An Introduction to Matrix Concentration Inequalities

Tropp, Joel A.

doi:10.1561/2200000048

Cited by 339 publications

(164 citation statements)

References 50 publications

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“…As in the case of classical concentration inequalities, the moment method essentially subsumes the matrix concentration approach and is often more powerful. We therefore do not discuss this approach further, but refer to [28] for a systematic development.…”

Section: Remark 25 a Variant On The Moment Methods Is To Use The Boundsmentioning

confidence: 99%

“…Recent years have seen impressive advances in this area, particularly in the understanding of universality phenomena that are exhibited by the spectra of classical random matrix models [8,26]. At the same time, random matrices have proved to be of major importance in contemporary applied mathematics, see, for example, [28,32] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…More recently, the study of structured random matrices has received renewed attention due to the needs of applied mathematics, cf. [28] and the references therein. However, significant new progress was made in the past few years.…”

Section: Introductionmentioning

confidence: 99%

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Structured Random Matrices

Handel

2017

The IMA Volumes in Mathematics and Its Applications

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Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact or approximate symmetries, such as matrices with i.i.d. entries, for which precise analytic results and limit theorems are available. Much less well understood are matrices that are endowed with an arbitrary structure, such as sparse Wigner matrices or matrices whose entries possess a given variance pattern. The challenge in investigating such structured random matrices is to understand how the given structure of the matrix is reflected in its spectral properties. This chapter reviews a number of recent results, methods, and open problems in this direction, with a particular emphasis on sharp spectral norm inequalities for Gaussian random matrices.

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Section: Remark 25 a Variant On The Moment Methods Is To Use The Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Structured Random Matrices

Handel

2017

The IMA Volumes in Mathematics and Its Applications

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“…To prove that Y 2 is small with high probability we use a stable rank (intrinsic dimension) matrix Bernstein inequality from [65] that was first proven in [53] following work in [37]. This inequality requires upper bounds on the spectral norm of each X j and on the variance of Y.…”

Section: Proof Following Lemma 31 We Haveτmentioning

confidence: 99%

Input Sparsity Time Low-rank Approximation via Ridge Leverage Score Sampling

Cohen

Musco

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present a new algorithm for finding a near optimal low-rank approximation of a matrix A in O(nnz(A)) time. Our method is based on a recursive sampling scheme for computing a representative subset of A's columns, which is then used to find a low-rank approximation.This approach differs substantially from prior O(nnz(A)) time algorithms, which are all based on fast Johnson-Lindenstrauss random projections.Our algorithm matches the guarantees of the random projection methods while offering a number of advantages.In addition to better performance on sparse and structured data, sampling algorithms can be applied in settings where random projections cannot. For example, we give new streaming algorithms for the column subset selection and projection-cost preserving sample problems. Our method has also been used in the fastest algorithms for provably accurate Nyström approximation of kernel matrices [56].

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“…We now provide some insights on (5). For this, we recall the matrix Bernstein inequality (see Theorem 6.6.1 in [13]). …”

Section: Introductionmentioning

confidence: 99%

On the Behaviour of the Estimated Fourth-Order Cumulants Matrix of a High-Dimensional Gaussian White Noise

Gouédard¹,

Loubaton²

2017

Latent Variable Analysis and Signal Separation

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Abstract. This paper is devoted to the study of the traditional estimator of the fourth-order cumulants matrix of a high-dimensional multivariate Gaussian white noise. If M represents the dimension of the noise and N the number of available observations, it is first established that this M 2 × M 2 matrix converges towards 0 in the spectral norm sensThe behaviour of the estimated fourth-order cumulants matrix is then evaluated in the asymptotic regime where M and N converge towards +∞ in such a way thatconverges towards a constant. In this context, it is proved that the matrix does not converge towards 0 in the spectral norm sense, and that its empirical eigenvalue distribution converges towards a shifted Marcenko-Pastur distribution. It is finally claimed that the largest and the smallest eigenvalue of the cumulant matrix converges almost surely towards the rightend and the leftend points of the support of the Marcenko-Pastur distribution.Keywords: Estimated joint fourth-order cumulants matrices, large random matrices, blind source separation in the high-dimensional context.

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An Introduction to Matrix Concentration Inequalities

Cited by 339 publications

References 50 publications

Structured Random Matrices

Structured Random Matrices

Input Sparsity Time Low-rank Approximation via Ridge Leverage Score Sampling

On the Behaviour of the Estimated Fourth-Order Cumulants Matrix of a High-Dimensional Gaussian White Noise

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