Taylor's theorem for matrix functions with applications to condition number estimation

Deadman, Edvin; Relton, Samuel D.

doi:10.1016/j.laa.2016.04.010

Cited by 18 publications

(16 citation statements)

References 12 publications

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“…Theorem 2.2,[16]). Let f have a power series expansion about the origin with radius of convergence r and let D ⊂ C be a simply connected open set within the circle of radius r centred at…”

mentioning

confidence: 97%

On The Stability of Polynomial Spectral Graph Filters

Kenlay

Thanou

Dong

2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Spectral graph filters are a key component in state-of-the-art machine learning models used for graph-based learning, such as graph neural networks. For certain tasks stability of the spectral graph filters is important for learning suitable representations. Understanding the type of structural perturbation to which spectral graph filters are robust lets us reason as to when we may expect them to be well suited to a learning task. In this work, we first prove that polynomial graph filters are stable with respect to the change in the normalised graph Laplacian matrix. We then show empirically that properties of a structural perturbation, specifically the relative locality of the edges removed in a binary graph, effect the change in the normalised graph Laplacian. Together, our results have implications on designing robust graph filters and representations under structural perturbation.

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mentioning

confidence: 97%

On The Stability of Polynomial Spectral Graph Filters

Kenlay

Thanou

Dong

2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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“…For the principal p-th root (and further discussion) we refer to [19,Theorem 7.2]. For a discussion of Taylor's theorem (for matrix functions) one should have a look in [13].…”

Section: Application: Newton Iterationmentioning

confidence: 99%

Free (rational) derivation

Schrempf

2021

Extr. Math.

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“…A further advantage of the proposed approach is reducing the current gap in the literature between a provably good low-rank approximation of the kernel matrix K to a provably accurate estimation of M. We derive an upper bound for the perturbation of the modified kernel matrix due to the Nyström method by making use of the Taylor series expansion for matrix functions [45]. Our analysis shows that a relatively small perturbation of the kernel matrix results in a practical upper bound for approximating the modified kernel matrix, or equivalently the normalized Laplacian matrix.…”

Section: B Main Contributionsmentioning

confidence: 99%

Scalable Spectral Clustering With Nyström Approximation: Practical and Theoretical Aspects

Pourkamali-Anaraki

2020

IEEE Open J. Signal Process.

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Spectral clustering techniques are valuable tools in signal processing and machine learning for partitioning complex data sets. The effectiveness of spectral clustering stems from constructing a non-linear embedding based on creating a similarity graph and computing the spectral decomposition of the Laplacian matrix. However, spectral clustering methods fail to scale to large data sets because of high computational cost and memory usage. A popular approach for addressing these problems utilizes the Nyström method, an efficient sampling-based algorithm for computing low-rank approximations to large positive semi-definite matrices. This paper demonstrates how the previously popular approach of Nyström-based spectral clustering has severe limitations. Existing time-efficient methods ignore critical information by prematurely reducing the rank of the similarity matrix associated with sampled points. Also, current understanding is limited regarding how utilizing the Nyström approximation will affect the quality of spectral embedding approximations. To address the limitations, this work presents a principled spectral clustering algorithm that exploits spectral properties of the similarity matrix associated with sampled points to regulate accuracy-efficiency trade-offs. We provide theoretical results to reduce the current gap and present numerical experiments with real and synthetic data. Empirical results demonstrate the efficacy and efficiency of the proposed method compared to existing spectral clustering techniques based on the Nyström method and other efficient methods. The overarching goal of this work is to provide an improved baseline for future research directions to accelerate spectral clustering.

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Taylor's theorem for matrix functions with applications to condition number estimation

Cited by 18 publications

References 12 publications

On The Stability of Polynomial Spectral Graph Filters

On The Stability of Polynomial Spectral Graph Filters

Free (rational) derivation

Scalable Spectral Clustering With Nyström Approximation: Practical and Theoretical Aspects

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