Spectra of sparse random matrices

Kühn, Reimer

doi:10.1088/1751-8113/41/29/295002

Cited by 126 publications

(262 citation statements)

References 52 publications

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“…For example, [182,183] describe a model for diffusion in a configuration space that combines features of infinite dimensionality and very low connectivity-for readers more familiar with the Erdős-Rényi G np random graph model [184] than with spin glass theory, the parameter region of interest in these applications corresponds to the extremely sparse region 1/n p log n/n. For ensembles of very sparse random matrices, there is a localization-delocalization transition which has been studied in detail [185,186,187,188]. In these applications, as a rule of thumb, eigenvector localization occurs when there is some sort of "structural heterogeneity," e.g., the degree (or coordination number) of a node is significantly higher or lower than average.…”

Section: Statistical Leverage In Large-scale Data Analysismentioning

confidence: 99%

Randomized Algorithms for Matrices and Data

Mahoney¹

2012

Chapman &Amp; Hall/CRC Data Mining and Knowledge Discovery Series

335

373

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Randomized algorithms for very large matrix problems have received a great deal of attention in recent years. Much of this work was motivated by problems in large-scale data analysis, largely since matrices are popular structures with which to model data drawn from a wide range of application domains, and this work was performed by individuals from many different research communities. While the most obvious benefit of randomization is that it can lead to faster algorithms, either in worst-case asymptotic theory and/or numerical implementation, there are numerous other benefits that are at least as important. For example, the use of randomization can lead to simpler algorithms that are easier to analyze or reason about when applied in counterintuitive settings; it can lead to algorithms with more interpretable output, which is of interest in applications where analyst time rather than just computational time is of interest; it can lead implicitly to regularization and more robust output; and randomized algorithms can often be organized to exploit modern computational architectures better than classical numerical methods.This monograph will provide a detailed overview of recent work on the theory of randomized matrix algorithms as well as the application of those ideas to the solution of practical problems in large-scale data analysis. Throughout this review, an emphasis will be placed on a few simple core ideas that underlie not only recent theoretical advances but also the usefulness of these tools in large-scale data applications. Crucial in this context is the connection with the concept of statistical leverage. This concept has long been used in statistical regression diagnostics to identify outliers; and it has recently proved crucial in the development of improved worst-case matrix algorithms that are also amenable to high-quality numerical implementation and that are useful to domain scientists. This connection arises naturally when one explicitly decouples the effect of randomization in these matrix algorithms from the underlying linear algebraic structure. This decoupling also permits much finer control in the application of randomization, as well as the easier exploitation of domain knowledge.Most of the review will focus on random sampling algorithms and random projection algorithms for versions of the linear least-squares problem and the low-rank matrix approximation problem. These two problems are fundamental in theory and ubiquitous in practice. Randomized methods solve these problems by constructing and operating on a randomized sketch of the input matrix Afor random sampling methods, the sketch consists of a small number of carefully-sampled and rescaled columns/rows of A, while for random projection methods, the sketch consists of a small number of linear combinations of the columns/rows of A. Depending on the specifics of the situation, when compared with the best previously-existing deterministic algorithms, the resulting randomized algorithms have worst-case running time that is asympt...

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Section: Statistical Leverage In Large-scale Data Analysismentioning

confidence: 99%

Randomized Algorithms for Matrices and Data

Mahoney¹

2012

Chapman &Amp; Hall/CRC Data Mining and Knowledge Discovery Series

335

373

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“…However, we expect that the properties, which we will investigate after this, do not depend on the details of the generation schemes. When the support of p(k) is not bounded from the above and values of the entries are kept finite, the first eigenvalue generally diverges as N → ∞ [11,12,13,14,15,16]. To avoid this possibility, we assume that p(k) = 0 for k, which is larger than a certain value, k max , unless infinitesimal entries are assumed.…”

Section: Model Definitionmentioning

confidence: 99%

Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices

Kabashima¹,

Takahashi²,

Watanabe³

2010

J. Phys.: Conf. Ser.

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Abstract.A methodology to analyze the properties of the first (largest) eigenvalue and its eigenvector is developed for large symmetric random sparse matrices utilizing the cavity method of statistical mechanics. Under a tree approximation, which is plausible for infinitely large systems, in conjunction with the introduction of a Lagrange multiplier for constraining the length of the eigenvector, the eigenvalue problem is reduced to a bunch of optimization problems of a quadratic function of a single variable, and the coefficients of the first and the second order terms of the functions act as cavity fields that are handled in cavity analysis. We show that the first eigenvalue is determined in such a way that the distribution of the cavity fields has a finite value for the second order moment with respect to the cavity fields of the first order coefficient. The validity and utility of the developed methodology are examined by applying it to two analytically solvable and one simple but non-trivial examples in conjunction with numerical justification.

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“…From a mathematical-physicist point of view, in the frame of RMT, in 1988 Rodgers and Bray [12] proposed an ensemble of sparse random matrices characterized by the connectivity ξ. Since then, several papers have been devoted to analytical and numerical studies of sparse symmetric random matrices (see for example [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]). Among the most relevant results of these studies we can mention that: (i) in the very sparse limit, ξ → 1, the density of states was found to deviate from the Wigner semicircle law with the appearance of singularities, around and at the band center, and tails beyond the semicircle [12][13][14][15][16][17][18][19][20][21]; (ii) a delocalization transition was found at ξ ≈ 1.4 [14][15][16]22]; (iii) the nearest-neighbor energy level spacing distribution P (s) was found to evolve from the Poisson to the Gaussian Orthogonal Ensemble (GOE) predictions for increasing ξ [11,14,16] (the same transition was reported for the number variance in Ref.…”

Section: Introductionmentioning

confidence: 99%

Universality in the spectral and eigenfunction properties of random networks

et al. 2015

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By the use of extensive numerical simulations we show that the nearest-neighbor energy level spacing distribution P (s) and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree ξ ≡ αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that Brody distribution characterizes well P (s) in the transition from α = 0, when the vertices in the network are isolated, to α = 1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.

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Spectra of sparse random matrices

Cited by 126 publications

References 52 publications

Randomized Algorithms for Matrices and Data

Randomized Algorithms for Matrices and Data

Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices

Universality in the spectral and eigenfunction properties of random networks

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