Estimating the norms of random circulant and Toeplitz matrices and their inverses

Pan, Victor Y.; Svadlenka, John; Zhao, Liang

doi:10.1016/j.laa.2014.06.027

Cited by 16 publications

(15 citation statements)

References 14 publications

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“…Unlike the latter papers, however, we state these basic estimates in a simpler form, refine them by following [7] rather than [46], and include their detailed proofs. On the related subject of estimating the norms and condition numbers of Gaussian matrices and random structured matrices see [9,12,13,7,46,20,47,39]. For a natural extension of our present work, one can combine randomized matrix multiplication with randomized augmentation and additive preprocessing of [31,32,38].…”

Section: Related Workmentioning

confidence: 99%

“…This motivates using Gaussian circulant multipliers H, that is, circulant matrices H whose first column vector is Gaussian. It has been proved in [39] that such matrices are expected to be well-conditioned, which is required for any multiplicative preconditioner.…”

Section: Supporting Genp With Gaussian Multipliersmentioning

confidence: 99%

“…Furthermore strong nonsingularity holds with a probability close to 1 if we fill the first column of a multiplier F or H with i.i.d. random variables defined under the uniform probability distribution over a sufficiently large finite set (see Appendix B and [39]). …”

Section: Example 62 Generation Of Random Unitary Circulant Matricesmentioning

confidence: 99%

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Random multipliers numerically stabilize Gaussian and block Gaussian elimination: Proofs and an extension to low-rank approximation

Pan

Qian

Yan

2015

Linear Algebra and its Applications

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We study two applications of standard Gaussian random multipliers. At first we prove that with a probability close to 1 such a multiplier is expected to numerically stabilize Gaussian elimination with no pivoting as well as block Gaussian elimination. Then, by extending our analysis, we prove that such a multiplier is also expected to support low-rank approximation of a matrix without customary oversampling. Our test results are in good accordance with this formal study. The results remain similar when we replace Gaussian multipliers with random circulant or Toeplitz multipliers, which involve fewer random parameters and enable faster multiplication. We formally support the observed efficiency of random structured multipliers applied to approximation, but we still continue our research in the case of elimination. We ✩ Some results of this paper have been presented at 203 Gaussian elimination Pivoting Block Gaussian elimination Low-rank approximation SRFT matrices Random circulant matrices specify a narrow class of unitary inputs for which Gaussian elimination with no pivoting is numerically unstable and then prove that, with a probability close to 1, a Gaussian random circulant multiplier does not fix numerical stability problems for such inputs. We also prove that the power of the random circulant preprocessing increases if we also include random permutations.

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Section: Related Workmentioning

confidence: 99%

Section: Supporting Genp With Gaussian Multipliersmentioning

confidence: 99%

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Random multipliers numerically stabilize Gaussian and block Gaussian elimination: Proofs and an extension to low-rank approximation

Pan

Qian

Yan

2015

Linear Algebra and its Applications

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“…An n × n random circulant matrix Z = Z(z) tends to be well-conditioned [PSZ15], and hence so do its n × k and k × n Toeplitz blocks B (we call them subcirculant), defined by the n entries of their first row or column. Indeed, κ(B) ≤ κ(Z(z)) for such blocks B.…”

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confidence: 99%

New studies of randomized augmentation and additive preprocessing

Pan

Zhao

2017

Linear Algebra and its Applications

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Abstract• A standard Gaussian random matrix has full rank with probability 1 and is wellconditioned with a probability quite close to 1 and converging to 1 fast as the matrix deviates from square shape and becomes more rectangular.• If we append sufficiently many standard Gaussian random rows or columns to any matrix A, such that ||A|| = 1, then the augmented matrix has full rank with probability 1 and is well-conditioned with a probability close to 1, even if the matrix A is rank deficient or ill-conditioned.• We specify and prove these properties of augmentation and extend them to additive preprocessing, that is, to adding a product of two rectangular Gaussian matrices.• By applying our randomization techniques to a matrix that has numerical rank ρ, we accelerate the known algorithms for the approximation of its leading and trailing singular spaces associated with its ρ largest and with all its remaining singular values, respectively.• Our algorithms use much fewer random parameters and run much faster when various random sparse and structured preprocessors replace Gaussian. Empirically the outputs of the resulting algorithms is as accurate as the outputs under Gaussian preprocessing.• Our novel duality techniques provides formal support, so far missing, for these empirical observations and opens door to derandomization of our preprocessing and to further acceleration and simplification of our algorithms by using more efficient sparse and structured preprocessors.• Our techniques and our progress can be applied to various other fundamental matrix computations such as the celebrated low-rank approximation of a matrix by means of random sampling. 1Key Words: Randomized matrix algorithms; Gaussian random matrices; Singular spaces of a matrix; Duality; Derandomization; Sparse and structured preprocessors 1 Introduction Randomized augmentation: outlineA standard Gaussian m × n random matrix, G (hereafter referred to just as Gaussian), has full rank with probability 1 (see Theorem B.1). Furthermore the expected spectral norms ||G|| and ||G + ||, G + denoting the Moore-Penrose generalized inverse, satisfy the following estimates (see Theorems B.2 and B.3):• E(||G||) ≈ 2 √ h, for h = max{m, n}, and|m−n| provided that l = min{m, n}, m = n, and e = 2.71828 . . . . Thus, for moderate or reasonably large integers m and n, the matrix G can be considered wellconditioned with the confidence growing fast as the integer |m − n| increases from 0. By virtue of part 2 of Theorem B.3, the matrix G can be viewed as well-conditioned even for m = n, although with a grain of salt, depending on context. Motivated by this information, we append sufficiently but reasonably many Gaussian rows or columns to any matrix A, possibly rank deficient or ill-conditioned, but normalized, such that ||A|| = 1. (Our approach requires attention to various pitfalls, and in particular it fails without normalization of an input matrix.) Then we prove that the cited properties of a Gaussian matrix also hold for the augmented matrix K and similarly for th...

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“…Non-asymptotic estimates for the condition number for random circulant and Toeplitz matrices with i.i.d. standard Gaussian random entries are given in [28,29]. The approach and results of [28,29] are different in nature from our results given in Theorem 1.2.…”

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confidence: 75%

The asymptotic distribution of the condition number for random circulant matrices

Barrera¹,

Manrique-Mirón²

2021

Preprint

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In this manuscript, we study the limiting distribution for the joint law of the largest and the smallest singular values for random circulant matrices with generating sequence given by independent and identically distributed random elements satisfying the so-called Lyapunov condition. Under an appropriated normalization, the joint law of the extremal singular values converges in distribution, as the matrix dimension tends to infinity, to an independent product of Rayleigh and Gumbel laws. The latter implies that a normalized condition number converges in distribution to a Fréchet law as the dimension of the matrix increases.

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Estimating the norms of random circulant and Toeplitz matrices and their inverses

Cited by 16 publications

References 14 publications

Random multipliers numerically stabilize Gaussian and block Gaussian elimination: Proofs and an extension to low-rank approximation

Random multipliers numerically stabilize Gaussian and block Gaussian elimination: Proofs and an extension to low-rank approximation

New studies of randomized augmentation and additive preprocessing

The asymptotic distribution of the condition number for random circulant matrices

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