Scaled relative graphs for system analysis

Chaffey, Thomas; Forni, Fulvio; Sepulchre, Rodolphe

doi:10.1109/cdc45484.2021.9683092

Cited by 8 publications

(5 citation statements)

References 18 publications

(25 reference statements)

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“…The promise of the SRG extends far beyond the analysis of algorithms from convex optimization. As already noted in [2], the modular fashion in which the SRG can be manipulated makes it an ideal candidate for dynamical system analysis, and the authors additionally give preliminary results connecting the SRG to classical tools from control theory. In order to unlock this potential, a better understanding of how to determine the SRG of an operator is required.…”

Section: Introductionmentioning

confidence: 78%

The Scaled Relative Graph of a Linear Operator

Pates¹

2021

Preprint

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The scaled relative graph (SRG) of an operator is a subset of the extended complex plane, that generalises the concept of the spectrum of a linear operator. It captures several additional features of an operator, such as contractiveness, and can be used to reveal the geometric nature of many of the inequality based arguments used in the convergence analyses of fixed point iterations. In this paper we show that the SRG of a linear operator can be determined from the numerical range of a closely related bounded linear operator. This allows much of the theory of the numerical range to applied in the SRG setting. We illustrate this by showing that algorithms developed for the numerical range can be re-purposed to plot the boundary of the SRG, and demonstrating that the SRG satisfies analogues of the Toeplitz-Hausdorff theorem and Hildebrant's theorem.

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Section: Introductionmentioning

confidence: 78%

The Scaled Relative Graph of a Linear Operator

Pates¹

2021

Preprint

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“…Suppose that A satisfies µ ∞,[η] −1 (A) = γ < 1 for some η ∈ R n >0 and that φ = prox f for some proper, l.s.c., convex f : R → ]−∞, +∞]. Define f N : R m → R n by f N (u) = x * u where x * u solves the fixed point problem x * u = Φ(Ax * u + Bu + b) 11 . Then for η max = max i∈{1,...,n} η i , η min = min i∈{1,...,n} η i , and…”

Section: Proofmentioning

confidence: 99%

“…Problem description and motivation: Monotone operator theory is a fertile field of nonlinear functional analysis that extends the notion of monotone functions on R to mappings on Hilbert spaces. Monotone operator methods are widely used to solve problems in optimization and control [48,6], game theory [42], systems analysis [11], data science [16], and explainable machine learning [14,52]. A crucial part of this theory is the design of algorithms for computing zeros of monotone operators.…”

Section: Introductionmentioning

confidence: 99%

Non-Euclidean Monotone Operator Theory and Applications

Davydov¹,

Jafarpour²,

Proskurnikov³

et al. 2023

Preprint

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While monotone operator theory is traditionally studied on Hilbert spaces, many interesting problems in data science and machine learning arise naturally in finite-dimensional vector spaces endowed with non-Euclidean norms, such as diagonally-weighted 1 or ∞ norms. This paper provides a natural generalization of monotone operator theory to finite-dimensional non-Euclidean spaces. The key tools are weak pairings and logarithmic norms. We show that the resolvent and reflected resolvent operators of non-Euclidean monotone mappings exhibit similar properties to their counterparts in Hilbert spaces. Furthermore, classical iterative methods and splitting methods for finding zeros of monotone operators are shown to converge in the non-Euclidean case. We apply our theory to equilibrium computation and Lipschitz constant estimation of recurrent neural networks, obtaining novel iterations and tighter upper bounds via forward-backward splitting.

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“…Follow-up work has extended the theory and applied it to analyze nonlinear operators: Huang, Ryu, and Yin characterized the SRG of normal matrices [18], Pates further characterized the SRG of linear operators using the Toeplitz-Hausdorff theorem [23], and Huang, Ryu, and Yin [19] established the tightness of the averagedness coefficient of the composition of averaged operators [21] and the DYS operator [13]. SRG has also found applications in control theory: Chaffey, Forni, and Rodolphe utilized the SRG to analyze input-output properties of feedback systems [7,8], and Chaffey and Sepulchre furthermore used it as an experimental tool to determine properties of a given model [6,9,10].…”

Section: Prior Workmentioning

confidence: 99%

Convergence Analyses of Davis-Yin Splitting via Scaled Relative Graphs

Lee¹,

Soheun²,

Ryu³

2022

Preprint

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Davis-Yin splitting (DYS) has found a wide range of applications in optimization, but its linear rates of convergence have not been studied extensively. The scaled relative graph (SRG) simplifies the convergence analysis of operator splitting methods by mapping the action of the operator onto the complex plane, but the prior SRG theory did not fully apply to the DYS operator. In this work, we formalize an SRG theory for the DYS operator and use it to obtain tighter contraction factors.

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Scaled relative graphs for system analysis

Cited by 8 publications

References 18 publications

The Scaled Relative Graph of a Linear Operator

The Scaled Relative Graph of a Linear Operator

Non-Euclidean Monotone Operator Theory and Applications

Convergence Analyses of Davis-Yin Splitting via Scaled Relative Graphs

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