G(A) enlarges Λ(A).…”

Section: Contributionsmentioning

confidence: 99%

Scaled Relative Graph of Normal Matrices

Huang

Ryu

Yin

2020

Preprint

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The Scaled Relative Graph (SRG) by Ryu, Hannah, and Yin (arXiv:1902.09788, 2019) is a geometric tool that maps the action of a multi-valued nonlinear operator onto the 2D plane, used to analyze the convergence of a wide range of iterative methods. As the SRG includes the spectrum for linear operators, we can view the SRG as a generalization of the spectrum to multi-valued nonlinear operators. In this work, we further study the SRG of linear operators and characterize the SRG of block-diagonal and normal matrices.

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Section: Contributionsmentioning

confidence: 99%

Scaled Relative Graph of Normal Matrices

Huang

Ryu

Yin

2020

Preprint

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“…This paper proposes the Scaled Relative Graph (SRG) as a tool for quantifying the errors introduced by an approximation. The SRG has recently been introduced in the theory of convex optimization [19], and allows simple, graphical proofs of algorithm convergence, and the derivation of tight convergence bounds [20]. The SRG gives a graphical representation of the incremental behavior of a nonlinear operator, and generalizes the Nyquist diagram of an LTI transfer function [21].…”

Section: Introductionmentioning

confidence: 99%

Circuit Model Reduction with Scaled Relative Graphs

Chaffey¹,

Padoan²

2022

Preprint

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Continued fractions are classical representations of complex objects (for example, real numbers) as sums and inverses of simpler objects (for example, integers). The analogy in linear circuit theory is a chain of series/parallel one-ports: the port behavior is a continued fraction containing the port behaviors of its elements. Truncating a continued fraction is a classical method of approximation, which corresponds to deleting the circuit elements furthest from the port. We apply this idea to chains of series/parallel one-ports composed of arbitrary nonlinear relations. This gives a model reduction method which automatically preserves properties such as incremental positivity. The Scaled Relative Graph (SRG) gives a graphical representation of the original and truncated port behaviors. The difference of these SRGs gives a bound on the approximation error, which is shown to be competitive with existing methods.

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“…This tool allows simple, intuitive and rigorous proofs of the convergence of many algorithms in convex optimization. Furthermore, the graphical method is particularly suitable for proving tightness of convergence bounds, with several novel tightness results being proved [1], [2].…”

Section: Introductionmentioning

confidence: 99%

Scaled relative graphs for system analysis

Chaffey

Forni

Sepulchre

2021

2021 60th IEEE Conference on Decision and Control (CDC)

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Scaled relative graphs were recently introduced to analyze the convergence of optimization algorithms using two dimensional Euclidean geometry. In this paper, we connect scaled relative graphs to the classical theory of input/output systems. It is shown that the Nyquist diagram of an LTI system on L 2 is the convex hull of its scaled relative graph under a particular change of coordinates. The SRG may be used to visualize approximations of static nonlinearities such as the describing function and quadratic constraints, allowing system properties to be verified or disproved. Interconnections of systems correspond to graphical manipulations of their SRGs. This is used to provide a simple, graphical proof of the classical incremental passivity theorem.The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n. 670645.

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Tight Coefficients of Averaged Operators via Scaled Relative Graph

Cited by 3 publications

References 22 publications

Scaled Relative Graph of Normal Matrices

Scaled Relative Graph of Normal Matrices

Circuit Model Reduction with Scaled Relative Graphs

Scaled relative graphs for system analysis

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